I cannot subscribe to any such interpretation of the Due Process Clause. Nothing in its words or its history permits it, and “fair distillations of relevant judicial history” are no substitute for the words and history of the clause itself. The phrase “due process of law” has through the years evolved as the successor in purpose and meaning to the words “law of the land” in Magna Charta which more plainly intended to call for a trial according to the existing law of the land in effect at the time an alleged offense had been committed. That provision in Magna Charta was designed to prevent defendants from being tried according to criminal laws or proclamations specifically promulgated to fit particular cases or to attach new consequences to old conduct. Nothing done since Magna Charta can be pointed to as intimating that the Due Process Clause gives courts power to fashion laws in order to meet new conditions, to fit the “decencies” of changed conditions, or to keep their consciences from being shocked by legislation, state or federal.
And, of course, the existence of such awesome judicial power cannot be buttressed or created by relying on the word “procedural.” Whether labeled as “procedural” or “substantive,” the Bill of Rights safeguards, far from
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being mere “tools with which” other unspecified “rights could be fully vindicated,” are the very vitals of a sound constitutional legal system designed to protect and safeguard the most cherished liberties of a free people. These safeguards were written into our Constitution not by judges, but by Constitution makers. Freedom in this Nation will be far less secure the very moment that it is decided that judges can determine which of these safeguards “should” or “should not be imposed” according to their notions of what constitutional provisions are consistent with the “traditions and conscience of our people.” Judges with such power, even though they profess to “proceed with restraint,” will be above the Constitution, with power to write it, not merely to interpret it, which I believe to be the only power constitutionally committed to judges.
There is one ominous sentence, if not more, in my Brother HARLAN’s opinion which bodes ill, in my judgment, both for legislative programs and constitutional commands. Speaking of procedural safeguards in the Bill of Rights, he says:
“These factors in combination suggest that legislatures may properly expect only a cautious deference for their procedural judgments, but that, conversely, courts must exercise their special responsibility for procedural guarantees with care to permit ample scope for achieving the purposes of legislative programs. . . . [T]he court should necessarily proceed with restraint.”
It is to be noted here that this case concerns Bill of Rights Amendments; that the “procedure” power my Brother HARLAN claims for the Court here relates solely to Bill of Rights safeguards, and that he is here claiming for the Court a supreme power to fashion new Bill of Rights safeguards according to the Court’s notions of
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what fits tradition and conscience. I do not believe that the Constitution vests any such power in judges, either in the Due Process Clause or anywhere else. Consequently, I do not vote to invalidate this Arizona law on the ground that it is “unfair,” but solely on the ground that it violates the Fifth and Sixth Amendments made obligatory on the States by the Fourteenth Amendment. Cf. Pointer v. Texas, 380 U. S. 400 , 380 U. S. 412 (Goldberg, J., concurring). It is enough for me that the Arizona law as here applied collides head-on with the Fifth and Sixth Amendments in the four respects mentioned. The only relevance to me of the Due Process Clause is that it would, of course, violate due process or the “law of the land” to enforce a law that collides with the Bill of Rights.
[ Footnote 2/1 ]
“In all criminal prosecutions, the accused shall enjoy the right . . . to be informed of the nature and cause of the accusation. . . .’ Also requiring notice is the Fifth Amendment’s provision that ‘No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a Grand Jury. . . .”
[ Footnote 2/2 ]
“In all criminal prosecutions, the accused shall . . . have the Assistance of Counsel in his defence.”
[ Footnote 2/3 ]
“No person . . . shall be compelled in any criminal case to be a witness against himself. . . .”
[ Footnote 2/4 ]
“In all criminal prosecutions, the accused shall enjoy the right . . . to be confronted with the witnesses against him. . . .”
MR. JUSTICE WHITE, concurring.
I join the Court’s opinion except for 387 U. S. I also agree that the privilege against compelled self-incrimination applies at the adjudicatory stage of juvenile court proceedings. I do not, however, find an adequate basis in the record for determining whether that privilege was violated in this case. The Fifth Amendment protects a person from being “compelled” in any criminal proceeding to be a witness against himself. Compulsion is essential to a violation. It may be that, when a judge, armed with the authority he has or which people think he has, asks questions of a party or a witness in an adjudicatory hearing, that person, especially if a minor, would feel compelled to answer, absent a warning to the contrary or similar information from some other source. The difficulty is that the record made at the habeas corpus hearing, which is the only information we have concerning the proceedings in the juvenile court, does not directly inform us whether Gerald Gault or his parents were told of Gerald’s right to remain silent; nor does it reveal whether the parties
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were aware of the privilege from some other source, just as they were already aware that they had the right to have the help of counsel and to have witnesses on their behalf. The petition for habeas corpus did not raise the Fifth Amendment issue, nor did any of the witnesses focus on it.
I have previously recorded my views with respect to what I have deemed unsound applications of the Fifth Amendment. See, for example, Miranda v. Arizona, 384 U. S. 436 , 384 U. S. 526 , and Malloy v. Hogan, 378 U. S. 1 , 378 U. S. 33 , dissenting opinions. These views, of course, have not prevailed. But I do hope that the Court will proceed with some care in extending the privilege, with all its vigor, to proceedings in juvenile court, particularly the nonadjudicatory stages of those proceedings.
In any event, I would not reach the Fifth Amendment issue here. I think the Court is clearly ill-advised to review this case on the basis of Miranda v. Arizona, since the adjudication of delinquency took place in 1964, long before the Miranda decision. See Johnson v. New Jersey, 384 U. S. 719 . Under these circumstances, this case is a poor vehicle for resolving a difficult problem. Moreover, no prejudice to appellants is at stake in this regard. The judgment below must be reversed on other grounds, and, in the event further proceedings are to be had, Gerald Gault will have counsel available to advise him.
For somewhat similar reasons, I would not reach the questions of confrontation and cross-examination, which are also dealt with in Part V of the opinion.
MR. JUSTICE HARLAN, concurring in part and dissenting in part.
Each of the 50 States has created a system of juvenile or family courts, in which distinctive rules are employed and special consequences imposed. The jurisdiction of
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these courts commonly extends both to cases which the States have withdrawn from the ordinary processes of criminal justice and to cases which involve acts that, if performed by an adult, would not be penalized as criminal. Such courts are denominated civil, not criminal, and are characteristically said not to administer criminal penalties. One consequence of these systems, at least as Arizona construes its own, is that certain of the rights guaranteed to criminal defendants by the Constitution are withheld from juveniles. This case brings before this Court for the first time the question of what limitations the Constitution places upon the operation of such tribunals. [ Footnote 3/1 ] For reasons which follow, I have concluded that the Court has gone too far in some respects, and fallen short in others, in assessing the procedural requirements demanded by the Fourteenth Amendment.
I
I must first acknowledge that I am unable to determine with any certainty by what standards the Court decides that Arizona’s juvenile courts do not satisfy the obligations of due process. The Court’s premise, itself the product of reasoning which is not described, is that the “constitutional and theoretical basis” of state systems of juvenile and family courts is “debatable”; it buttresses these doubts by marshaling a body of opinion which suggests that the accomplishments of these courts have often fallen short of expectations. [ Footnote 3/2 ] The Court does not
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indicate at what points or for what purposes such views, held either by it or by other observers, might be pertinent to the present issues. Its failure to provide any discernible standard for the measurement of due process in relation to juvenile proceedings unfortunately might be understood to mean that the Court is concerned principally with the wisdom of having such courts at all.
If this is the source of the Court’s dissatisfaction, I cannot share it. I should have supposed that the constitutionality of juvenile courts was beyond proper question under the standards now employed to assess the substantive validity of state legislation under the Due Process Clause of the Fourteenth Amendment. It can scarcely be doubted that it is within the State’s competence to adopt measures reasonably calculated to meet more effectively the persistent problems of juvenile delinquency; as the opinion for the Court makes abundantly plain, these are among the most vexing and ominous of the concerns which now face communities throughout the country.
The proper issue here is, however, not whether the State may constitutionally treat juvenile offenders through a system of specialized courts, but whether the proceedings in Arizona’s juvenile courts include procedural guarantees which satisfy the requirements of the Fourteenth Amendment. Among the first premises of our constitutional system is the obligation to conduct any proceeding in which an individual may be deprived of liberty or property in a fashion consistent with the “traditions and conscience of our people.” Snyder v. Massachusetts, 291 U. S. 97 , 291 U. S. 105 . The importance of these procedural guarantees is doubly intensified here. First, many of the problems with which Arizona is concerned
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are among those traditionally confined to the processes of criminal justice; their disposition necessarily affects in the most direct and substantial manner the liberty of individual citizens. Quite obviously, systems of specialized penal justice might permit erosion, or even evasion, of the limitations placed by the Constitution upon state criminal proceedings. Second, we must recognize that the character and consequences of many juvenile court proceedings have, in fact, closely resembled those of ordinary criminal trials. Nothing before us suggests that juvenile courts were intended as a device to escape constitutional constraints, but I entirely agree with the Court that we are nonetheless obliged to examine with circumspection the procedural guarantees the State has provided.
The central issue here, and the principal one upon which I am divided from the Court, is the method by which the procedural requirements of due process should be measured. It must at the outset be emphasized that the protections necessary here cannot be determined by resort to any classification of juvenile proceedings either as criminal or as civil, whether made by the State or by this Court. Both formulae are simply too imprecise to permit reasoned analysis of these difficult constitutional issues. The Court should instead measure the requirements of due process by reference both to the problems which confront the State and to the actual character of the procedural system which the State has created. The Court has for such purposes chiefly examined three connected sources: first, the “settled usages and modes of proceeding,” Murray’s Lessee v. Hoboken Land & Improvement Co., 18 How. 272, 59 U. S. 277 ; second, the “fundamental principles of liberty and justice which lie at the base of all our civil and political institutions,” Hebert v. Louisiana, 272 U. S. 312 , 272 U. S. 316 , and third, the character and requirements of the circumstances presented in each situation. FCC v. WJR, 337 U. S. 265 , 337 U. S. 277 ; Yakus v.
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United States, 321 U. S. 414 . See, further, my dissenting opinion in Poe v. Ullman, 367 U. S. 497 , 367 U. S. 522 , and compare my opinion concurring in the result in Pointer v. Texas, 380 U. S. 400 , 380 U. S. 408 . Each of these factors is relevant to the issues here, but it is the last which demands particular examination.
The Court has repeatedly emphasized that determination of the constitutionally required procedural safeguards in any situation requires recognition both of the “interests affected” and of the “circumstances involved.” FCC v. WJR, supra, at 337 U. S. 277 . In particular, a “compelling public interest” must, under our cases, be taken fully into account in assessing the validity under the due process clauses of state or federal legislation and its application. See, e.g., Yakus v. United States, supra, at 321 U. S. 442 ; Bowles v. Willingham, 321 U. S. 503 , 321 U. S. 520 ; Miller v. Schoene, 276 U. S. 272 , 276 U. S. 279 . Such interests would never warrant arbitrariness or the diminution of any specifically assured constitutional right, Home Bldg. & Loan Assn. v. Blaisdell, 290 U. S. 398 , 290 U. S. 426 , but they are an essential element of the context through which the legislation and proceedings under it must be read and evaluated.
No more evidence of the importance of the public interests at stake here is required than that furnished by the opinion of the Court; it indicates that “some 601,000 children under 18, or 2% of all children between 10 and 17, came before juvenile courts” in 1965, and that “about one-fifth of all arrests for serious crimes” in 1965 were of juveniles. The Court adds that the rate of juvenile crime is steadily rising. All this, as the Court suggests, indicates the importance of these due process issues, but it mirrors no less vividly that state authorities are confronted by formidable and immediate problems involving the most fundamental social values. The state legislatures have determined that the most hopeful solution for
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these problems is to be found in specialized courts, organized under their own rules and imposing distinctive consequences. The terms and limitations of these systems are not identical, nor are the procedural arrangements which they include, but the States are uniform in their insistence that the ordinary processes of criminal justice are inappropriate, and that relatively informal proceedings, dedicated to premises and purposes only imperfectly reflected in the criminal law, are instead necessary.
It is well settled that the Court must give the widest deference to legislative judgments that concern the character and urgency of the problems with which the State is confronted. Legislatures are, as this Court has often acknowledged, the “main guardian” of the public interest, and, within their constitutional competence, their understanding of that interest must be accepted as “well nigh” conclusive. Berman v. Parker, 348 U. S. 26 , 348 U. S. 32 . This principle does not, however, reach all the questions essential to the resolution of this case. The legislative judgments at issue here embrace assessments of the necessity and wisdom of procedural guarantees; these are questions which the Constitution has entrusted at least in part to courts, and upon which courts have been understood to possess particular competence. The fundamental issue here is, therefore, in what measure and fashion the Court must defer to legislative determinations which encompass constitutional issues of procedural protection.
It suffices for present purposes to summarize the factors which I believe to be pertinent. It must first be emphasized that the deference given to legislators upon substantive issues must realistically extend in part to ancillary procedural questions. Procedure at once reflects and creates substantive rights, and every effort of courts since the beginnings of the common law to separate the two has proved essentially futile. The distinction between them is particularly inadequate here, where the
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legislature’s substantive preferences directly and unavoidably require judgments about procedural issues. The procedural framework is here a principal element of the substantive legislative system; meaningful deference to the latter must include a portion of deference to the former. The substantive-procedural dichotomy is, nonetheless, an indispensable tool of analysis, for it stems from fundamental limitations upon judicial authority under the Constitution. Its premise is ultimately that courts may not substitute for the judgments of legislators their own understanding of the public welfare, but must instead concern themselves with the validity under the Constitution of the methods which the legislature has selected. See, e.g., McLean v. Arkansas, 211 U. S. 539 , 211 U. S. 547 ; Olsen v. Nebraska, 313 U. S. 236 , 313 U. S. 246 -247. The Constitution has in this manner created for courts and legislators areas of primary responsibility which are essentially congruent to their areas of special competence. Courts are thus obliged both by constitutional command and by their distinctive functions to bear particular responsibility for the measurement of procedural due process. These factors, in combination, suggest that legislatures may properly expect only a cautious deference for their procedural judgments, but that, conversely, courts must exercise their special responsibility for procedural guarantees with care to permit ample scope for achieving the purposes of legislative programs. Plainly, courts can exercise such care only if they have in each case first studied thoroughly the objectives and implementation of the program at stake; if, upon completion of those studies, the effect of extensive procedural restrictions upon valid legislative purposes cannot be assessed with reasonable certainty, the court should necessarily proceed with restraint.
The foregoing considerations, which I believe to be fair distillations of relevant judicial history, suggest
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three criteria by which the procedural requirements of due process should be measured here: first, no more restrictions should be imposed than are imperative to assure the proceedings’ fundamental fairness; second, the restrictions which are imposed should be those which preserve, so far as possible, the essential elements of the State’s purpose, and finally, restrictions should be chosen which will later permit the orderly selection of any additional protections which may ultimately prove necessary. In this way, the Court may guarantee the fundamental fairness of the proceeding, and yet permit the State to continue development of an effective response to the problems of juvenile crime.
II
Measured by these criteria, only three procedural requirements should, in my opinion, now be deemed required of state juvenile courts by the Due Process Clause of the Fourteenth Amendment: first, timely notice must be provided to parents and children of the nature and terms of any juvenile court proceeding in which a determination affecting their rights or interests may be made; second, unequivocal and timely notice must be given that counsel may appear in any such proceeding in behalf of the child and its parents, and that, in cases in which the child may be confined in an institution, counsel may, in circumstances of indigency, be appointed for them, and third, the court must maintain a written record, or its equivalent, adequate to permit effective review on appeal or in collateral proceedings. These requirements would guarantee to juveniles the tools with which their rights could be fully vindicated, and yet permit the States to pursue without unnecessary hindrance the purposes which they believe imperative in this field. Further, their imposition now would later
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permit more intelligent assessment of the necessity under the Fourteenth Amendment of additional requirements, by creating suitable records from which the character and deficiencies of juvenile proceedings could be accurately judged. I turn to consider each of these three requirements.
The Court has consistently made plain that adequate and timely notice is the fulcrum of due process, whatever the purposes of the proceeding. See, e.g., Roller v. Holly, 176 U. S. 398 , 176 U. S. 409 .; Coe v. Armour Fertilizer Works, 237 U. S. 413 , 237 U. S. 424 . Notice is ordinarily the prerequisite to effective assertion of any constitutional or other rights; without it, vindication of those rights must be essentially fortuitous. So fundamental a protection can neither be spared here nor left to the “favor or grace” of state authorities. Central of Georgia Ry. v. Wright, 207 U. S. 127 , 207 U. S. 138 ; Coe v. Armour Fertilizer Works, supra, at 237 U. S. 425 .
Provision of counsel and of a record, like adequate notice, would permit the juvenile to assert very much more effectively his rights and defenses, both in the juvenile proceedings and upon direct or collateral review. The Court has frequently emphasized their importance in proceedings in which an individual may be deprived of his liberty, see Gideon v. Wainwright, 372 U. S. 335 , and Griffin v. Illinois, 351 U. S. 12 ; this reasoning must include with special force those who are commonly inexperienced and immature. See Powell v. Alabama, 287 U. S. 45 . The facts of this case illustrate poignantly the difficulties of review without either an adequate record or the participation of counsel in the proceeding’s initial stages. At the same time, these requirements should not cause any substantial modification in the character of juvenile court proceedings: counsel, although now present in only a small percentage of juvenile cases, have apparently already appeared without
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incident in virtually all juvenile courts; [ Footnote 3/3 ] and the maintenance of a record should not appreciably alter the conduct of these proceedings.
The question remains whether certain additional requirements, among them the privilege against self-incrimination, confrontation, and cross-examination, must now, as the Court holds, also be imposed. I share in part the views expressed in my Brother WHITE’s concurring opinion, but believe that there are other, and more deep-seated, reasons to defer, at least for the present, the imposition of such requirements.
Initially, I must vouchsafe that I cannot determine with certainty the reasoning by which the Court concludes that these further requirements are now imperative. The Court begins from the premise, to which it gives force at several points, that juvenile courts need not satisfy “all of the requirements of a criminal trial.” It therefore scarcely suffices to explain the selection of these particular procedural requirements for the Court to declare that juvenile court proceedings are essentially criminal, and thereupon to recall that these are requisites for a criminal trial. Nor does the Court’s voucher of “authoritative opinion,” which consists of four extraordinary juvenile cases, contribute materially to the solution of these issues. The Court has, even under its own premises, asked the wrong questions: the problem here is to determine what forms of procedural protection are necessary to guarantee the fundamental fairness of juvenile proceedings, and not which of the procedures now employed in criminal trials should be transplanted intact to proceedings in these specialized courts.
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In my view, the Court should approach this question in terms of the criteria, described above, which emerge from the history of due process adjudication. Measured by them, there are compelling reasons at least to defer imposition of these additional requirements. First, quite unlike notice, counsel, and a record, these requirements might radically alter the character of juvenile court proceedings. The evidence from which the Court reasons that they would not is inconclusive, [ Footnote 3/4 ] and other available evidence suggests that they very likely would. [ Footnote 3/5 ] At the least, it is plain that these additional requirements would contribute materially to the creation in these proceedings of the atmosphere of an ordinary criminal trial, and would, even if they do no more, thereby largely frustrate a central purpose of these specialized courts. Further, these are restrictions intended to conform to the demands of an intensely adversary system of criminal justice; the broad purposes which they represent might be served in juvenile courts with equal effectiveness by procedural devices more consistent with the premises of proceedings
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in those courts. As the Court apparently acknowledges, the hazards of self-accusation, for example, might be avoided in juvenile proceedings without the imposition of all the requirements and limitations which surround the privilege against self-incrimination. The guarantee of adequate notice, counsel, and a record would create conditions in which suitable alternative procedures could be devised; but, unfortunately, the Court’s haste to impose restrictions taken intact from criminal procedure may well seriously hamper the development of such alternatives. Surely this illustrates that prudence and the principles of the Fourteenth Amendment alike require that the Court should now impose no more procedural restrictions than are imperative to assure fundamental fairness, and that the States should instead be permitted additional opportunities to develop without unnecessary hindrance their systems of juvenile courts.
I find confirmation for these views in two ancillary considerations. First, it is clear that an uncertain, but very substantial, number of the cases brought to juvenile courts involve children who are not in any sense guilty of criminal misconduct. Many of these children have simply the misfortune to be in some manner distressed; others have engaged in conduct, such as truancy, which is plainly not criminal. [ Footnote 3/6 ] Efforts are now being made to develop effective, and entirely noncriminal, methods of treatment for these children. [ Footnote 3/7 ] In such cases, the state authorities
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are in the most literal sense acting in loco parentis; they are, by any standard, concerned with the child’s protection, and not with his punishment. I do not question that the methods employed in such cases must be consistent with the constitutional obligation to act in accordance with due process, but certainly the Fourteenth Amendment does not demand that they be constricted by the procedural guarantees devised for ordinary criminal prosecutions. Cf. Minnesota ex rel. Pearson v. Probate Court, 309 U. S. 270 . It must be remembered that the various classifications of juvenile court proceedings are, as the vagaries of the available statistics illustrate, often arbitrary or ambiguous; it would therefore be imprudent, at the least, to build upon these classifications rigid systems of procedural requirements which would be applicable, or not, in accordance with the descriptive label given to the particular proceeding. It is better, it seems to me, to begin by now requiring the essential elements of fundamental fairness in juvenile courts, whatever the label given by the State to the proceeding; in this way, the Court could avoid imposing unnecessarily rigid restrictions, and yet escape dependence upon classifications which may often prove to be illusory. Further, the provision of notice, counsel, and a record would permit orderly efforts to determine later whether more satisfactory classifications can be devised, and, if they can, whether additional procedural requirements are necessary for them under the Fourteenth Amendment.
Second, it should not be forgotten that juvenile crime and juvenile courts are both now under earnest study throughout the country. I very much fear that this Court, by imposing these rigid procedural requirements, may inadvertently have served to discourage these efforts to find more satisfactory solutions for the problems of juvenile crime, and may thus now hamper enlightened development of the systems of juvenile courts. It is
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appropriate to recall that the Fourteenth Amendment does not compel the law to remain passive in the midst of change; to demand otherwise denies “every quality of the law but its age.” Hurtado v. California, 110 U. S. 516 , 110 U. S. 529 .
III
Finally, I turn to assess the validity of this juvenile court proceeding under the criteria discussed in this opinion. Measured by them, the judgment below must, in my opinion, fall. Gerald Gault and his parents were not provided adequate notice of the terms and purposes of the proceedings in which he was adjudged delinquent; they were not advised of their rights to be represented by counsel, and no record in any form was maintained of the proceedings. It follows, for the reasons given in this opinion, that Gerald Gault was deprived of his liberty without due process of law, and I therefore concur in the judgment of the Court.
[ Footnote 3/1 ]
Kent v. United States, 383 U. S. 541 , decided at the 1965 Term, did not purport to rest on constitutional grounds.
[ Footnote 3/2 ]
It is appropriate to observe that, whatever the relevance the Court may suppose that this criticism has to present issues, many of the critics have asserted that the deficiencies of juvenile courts have stemmed chiefly from the inadequacy of the personnel and resources available to those courts. See, e.g., Paulsen, Kent v. United States: The Constitutional Context of Juvenile Cases, 1966 Sup.Ct.Rev. 167, 191-192; Handler, The Juvenile Court and the Adversary System: Problems of Function and Form, 1965 Wis.L.Rev. 7, 46.
[ Footnote 3/3 ]
The statistical evidence here is incomplete, but see generally Skoler & Tenney, Attorney Representation in Juvenile Court, 4 J.Fam.Law 77. They indicate that some 91% of the juvenile court judges whom they polled favored representation by counsel in their courts. Id. at 88.
[ Footnote 3/4 ]
Indeed, my Brother BLACK candidly recognizes that such is apt to be the effect of today’s decision, ante, p. 387 U. S. 60 . The Court itself is content merely to rely upon inapposite language from the recommendations of the Children’s Bureau, plus the terms of a single statute.
[ Footnote 3/5 ]
The most cogent evidence, of course, consists of the steady rejection of these requirements by state legislatures and courts. The wide disagreement and uncertainty upon this question are also reflected in Paulsen, Kent v. United States: The Constitutional Context of Juvenile Cases, 1966 Sup.Ct.Rev. 167, 186, 191. See also Paulsen, Fairness to the Juvenile Offender, 41 Minn.L.Rev. 547, 561-562; McLean, An Answer to the Challenge of Kent, 53 A.B.A.J. 456, 457; Alexander, Constitutional Rights in Juvenile Court, 46 A.B.A.J. 1206; Shears, Legal Problems Peculiar to Children’s Courts, 48 A.B.A.J. 719; Siler, The Need for Defense Counsel in the Juvenile Court, 11 Crime & Delin. 45, 57-58. Compare Handler, The Juvenile Court and the Adversary System: Problems of Function and Form, 1965 Wis.L.Rev. 7, 32.
[ Footnote 3/6 ]
Estimates of the number of children in this situation brought before juvenile courts range from 26% to some 48%; variation seems chiefly a product both of the inadequacy of records and of the difficulty of categorizing precisely the conduct with which juveniles are charged. See generally Sheridan, Juveniles Who Commit Noncriminal Acts: Why Treat in a Correctional System? 31 Fed. Probation 26, 27. By any standard, the number of juveniles involved is “considerable.” Ibid.
[ Footnote 3/7 ]
Id. at 28-30.
MR. JUSTICE STEWART, dissenting.
The Court today uses an obscure Arizona case as a vehicle to impose upon thousands of juvenile courts throughout the Nation restrictions that the Constitution made applicable to adversary criminal trials. [ Footnote 4/1 ] I believe the Court’s decision is wholly unsound as a matter of constitutional law, and sadly unwise as a matter of judicial policy.
Juvenile proceedings are not criminal trials. They are not civil trials. They are simply not adversary proceedings. Whether treating with a delinquent child, a neglected
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child, a defective child, or a dependent child, a juvenile proceeding’s whole purpose and mission is the very opposite of the mission and purpose of a prosecution in a criminal court. The object of the one is correction of a condition. The object of the other is conviction and punishment for a criminal act.
In the last 70, years many dedicated men and women have devoted their professional lives to the enlightened task of bringing us out of the dark world of Charles Dickens in meeting our responsibilities to the child in our society. The result has been the creation in this century of a system of juvenile and family courts in each of the 50 States. There can be no denying that, in many areas the performance of these agencies has fallen disappointingly short of the hopes and dreams of the courageous pioneers who first conceived them. For a variety of reasons, the reality has sometimes not even approached the ideal, and much remains to be accomplished in the administration of public juvenile and family agencies — in personnel, in planning, in financing, perhaps in the formulation of wholly new approaches.
I possess neither the specialized experience nor the expert knowledge to predict with any certainty where may lie the brightest hope for progress in dealing with the serious problems of juvenile delinquency. But I am certain that the answer does not lie in the Court’s opinion in this case, which serves to convert a juvenile proceeding into a criminal prosecution.
The inflexible restrictions that the Constitution so wisely made applicable to adversary criminal trials have no inevitable place in the proceedings of those public social agencies known as juvenile or family courts. And to impose the Court’s long catalog of requirements upon juvenile proceedings in every area of the country is to invite a long step backwards into the nineteenth century. In that era, there were no juvenile proceedings, and a
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child was tried in a conventional criminal court with all the trappings of a conventional criminal trial. So it was that a 12-year-old boy named James Guild was tried in New Jersey for killing Catharine Beakes. A jury found him guilty of murder, and he was sentenced to death by hanging. The sentence was executed. It was all very constitutional. [ Footnote 4/2 ]
A State in all its dealings must, of course, accord every person due process of law. And due process may require that some of the same restrictions which the Constitution has placed upon criminal trials must be imposed upon juvenile proceedings. For example, I suppose that all would agree that a brutally coerced confession could not constitutionally be considered in a juvenile court hearing. But it surely does not follow that the testimonial privilege against self-incrimination is applicable in all juvenile proceedings. [ Footnote 4/3 ] Similarly, due process clearly
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requires timely notice of the purpose and scope of any proceedings affecting the relationship of parent and child. Armstrong v. Manzo, 380 U. S. 545 . But it certainly does not follow that notice of a juvenile hearing must be framed with all the technical niceties of a criminal indictment. See Russell v. United States, 369 U. S. 749 .
In any event, there is no reason to deal with issues such as these in the present case. The Supreme Court of Arizona found that the parents of Gerald Gault
“knew of their right to counsel, to subpoena and cross-examine witnesses, of the right to confront the witnesses against Gerald, and the possible consequences of a finding of delinquency.”
99 Ariz. 181, 185, 407 P.2d 760, 763. It further found that “Mrs. Gault knew the exact nature of the charge against Gerald from the day he was taken to the detention home.” 99 Ariz. at 193, 407 P.2d at 768. And, as MR. JUSTICE WHITE correctly points out, pp. 387 U. S. 64 -65, ante, no issue of compulsory self-incrimination is presented by this case.
I would dismiss the appeal.
[ Footnote 4/1 ]
I find it strange that a Court so intent upon fastening an absolute right to counsel upon nonadversary juvenile proceeding has not been willing even to consider whether the Constitution requires a lawyer’s help in a criminal prosecution upon a misdemeanor charge. See Winters v. Beck, 385 U. S. 907 ; DeJoseph v. Connecticut, 385 U. S. 982 .
[ Footnote 4/2 ]
State v. Guild, 5 Halst. 163, 18 Am. Dec. 404 (N.J.Sup.Ct.).
“Thus, also, in very modern times, a boy of ten years old was convicted on his own confession of murdering his bed-fellow, there appearing in his whole behavior plain tokens of a mischievous discretion, and as the sparing this boy merely on account of his tender years might be of dangerous consequence to the public, by propagating a notion that children might commit such atrocious crimes with impunity, it was unanimously agreed by all the judges that he was a proper subject of capital punishment.”
4 Blackstone, Commentaries 23 (Wendell ed. 1847).
[ Footnote 4/3 ]
Until June 13, 1966, it was clear that the Fourteenth Amendment’s ban upon the use of a coerced confession is constitutionally quite a different thing from the Fifth Amendment’s testimonial privilege against self-incrimination. See, for example, the Court’s unanimous opinion in Brown v. Mississippi, 297 U. S. 278 , at 297 U. S. 285 -286, written by Chief Justice Hughes and joined by such distinguished members of this Court as Mr. Justice Brandeis, Mr. Justice Stone, and Mr. Justice Cardozo. See also Tehan v. Shott, 382 U. S. 406 , decided January 19, 1966, where the Court emphasized the “contrast” between “the wrongful use of a coerced confession” and “the Fifth Amendment’s privilege against self-incrimination.” 382 U.S. at 382 U. S. 416 . The complete confusion of these separate constitutional doctrines in Part V of the Court’s opinion today stems, no doubt, from Miranda v. Arizona, 384 U. S. 436 , a decision which I continue to believe was constitutionally erroneous.
Contributors and Attributions
- Authored by : US Government . Located at : https://supreme.justia.com/cases/federal/us/387/1/case.html . License : Public Domain: No Known Copyright


16.15: Introduction to Continuous-Time Markov Chains
This section begins our study of Markov processes in continuous time and with discrete state spaces. Recall that a Markov process with a discrete state space is called a Markov chain , so we are studying continuous-time Markov chains . It will be helpful if you review the section on general Markov processes, at least briefly, to become familiar with the basic notation and concepts. Also, discrete-time chains plays a fundamental role, so you will need review this topic also.
We will study continuous-time Markov chains from different points of view. Our point of view in this section, involving holding times and the embedded discrete-time chain, is the most intuitive from a probabilistic point of view, and so is the best place to start. In the next section, we study the transition probability matrices in continuous time. This point of view is somewhat less intuitive, but is closest to how other types of Markov processes are treated. Finally, in the third introductory section we study the Markov chain from the view point of potential matrices. This is the least intuitive approach, but analytically one of the best. Naturally, the interconnections between the various approaches are particularly important.
Preliminaries
As usual, we start with a probability space \( (\Omega, \mathscr{F}, \P) \), so that \( \Omega \) is the set of outcomes, \( \mathscr{F} \) the \( \sigma \)-algebra of events, and \( \P \) the probability measure on the sample space \( (\Omega, \mathscr{F}) \). The time space is \( ([0, \infty), \mathscr{T}) \) where as usual, \( \mathscr{T} \) is the Borel \( \sigma \)-algebra on \( [0, \infty) \) corresponding to the standard Euclidean topology. The state space is \( (S, \mathscr{S}) \) where \( S \) is countable and \( \mathscr{S} \) is the power set of \( S \). So every subset of \( S \) is measurable, as is every function from \( S \) to another measurable space. Recall that \( \mathscr{S} \) is also the Borel \( \sigma \) algebra corresponding to the discrete topology on \( S \). With this topology, every function from \( S \) to another topological space is continuous. Counting measure \( \# \) is the natural measure on \( (S, \mathscr{S}) \), so in the context of the general introduction, integrals over \( S \) are simply sums. Also, kernels on \( S \) can be thought of as matrices, with rows and sums indexed by \( S \). The left and right kernel operations are generalizations of matrix multiplication.
Suppose now that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is stochastic process with state space \( (S, \mathscr{S}) \). For \( t \in [0, \infty) \), let \( \mathscr{F}^0_t = \sigma\{X_s: s \in [0, t]\} \), so that \( \mathscr{F}^0_t \) is the \( \sigma \)-algebra of events defined by the process up to time \( t \). The collection of \( \sigma \)-algebras \( \mathfrak{F}^0 = \{\mathscr{F}^0_t: t \in [0, \infty)\} \) is the natural filtration associated with \( \bs{X} \). For technical reasons, it's often necessary to have a filtration \( \mathfrak{F} = \{\mathscr{F}_t: t \in [0, \infty)\} \) that is slightly finer than the natural one, so that \( \mathscr{F}^0_t \subseteq \mathscr{F}_t \) for \( t \in [0, \infty) \) (or in equivlaent jargon, \( \bs{X} \) is adapted to \( \mathfrak{F} \)). See the general introduction for more details on the common ways that the natural filtration is refined. We will also let \(\mathscr{G}_t = \sigma\{X_s: s \in [t, \infty)\}\), the \( \sigma \)-algebra of events defined by the process from time \( t \) onward. If \( t \) is thought of as the present time, then \( \mathscr{F}_t \) is the collection of events in the past and \( \mathscr{G}_t \) is the collection of events in the future.
It's often necessary to impose assumptions on the continuity of the process \( \bs{X} \) in time. Recall that \( \bs{X} \) is right continuous if \( t \mapsto X_t(\omega) \) is right continuous on \( [0, \infty) \) for every \( \omega \in \Omega \), and similarly \( \bs{X} \) has left limits if \( t \mapsto X_t(\omega) \) has left limits on \( (0, \infty) \) for every \( \omega \in \Omega \). Since \( S \) has the discrete topology, note that if \( \bs{X} \) is right continuous, then for every \( t \in [0, \infty) \) and \( \omega \in \Omega \), there exists \( \epsilon \) (depending on \( t \) and \( \omega \)) such that \( X_{t+s}(\omega) = X_t(\omega) \) for \( s \in [0, \epsilon) \). Similarly, if \( \bs{X} \) has left limits, then for every \( t \in (0, \infty) \) and \( \omega \in \Omega \) there exists \( \delta \) (depending on \( t \) and \( \omega \)) such that \( X_{t - s}(\omega) \) is constant for \( s \in (0, \delta) \).
The Markov Property
There are a number of equivalent ways to state the Markov property. At the most basic level, the property states that the past and future are conditionally independent, given the present .
The process \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a Markov chain on \( S \) if for every \( t \in [0, \infty) \), \( A \in \mathscr{F}_t \), and \( B \in \mathscr{G}_t \), \[ \P(A \cap B \mid X_t) = \P(A \mid X_t) \P(B \mid X_t) \]
Another version is that the conditional distribution of a state in the future, given the past, is the same as the conditional distribution just given the present state.
The process \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a Markov chain on \( S \) if for every \( s, \, t \in [0, \infty) \), and \( x \in S \), \[ \P(X_{s + t} = x \mid \mathscr{F}_s) = \P(X_{s + t} = x \mid X_s) \]
Technically, in the last two definitions, we should say that \( \bs{X} \) is a Markov process relative to the filtration \( \mathfrak{F} \). But recall that if \( \bs{X} \) satisfies the Markov property relative to a filtration, then it satisfies the Markov property relative to any coarser filtration, and in particular, relative to the natural filtration. For the natural filtration, the Markov property can also be stated without explicit reference to \( \sigma \)-algebras, although at the cost of additional clutter:
The process \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a Markov chain on \( S \) if and only if for every \( n \in \N_+ \), time sequence \( (t_1, t_2, \ldots, t_n) \in [0, \infty)^n \) with \( t_1 \lt t_2 \lt \cdots \lt t_n \), and state sequence \( (x_1, x_2, \ldots, x_n) \in S^n \), \[ \P\left(X_{t_n} = x_n \mid X_{t_1} = x_1, X_{t_2} = x_2, \ldots X_{t_{n-1}} = x_{n-1}\right) = \P\left(X_{t_n} = x_n \mid X_{t_{n-1}} = x_{n-1}\right) \]
As usual, we also assume that our Markov chain \( \bs{X} \) is time homogeneous , so that \( \P(X_{s + t} = y \mid X_s = x) = \P(X_t = y \mid X_0 = x) \) for \( s, \, t \in [0, \infty) \) and \( x, \, y \in S \). So, for a homogeneous Markov chain on \( S \), the process \( \{X_{s+t}: t \in [0, \infty)\} \) given \( X_s = x \), is independent of \( \mathscr{F}_s \) and equivalent to the process \( \{X_t: t \in [0, \infty)\} \) given \( X_0 = x \), for every \( s \in [0, \infty) \) and \( x \in S \). That is, if the chain is in state \( x \in S \) at a particular time \( s \in [0, \infty) \), it does not matter how the chain got to \( x \); the chain essentially starts over in state \( x \).
The Strong Markov Property
Random times play an important role in the study of continuous-time Markov chains. It's often necessary to allow random times to take the value \( \infty \), so formally, a random time \( \tau \) is a random variable on the underlying sample space \( (\Omega, \mathscr{F}) \) taking values in \( [0, \infty] \). Recall also that a random time \( \tau \) is a stopping time (also called a Markov time or an optional time ) if \( \{\tau \le t\} \in \mathscr{F}_t \) for every \( t \in [0, \infty) \). If \( \tau \) is a stopping time, the \( \sigma \)-algebra associated with \( \tau \) is \[ \mathscr{F}_\tau = \{A \in \mathscr{F}: A \cap \{\tau \le t\} \in \mathscr{F}_t \text{ for all } t \in [0, \infty)\} \] So \( \mathscr{F}_\tau \) is the collection of events up to the random time \( \tau \) in the same way that \( \mathscr{F}_t \) is the collection of events up to the deterministic time \( t \in [0, \infty) \). We usually want the Markov property to extend from deterministic times to stopping times.
The process \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a strong Markov chain on \( S \) if for every stopping time \( \tau \), \(t \in [0, \infty) \), and \( x \in S \), \[ \P(X_{\tau + t} = x \mid \mathscr{F}_\tau) = \P(X_{\tau + t} = x \mid X_\tau) \]
So, for a homogeneous strong Markov chain on \( S \), the process \( \{X_{\tau + t}: t \in [0, \infty)\} \) given \( X_\tau = x \), is independent of \( \mathscr{F}_\tau \) and equivalent to the process \( \{X_t: t \in [0, \infty)\} \) given \( X_0 = x \), for every stopping time \( \tau \) and \( x \in S \). That is, if the chain is in state \( x \in S \) at a stopping time \( \tau \), then the chain essentially starts over at \( x \), independently of the past.
Holding Times and the Jump Chain
For our first point of view, we sill study when and how our Markov chain \( \bs{X} \) changes state. The discussion depends heavily on properties of the exponential distribution, so we need a quick review.
The Exponential Distribution
A random variable \( \tau \) has the exponential distribution with rate parameter \( r \in (0, \infty) \) if \( \tau \) has a continuous distribution on \( [0, \infty) \) with probability density function \( f \) given by \( f(t) = r e^{-r t} \) for \( t \in [0, \infty) \). Equivalently, the right distribution function \( F^c \) is given by \[ F^c(t) = \P(\tau \gt t) = e^{-r t}, \quad t \in [0, \infty) \] The mean of the distribution is \( 1 / r \) and the variance is \( 1 / r^2 \). The exponential distribution has an amazing number of characterizations. One of the most important is the memoryless property which states that a random variable \( \tau \) with values in \( [0, \infty) \) has an exponential distribution if and only if the conditional distribution of \( \tau - s \) given \( \tau \gt s \) is the same as the distribution of \( \tau \) itself, for every \( s \in [0, \infty) \). It's easy to see that the memoryless property is equivalent to the law of exponents for right distribution function \( F^c \), namely \( F^c(s + t) = F^c(s) F^c(t) \) for \( s, \, t \in [0, \infty) \). Since \( F^c \) is right continuous, the only solutions are exponential functions.
For our study of continuous-time Markov chains, it's helpful to extend the exponential distribution to two degenerate cases, \( \tau = 0 \) with probability 1, and \( \tau = \infty \) with probability 1. In terms of the parameter, the first case corresponds to \( r = \infty \) so that \( F(t) = \P(\tau \gt t) = 0 \) for every \( t \in [0, \infty) \), and the second case corresponds to \( r = 0 \) so that \( F(t) = \P(\tau \gt t) = 1 \) for every \( t \in [0, \infty) \). Note that in both cases, the function \( F \) satisfies the law of exponents, and so corresponds to a memoryless distribution in a general sense. In all cases, the mean of the exponential distribution with parameter \( r \in [0, \infty] \) is \( 1 / r \), where we interpret \( 1/0 = \infty \) and \( 1/\infty = 0 \).
Holding Times
The Markov property implies the memoryless property for the random time when a Markov process first leaves its initial state. It follows that this random time must have an exponential distribution.
Suppose that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a Markov chain on \( S \), and let \( \tau = \inf\{t \in [0, \infty): X_t \ne X_0\} \). For \( x \in S \), the conditional distribution of \( \tau \) given \( X_0 = x \) is exponential with parameter \( \lambda(x) \in [0, \infty] \).
Proof
Let \( x \in S \) and \( s \in [0, \infty) \). The events \( X_0 = x \) and \( \tau \gt s \) imply \( X_s = x \). By the Markov property, given \( X_s = x \), the chain starts over at time \( s \) in state \( x \), independent of \( \{X_0 = x\} \) and \( \{\tau \gt s\} \), since both events are in \( \mathscr{F}_s \). Hence for \( t \in [0, \infty) \), \[ \P(\tau \gt t + s \mid X_0 = x, \tau \gt s) = \P(\tau \gt t + s \mid X_0 = x, X_s = x, \tau \gt s) = \P(\tau \gt t \mid X_0 = x)\] It follows that \( \tau \) has the memoryless property, and hence has an exponential distribution with parameter \( \lambda(x) \in [0, \infty] \).
So, associated with the Markov chain \( \bs{X} \) on \( S \) is a function \( \lambda: S \to [0, \infty] \) that gives the exponential parameters for the holding times in the states. Considering the ordinary exponential distribution, and the two degenerate versions, we are led to the following classification of states:
Suppose again that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a Markov chain on \( S \) with exponential parameter function \( \lambda \). Let \( x \in S \).
- If \( \lambda(x) = 0 \) then \( \P(\tau = \infty \mid X_0 = x) = 1 \), and \( x \) is said to be an absorbing state .
- If \( \lambda(x) \in (0, \infty) \) then \( \P(0 \lt \tau \lt \infty \mid X_0 = x) = 1 \) and \( x \) is said to be an stable state .
- If \( \lambda(x) = \infty \) then \( \P(\tau = 0 \mid X_0 = x) = 1 \), and \( x \) is said to be an instantaneous state .
As you can imagine, an instantaneous state corresponds to weird behavior, since the chain starting in the state leaves the state at times arbitrarily close to 0. While mathematically possible, instantaneous states make no sense in most applications, and so are to be avoided. Also, the proof of the last result has some technical holes. We did not really show that \( \tau \) is a valid random time, let alone a stopping time. Fortunately, one of our standard assumptions resolves these problems.
Suppose again that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a Markov chain on \( S \). If the process \( \bs{X} \) and the filtration \( \mathfrak{F} \) are right continuous, then
- \( \tau \) is a stopping time.
- \( \bs{X} \) has no instantaneous states.
- \( \P(X_\tau \ne x \mid X_0 = x) = 1 \) if \( x \in S \) is stable.
- \( \bs{X} \) is a strong Markov process.
Proof
- Let \( t \in [0, \infty) \). By right continuity, \[ \{\tau \lt t\} = \{X_s \ne X_0 \text{ for some } s \in (0, t)\} = \{X_s \ne X_0 \text{ for some rational } s \in (0, t)\} \] But for \( s \in (0, t) \), \( \{X_s \ne X_0\} \in \mathscr{F}_s \subseteq \mathscr{F}_t \). The last event in the displayed equation is a countable union, so \( \{\tau \lt t\} \in \mathscr{F}_t \). Since \( \mathfrak{F} \) is right continuous, \( \tau \) is a stopping time.
- Suppose that \( \omega \in \Omega \) and \( X_0(\omega) = x \). Since \( \bs{X} \) is right continuous, there exists \( \epsilon \gt 0 \) such that \( X_t(\omega) = x \) for \( 0 \le t \lt \epsilon \) and hence \( \tau(\omega) \ge \epsilon \gt 0 \). So \( \P(\tau \gt 0 \mid X_0 = x) = 1 \).
- Similarly, suppose that \( \omega \in \Omega \) and that \( X_0(\omega) = x \) and \( X_{\tau(\omega)}(\omega) = y \). Since \( \bs{X} \) is right continuous, there exists \( \epsilon \gt 0 \) such that \( X_t(\omega) = y \) for \( \tau(\omega) \le t \lt \tau(\omega) + \epsilon \). But by definition of \( \tau(\omega) \), there exists \( t \in (\tau(\omega), \tau(\omega) + \epsilon) \) with \( X_t(\omega) \ne x \). Hence \( \P(X_\tau \ne x \mid X_0 = x) = 1 \).
There is actually a converse to part (b) that states that if \( \bs{X} \) has no instantaneous states, then there is a version of \( \bs{X} \) that is right continuous. From now on, we will assume that our Markov chains are right continuous with probability 1, and hence have no instantaneous states. On the other hand, absorbing states are perfectly reasonable and often do occur in applications. Finally, if the chain enters a stable state, it will stay there for a (proper) exponentially distributed time, and then leave.
The Jump Chain
Without instantaneous states, we can now construct a sequence of stopping times. Basically, we let \( \tau_n \) denote the \( n \)th time that the chain changes state for \( n \in \N_+ \), unless the chain has previously been caught in an absorbing state. Here is the formal construction:
Suppose again that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a Markov chain on \( S \). Let \( \tau_0 = 0 \) and \( \tau_1 = \inf\{t \in [0, \infty): X_t \ne X_0\} \). Recursively, suppose that \( \tau_n \) is defined for \( n \in \N_+ \). If \( \tau_n = \infty \) let \( \tau_{n+1} = \infty \). Otherwise, let \[ \tau_{n+1} = \inf\left\{t \in [\tau_n, \infty): X_t \ne X_{\tau_n}\right\} \] Let \( M = \sup\{n \in \N: \tau_n \lt \infty\} \).
In the definition of \( M \), of course, \( \sup(\N) = \infty \), so \( M \) is the number of changes of state. If \( M \lt \infty \), the chain was sucked into an absorbing state at time \( \tau_M \). Since we have ruled out instantaneous states, the sequence of random times in strictly increasing up until the (random) term \( M \). That is, with probability 1, if \( n \in \N \) and \( \tau_n \lt \infty \) then \( \tau_n \lt \tau_{n+1} \). Of course by construction, if \( \tau_n = \infty \) then \( \tau_{n+1} = \infty \). The increments \( \tau_{n+1} - \tau_n \) for \( n \in \N \) with \( n \lt M \) are the times spent in the states visited by \( \bs{X} \). The process at the random times when the state changes forms an embedded discrete-time Markov chain.
Suppose again that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a Markov chain on \( S \). Let \( \{\tau_n: n \in \N\} \) denote the stopping times and \( M \) the random index, as defined above. For \( n \in \N \), let \( Y_n = X_{\tau_n} \) if \( n \le M \) and \( Y_n = X_{\tau_M} \) if \( n \gt M \). Then \( \bs{Y} = \{Y_n: n \in \N\} \) is a (homogenous) discrete-time Markov chain on \( S \), known as the jump chain of \( \bs{X} \).
Proof
For \( n \in \N \) let \( \mathscr{G}_n = \sigma\{Y_0, Y_1, \ldots, Y_n\} \), the \( \sigma \)-algebra of events for the process \( \bs{Y} \), up to the discrete time \( n \). Let \( x \in S \). If \( x \) is stable, then given \( Y_n = x \), the random times \( \tau_n \) and \( \tau_{n+1} \) are finite with probability 1. (Note that we cannot get to \( x \) from an absorbing state.) So \[ \P(Y_{n+1} = y \mid Y_n = x, \mathscr{G}_n) = \P\left(X_{\tau_{n+1}} = y \mid X_{\tau_n} = x, \mathscr{G}_n\right), \quad y \in S \] But by the strong Markov property, given \( X_{\tau_n} = x \), the chain starts over at time \( \tau_n \) in state \( x \), independent of \( \mathscr{G}_n \subseteq \mathscr{F}_{\tau_n} \). Hence \[ \P(Y_{n+1} = y \mid Y_n = x, \mathscr{G}_n) = \P(X_\tau = y \mid X_0 = x), \quad y \in S\] On the other hand, if \( x \) is an absorbing state, then by construction, \[ \P(Y_{n+1} = y \mid Y_n = x, \mathscr{G}_n) = I(x, y), \quad y \in S \] where \( I \) is the identity matrix on \( S \).
As noted in the proof, the one-step transition probability matrix \( Q \) for the jump chain \( \bs{Y} \) is given for \( (x, y) \in S^2 \) by \[ Q(x, y) = \begin{cases} \P(X_\tau = y \mid X_0 = x), & x \text{ stable} \\ I(x, y), & x \text{ absorbing} \end{cases} \] where \( I \) is the identity matrix on \( S \). Of course \( Q \) satisfies the usual properties of a probability matrix on \( S \), namely \( Q(x, y) \ge 0 \) for \( (x, y) \in S^2 \) and \( \sum_{y \in S} Q(x, y) = 1 \) for \( x \in S \). But \( Q \) satisfies another interesting property as well. Since the the state actually changes at time \( \tau \) starting in a stable state, we must have \( Q(x, x) = 0 \) if \( x \) is stable and \( Q(x, x) = 1 \) if \( x \) is absorbing.
Given the initial state, the holding time and the next state are independent.
If \( x, \, y \in S \) and \( t \in [0, \infty) \) then \( \P(Y_1 = y, \tau_1 \gt t \mid Y_0 = x) = Q(x, y) e^{-\lambda(x) t} \)
Proof
Suppose that \( x \) is a stable state, so that given \( Y_0 = X_0 = x \), the stopping time \( \tau_1 = \tau\) has a proper exponential distribution with parameter \( \lambda(x) \in (0, \infty) \). Note that \[ \P(Y_1 = y, \tau_1 \gt t \mid Y_0 = x) = \P(X_{\tau} = y, \tau \gt t \mid X_0 = x) = \P(X_\tau = y \mid \tau \gt t, X_0 = x) \P(\tau \gt t \mid X_0 = x) \] Note that if \( X_0 = x \) and \( \tau \gt t \) then \( X_t = x \) also. By the Markov property, given \( X_t = x \), the chain starts over at time \( t \) in state \( x \), independent of \( \{X_0 = x\} \) and \( \{\tau \gt t\} \), both events in \( \mathscr{F}_t \). Hence \[ \P(X_\tau = y \mid \tau \gt t, X_0 = x) = \P(X_\tau = y \mid X_t = x, \tau \gt t, X_0 = x) = \P(X_\tau = y \mid X_0 = x) = Q(x, y) \] Of course \( \P(\tau \gt t \mid X_0 = x) = e^{-\lambda(x) t} \).
If \( x \) is an absorbing state then \( \P(\tau = \infty \mid X_0 = x) = 1 \), \( \P(Y_1 = x \mid Y_0 = x) = 1 \), and \( \lambda(x) = 0 \). Hence \[ \P(Y_1 = y, \tau_1 \gt t \mid Y_0 = x) = I(x, y) = Q(x, y) e^{-\lambda(x) t} \]
The following theorem is a generalization. The changes in state and the holding times are independent, given the initial state.
Suppose that \( n \in \N_+ \) and that \( (x_0, x_1, \ldots, x_n) \) is a sequence of stable states and \( (t_1, t_2, \ldots, t_n) \) is a sequence in \( [0, \infty) \). Then \begin{align*} & \P(Y_1 = x_1, \tau_1 \gt t_1, Y_2 = x_2, \tau_2 - \tau_1 \gt t_2, \ldots, Y_n = x_n, \tau_n - \tau_{n-1} \gt t_n \mid Y_0 = x_0) \\ & = Q(x_0, x_1) e^{-\lambda(x_0) t_1} Q(x_1, x_2) e^{-\lambda(x_1) t_2} \cdots Q(x_{n-1}, x_n) e^{-\lambda(x_{n-1}) t_n} \end{align*}
Proof
The proof is by induction, and the essence is captured in the case \( n = 2 \). So suppose that \( x_0, \, x_1, \, x_2 \) are stable states and \( t_1, \, t_2 \in [0, \infty) \). Then \begin{align*} & \P(Y_1 = x_1, \tau_1 \gt t_1, Y_2 = x_2, \tau_2 - \tau_1 \gt t_2 \mid Y_0 = x_0) \\ & = \P(Y_2 = x_2, \tau_2 - \tau_1 \gt t_2 \mid X_0 = x, Y_1 = x_1, \tau_1 \gt t_1) \P(Y_1 = x_1, \tau_1 \gt t_1 \mid Y_0 = x_0) \end{align*} But \( \P(Y_1 = x_1, \tau_1 \gt t_1 \mid Y_0 = x_0) = Q(x_0, x_1) e^{-\lambda(x_0) t_1} \) by the previous theorem. Next, by definition, \[\P(Y_2 = x_2, \tau_2 - \tau_1 \gt t_2 \mid X_0 = x, Y_1 = x_1, \tau_1 \gt t_1) = \P\left(X_{\tau_2} = x_2, \tau_2 - \tau_1 \gt t_2 \mid X_0 = x_0, X_{\tau_1} = x_1, \tau_1 \gt t_1\right) \] But by the strong Markov property, given \( X_{\tau_1} = x_1 \), the chain starts over at time \( \tau_1 \) in state \( x \), independent of the events \( \{X_0 = x_0\} \) and \( \{\tau_1 \gt t_1\} \) (both events in \( \mathscr{F}_{\tau_1} \)). Hence using the previous theorem again, \[ \P(Y_2 = x_2, \tau_2 - \tau_1 \gt t_2 \mid X_0 = x, Y_1 = x_1, \tau_1 \gt t_1) = \P(X_\tau = x_2, \tau \gt t_2 \mid X_0 = x_1) = Q(x_1, x_2) e^{-\lambda(x_1)t_2} \]
Regularity
We now know quite a bit about the structure of a continuous-time Markov chain \( \bs{X} = \{X_t: t \in [0, \infty)\} \) (without instantaneous states). Once the chain enters a given state \( x \in S \), the holding time in state \( x \) has an exponential distribution with parameter \( \lambda(x) \in [0, \infty) \), after which the next state \( y \in S \) is chosen, independently of the holding time, with probability \( Q(x, y) \). However, we don't know everything about the chain. For the sequence \( \{\tau_n: n \in \N\} \) defined above, let \( \tau_\infty = \lim_{n \to \infty} \tau_n \), which exists in \( (0, \infty] \) of course, since the sequence is increasing. Even though the holding time in a state is positive with probability 1, it's possible that \( \tau_\infty \lt \infty \) with positive probability, in which case we know nothing about \( X_t \) for \(t \ge \tau_\infty \). The event \( \{\tau_\infty \lt \infty\} \) is known as explosion , since it means that the \( \bs{X} \) makes infinitely many transitions before the finite time \( \tau_\infty \). While not as pathological as the existence of instantaneous states, explosion is still to be avoided in most applications.
A Markov chain \( \bs{X} = \{X_t: t \in [0, \infty)\} \) on \( S \) is regular if each of the following events has probability 1:
- \( \bs{X} \) is right continuous.
- \( \tau_n \to \infty \) as \( n \to \infty \).
There is a simple condition on the exponential parameters and the embedded chain that is equivalent to condition (b).
Suppose that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a right-continuous Markov chain on \( S \) with exponential parameter function \( \lambda \) and embedded chain \( \bs{Y} = (Y_0, Y_1, \ldots) \). Then \( \tau_n \to \infty \) as \( n \to \infty \) with probability 1 if and only if \( \sum_{n=0}^\infty 1 \big/ \lambda(Y_n) = \infty \) with probability 1.
Proof
Given \( \bs{Y} = (y_0, y_1, \ldots) \), the distribution of \( \tau_\infty = \lim_{n \to \infty} \tau_n \) is the distribution of \( T_\infty = \sum_{n=0}^\infty T_n \) where \( (T_0, T_1, \ldots) \) are independent, and \( T_n \) has the exponential distribution with parameter \( \lambda(y_n) \). Note that \( \E(T_\infty) = \sum_{n=0}^\infty 1 \big/ \lambda(y_n) \). In the section on the exponential distribution, it's shown that \( \P(T_\infty = \infty) = 1 \) if and only if \( \E(T_\infty) = \infty \).
If \(\lambda \) is bounded, then \( \bs{X} \) is regular.
Suppose that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a Markov chain on \( S \) with exponential parameter function \( \lambda \). If \(\lambda \) is bounded, then \( \bs{X} \) is regular.
Proof
Suppose that \( \lambda(x) \le r \) for \( x \in S \), where \( r \in (0, \infty) \). Then in particular, \( \bs{X} \) has no instantaneous states and so is right continuous. Moreover, \( 1 / \lambda(x) \ge 1 / r \) for \( x \in S \) so \( \sum_{n=0}^\infty 1 \big / \lambda(Y_n) = \infty \) with probability 1, where as ususal, \( \bs{Y} = (Y_0, Y_1, \ldots) \) is the jump chain of \( \bs{X} \).
Here is another sufficient condition that is useful when the state space is infinite.
Suppose that \( \bs X = \{X_t: t \in [0, \infty)\} \) is a Markov chain on \( S \) with exponential parameter function \( \lambda: S \to [0, \infty) \). Let \( S_+ = \{x \in S: \lambda(x) \gt 0\} \). Then \( \bs X \) is regular if \[ \sum_{x \in S_+} \frac{1}{\lambda(x)} = \infty \]
Proof
By assumption, \( \lambda(x) \lt \infty \) for \( x \in S \), so there are no instantaneous states and so we can take \( \bs X \) to be right continuous. Next, \[ \sum_{n=0}^\infty \frac{1}{\lambda(Y_n)} = \sum_{n=0}^\infty \sum_{x \in S} \frac{1}{\lambda(x)} \bs{1}(Y_n = x) = \sum_{x \in S} \frac{1}{\lambda(x)} \sum_{n=0}^\infty \bs{1}(Y_n = x) = \sum_{x \in S} \frac{N_x}{\lambda(x)} \] where \( N_x = \sum_{n=0}^\infty \bs{1}(Y_n = x) \) is the number of times that the jump chain \( \bs Y \) is in state \( x \). Suppose that \( \sum_{x \in S_+} 1 / \lambda(x) = \infty \). Note that it must be the case that \( S_+ \), and hence \( S \), is infinite. With probability 1, either \( \bs Y \) enters an absorbing state (a state \( x \in S \) with \( \lambda(x) = 0 \)), or \( N_x = \infty \) for some \( x \in S_+ \), or \( N_x \ge 1 \) for infinitely many \( x \in S_+ \). In any case, \[ \sum_{n=0}^\infty \frac{1}{\lambda(Y_n)} = \sum_{x \in S} \frac{N_x}{\lambda(x)} = \infty\]
As a corollary, note that if \( S \) is finite then \( \lambda \) is bounded, so a continuous-time Markov chain on a finite state space is regular. So to review, if the exponential parameter function \( \lambda \) is finite, the chain \( \bs{X} \) has no instantaneous states. Even better, if \( \lambda \) is bounded or if the conditions in the last theorem are satisfied, then \( \bs{X} \) is regular. A continuous-time Markov chain with bounded exponential parameter function \( \lambda \) is called uniform , for reasons that will become clear in the next section on transition matrices. As we will see in later section, a uniform continuous-time Markov chain can be constructed from a discrete-time chain and an independent Poisson process. For the next result, recall that to say that \( \bs{X} \) has left limits with probability 1 means that the random function \( t \mapsto X_t \) has limits from the left on \( (0, \infty) \) with probability 1.
If \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is regular then \( \bs{X} \) has left limits with probability 1.
Proof
Suppose first that there are no absorbing states. Under the assumptions, with probability 1, \( 0 \lt \tau_n \lt \infty \) for each \( n \in \N \) and \( \tau_n \to \infty \) as \( n \to \infty \). Moreover, \( X_t = Y_n \) for \( t \in [\tau_n, \tau_{n+1}) \) and \( n \in \N \). So \( t \mapsto X_t \) has left limits on \( (0, \infty) \) with probability 1. The same basic argument works with absorbing states, except that possibly \( \tau_{n+1} = \infty \).
Thus, our standard assumption will be that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a regular Markov chain on \( S \). For such a chain, the behavior of \( \bs{X} \) is completely determined by the exponential parameter function \( \lambda \) that governs the holding times, and the transition probability matrix \( Q \) of the jump chain \( \bs{Y} \). Conversely, when modeling real stochastic systems, we often start with \( \lambda \) and \( Q \). It's then relatively straightforward to construct the continuous-time Markov chain that has these parameters. For simplicity, we will assume that there are no absorbing states. The inclusion of absorbing states is not difficult, but mucks up the otherwise elegant exposition.
Suppose that \( \lambda: S \to (0, \infty) \) is bounded and that \( Q \) is a probability matrix on \( S \) with the property that \( Q(x, x) = 0 \) for every \( x \in S \). The regular, continuous-time Markov chain \( \bs X = \{X_t: t \in [0, \infty)\} \) with exponential parameter function \( \lambda \) and jump transition matrix \( Q \) can be constructed as follows:
- First construct the jump chain \( \bs Y = (Y_0, Y_1, \ldots) \) having transition matrix \( Q \).
- Next, given \( \bs Y = (x_0, x_1, \ldots) \), the transition times \( (\tau_1, \tau_2, \ldots) \) are constructed so that the holding times \( (\tau_1, \tau_2 - \tau_1, \ldots) \) are independent and exponentially distributed with parameters \( (\lambda(x_0), \lambda(x_1), \ldots) \)
- Again given \( \bs Y = (x_0, x_1, \ldots) \), define \( X_t = x_0 \) for \( 0 \le t \lt \tau_1 \) and for \( n \in \N_+ \), define \( X_t = x_n \) for \( \tau_n \le t \lt \tau_{n+1}) \).
Additional details
Using product sets and product measures, it's straightforward to construct a probability space \( (\Omega, \mathscr{F}, \P) \) with the following objects and properties:
- \( \bs{Y} = (Y_0, Y_1, \ldots) \) is a Markov chain on \( S \) with transition matrix \( Q \).
- \( \bs{T} = \{T_x: x \in S\} \) is a collection of independent random variables with values in \( [0, \infty) \) such that \( T_x \) has the exponential distribution with parameter \( \lambda(x) \) for each \( x \in S \).
- \( \bs{Y} \) and \( \bs{T} \) are independent.
Define \( \bs{X} = \{X_t: t \in [0, \infty)\} \) as follows: First, \( \tau_1 = T_{Y_0} \) and \( X_t = Y_0 \) for \( 0 \le t \lt \tau_1 \). Recursively, if \( X_t \) is defined on \( [0, \tau_n) \), let \( \tau_{n+1} = \tau_n + T_{Y_n} \) and then let \( X_t = Y_n \) for for \( \tau_n \le t \lt \tau_{n+1} \). Since \( \lambda \) is bounded, \( \tau_n \to \infty \) as \( n \to \infty \), so \( X_t \) is well defined for \( t \in [0, \infty) \). By construction, \( t \mapsto X_t \) is right continuous and has left limits. The Markov property holds by the memoryless property of the exponential distribution and the fact that \( \bs Y \) is a Markov chain. Finally, by construction, \( \bs X \) has exponential parameter function \( \lambda \) and jump chain \( \bs{Y} \).
Often, particularly when \( S \) is finite, the essential structure of a standard, continuous-time Markov chain can be succinctly summarized with a graph.
Suppose again that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a regular Markov chain on \( S \), with exponential parameter function \( \lambda \) and embedded transition matrix \( Q \). The state graph of \( \bs{X} \) is the graph with vertex set \( S \) and directed edge set \( E = \{(x, y) \in S^2: Q(x, y) \gt 0\} \). The graph is labeled as follows:
- Each vertex \( x \in S \) is labeled with the exponential parameter \( \lambda(x) \).
- Each edge \( (x, y) \in E \) is labeled with the transition probability \( Q(x, y) \).
So except for the labels on the vertices, the state graph of \( \bs{X} \) is the same as the state graph of the discrete-time jump chain \( \bs{Y} \). That is, there is a directed edge from state \( x \) to state \( y \) if and only if the chain, when in \( x \), can move to \( y \) after the random holding time in \( x \). Note that the only loops in the state graph correspond to absorbing states, and for such a state there are no outward edges.
Let's return again to the construction above of a continuous-time Markov chain from the jump transition matrix \( Q \) and the exponential parameter function \( \lambda \). Again for simplicity, assume there are no absorbing states. We assume that \( Q(x, x) = 0 \) for all \( x \in S \), so that the state really does change at the transition times. However, if we drop this assumption, the construction still produces a continuous-time Markov chain, but with an altered jump transition matrix and exponential parameter function.
Suppose that \( Q \) is a transition matrix on \( S \times S \) with \( Q(x, x) \lt 1 \) for \( x \in S \), and that \( \lambda: S \to (0, \infty) \) is bounded. The stochastic process \( \bs X = \{X_t: t \in [0, \infty)\} \) constructed above from \( Q \) and \( \lambda \) is a regular, continuous-time Markov chain with exponential parameter function \( \tilde \lambda \) and jump transition matrix \( \tilde Q \) given by \begin{align*} & \tilde \lambda(x) = \lambda(x)[1 - Q(x, x)], \quad x \in S \\ & \tilde Q(x, y) = \frac{Q(x, y)}{1 - Q(x, x)}, \quad (x, y) \in S^2, \, x \ne y \end{align*}
Proof 1
As before, the fact that \( \bs X \) is a continuous-time Markov chain follows from the memoryless property of the exponential distribution and the Markov property of the jump chain \( \bs Y \). By construction, \( t \mapsto X_t \) is right continuous and has left limits. The main point, however, is that \( (\tau_1, \tau_2, \ldots) \) is not necessarily the sequence of transition times, when the state actually changes. So we just need to determine the parameters. Suppose \( X_0 = x \in S \) and let \( \tau = \tau_1 \) have the exponential distribution with parameter \( \lambda(x) \), as in the construction. Let \( T \) denote the time when the state actually does change. For \( t \in [0, \infty) \), the event \( T \gt t \) can happen in two ways: either \( \tau \gt t \) or \( \tau = s \) for some \( s \in [0, t] \), the chain jumps back into state \( x \) at time \( s \), and the process then stays in \( x \) for a period of at least \( t - s \). Thus let \( F_x(t) = \P(T \gt t \mid X_0 = x) \). Taking the two cases, conditioning on \( \tau \), and using the Markov property gives \[ F_x(t) = e^{-\lambda(x) t} + \int_0^t \lambda(x) e^{-\lambda(x) s} Q(x, x) F_x(t - s) ds \] Using the change of variables \( u = t - s \) and simplifying gives \[ F_x(t) = e^{-\lambda(x) t} \left[1 + \lambda(x) Q(x, x) \int_0^t e^{\lambda(x) u} F_x(u) du\right] \] Differentiating with respect to \( t \) then gives \[ F_x^\prime(t) = -\lambda(x) [1 - Q(x, x)] F_x(t) \] with the initial condition \( F_x(0) = 1 \). The solution of course is \( F_x(t) = \exp\{-\lambda(x)[1 - Q(x, x)]\} \) for \( t \in [0, \infty) \). When the state does change, the new state \( y \ne x \) is chosen with probability \[ \P(Y_1 = y \mid Y_0 = x, Y_1 \ne x) = \frac{Q(x, y)}{1 - Q(x, x)} \]
Proof 2
As in the first proof, we just need to determine the parameters. Given \( X_0 = Y_0 = x \), the discrete time \( N \) when \( \ Y\) first changes state has the geometric distribution on \( \N_+ \) with success parameter \( 1 - Q(x, x) \). Hence the time until \( \bs X \) actually changes state has the distribution of \( T = \sum_{i=1}^N U_i \) where \( \bs U = (U_1, U_2, \ldots) \) is a sequence of independent variables, each exponentially distributed with parameter \( \lambda(x) \) and with \( \bs U \) independent of \( N \). In the section on the exponential distribution, it is shown that \( T \) also has the exponential distribution, but with parameter \( \lambda(x)[1 - Q(x, x)] \). (The proof is simple using generating functions.) As in the first proof, when the state does change, the new state \( y \ne x \) is chosen with probability \[ \P(Y_1 = y \mid Y_0 = x, Y_1 \ne x) = \frac{Q(x, y)}{1 - Q(x, x)} \]
This construction will be important in our study of chains subordinate to the Poisson process.
Transition Times
The structure of a regular Markov chain on \( S \), as described above, can be explained purely in terms of a family of independent, exponentially distributed random variables. The main tools are some additional special properties of the exponential distribution, that we need to restate in the setting of our Markov chain. Our interest is in how the process evolves among the stable states until it enters an absorbing state (if it does). Once in an absorbing state, the chain stays there forever, so the behavior from that point on is trivial.
Suppose that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a regular Markov chain on \( S \), with exponential parameter function \( \lambda \) and transition probability matrix \( Q \). Define \( \mu(x, y) = \lambda(x) Q(x, y) \) for \( (x, y) \in S^2 \). Then
- \( \lambda(x) = \sum_{y \in S} \mu(x, y) \) for \( x \in S \).
- \( Q(x, y) = \mu(x, y) \big/ \lambda(x) \) if \( (x, y) \in S^2 \) and \( x \) is stable.
The main point is that the new parameters \( \mu(x, y) \) for \( (x, y) \in S^2 \) determine the exponential parameters \( \lambda(x) \) for \( x \in S \), and the transition probabilities \( Q(x, y) \) when \( x \in S \) is stable and \( y \in S \). Of course we know that if \( \lambda(x) = 0 \), so that \( x \) is absorbing, then \( Q(x, x) = 1 \). So in fact, the new parameters, as specified by the function \( \mu \), completely determine the old parmeters, as specified by the functions \( \lambda \) and \( Q \). But so what?
Consider the functions \( \mu \), \( \lambda \), and \( Q \) as given in the previous result. Suppose that \( T_{x,y} \) has the exponential distribution with parameter \( \mu(x, y) \) for each \( (x, y) \in S^2 \) and that \( \left\{T_{x,y}: (x, y) \in S^2\right\} \) is a set of independent random variables. Then
- \( T_x = \inf\left\{T_{x,y}: y \in S\right\} \) has the exponential distribution with parameter \( \lambda(x) \) for \( x \in S \).
- \(\P\left(T_x = T_{x, y}\right) = Q(x, y)\) for \( (x, y) \in S^2 \).
Proof
These are basic results proved in the section on the exponential distribution.
So here's how we can think of a regular, continuous-time Markov chain on \( S \): There is a timer associated with each \( (x, y) \in S^2 \), set to the random time \( T_{x,y} \). All of the timers function independently. When the chain enters state \( x \in S \), the timers on \( (x, y) \) for \( y \in S \) are started simultaneously. As soon as the first alarm goes off for a particular \( (x, y) \), the chain immediately moves to state \( y \), and the process repeats. Of course, if \( \mu(x, y) = 0 \) then \( T_{x, y} = \infty \) with probability 1, so only the timers with \( \lambda(x) \gt 0 \) and \( Q(x, y) \gt 0 \) matter (these correspond to the non-loop edges in the state graph). In particular, if \( x \) is absorbing, then the timers on \( (x, y) \) are set to infinity for each \( y \), and no alarm ever sounds.
The new collection of exponential parameters can be used to give an alternate version of the state graph. Again, the vertex set is \( S \) and the edge set is \( E = \{(x, y) \in S^2: Q(x, y) \gt 0\} \). But now each edge \( (x, y) \) is labeled with the exponential rate parameter \( \mu(x, y) \). The exponential rate parameters are closely related to the generator matrix, a matrix of fundamental importance that we will study in the next section.
Examples and Exercises
The Two-State Chain
The two-state chain is the simplest non-trivial, continuous-time Markov chain, but yet this chain illustrates many of the important properties of general continuous-time chains. So consider the Markov chain \( \bs{X} = \{X_t: t \in [0, \infty)\} \) on the set of states \( S = \{0, 1\} \), with transition rate \( a \in [0, \infty) \) from 0 to 1 and transition rate \( b \in [0, \infty) \) from 1 to 0.
The transition matrix \( Q \) for the embedded chain is given below. Draw the state graph in each case.
- \( Q = \left[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}\right] \) if \( a \gt 0 \) and \( b \gt 0 \), so that both states are stable.
- \( Q = \left[\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}\right] \) if \( a = 0 \) and \( b \gt 0 \), so that \( a \) is absorbing and \( b \) is stable.
- \( Q = \left[\begin{matrix} 0 & 1 \\ 0 & 1 \end{matrix}\right] \) if \( a \gt 0 \) and \( b = 0 \), so that \( a \) is stable and \( b \) is absorbing.
- \( Q = \left[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right] \) if \( a = 0 \) and \( b = 0 \), so that both states are absorbing.
We will return to the two-state chain in subsequent sections.
Computational Exercises
Consider the Markov chain \( \bs{X} = \{X_t: t \in [0, \infty)\} \) on \( S = \{0, 1, 2\} \) with exponential parameter function \( \lambda = (4, 1, 3) \) and embedded transition matrix \[ Q = \left[\begin{matrix} 0 & \frac{1}{2} & \frac{1}{2} \\ 1 & 0 & 0 \\ \frac{1}{3} & \frac{2}{3} & 0\end{matrix}\right] \]
- Draw the state graph and classify the states.
- Find the matrix of transition rates.
- Classify the jump chain in terms of recurrence and period.
- Find the invariant distribution of the jump chain.
Answer
- The edge set is \( E = \{(0, 1), (0, 2), (1, 0), (2, 0), (2, 1)\} \). All states are stable.
- The matrix of transition rates is \[ \left[\begin{matrix} 0 & 2 & 2 \\ 1 & 0 & 0 \\ 1 & 2 & 0 \end{matrix}\right] \]
- The jump chain is irreducible, positive recurrent, and aperiodic.
- The invariant distribution for the jump chain has PDF \[ f = \left[\begin{matrix} \frac{6}{14} & \frac{5}{14} & \frac{3}{14}\end{matrix}\right] \]
Special Models
Read the introduction to chains subordinate to the Poisson process.
Read the introduction to birth-death chains.
Read the introduction to continuous-time queuing chains.
Read the introduction to continuous-time branching chains.


1.1: Bodies
Bodies in Engineering Mechanics
A body , for the purposes of engineering mechanics, is a collection of matter that is analyzed as a single object. This can be something simple like a rubber ball, or it can be something made of many parts such as a car. What can count as a body and what cannot count as a body is dependent on the circumstances of the analysis. In some circumstances in engineering mechanics, it is useful to make certain assumptions about the bodies being analyzed. We will usually need to assume the body is either rigid or deformable, and we will also need to assume that the body is either a particle or an extended body.
Rigid versus Deformable Bodies
Rigid bodies do not deform (stretch, compress, or bend) when subjected to loads, while deformable bodies do deform. In actuality, no physical body is completely rigid, but most bodies deform so little that this deformation has a minimal impact on the analysis. For this reason, we usually assume in the statics and dynamics courses that the bodies discussed are rigid. In the strength of materials course we specifically remove this assumption and examine how bodies deform and eventually fail under loading.
There is no set boundary for determining if a body can be approximated as rigid, but there are two factors to look for that indicate that a rigid body assumption is not appropriate. First, if the body is being significantly stretched, compressed, or bent during the period of analysis, then the body should not be analyzed as a rigid body. Second, if the body has parts that are free to move relative to one another, then the body as a whole should not be analyzed as a rigid body; this is instead a machine, comprised of multiple connected bodies that will each need to be analyzed separately.
Particles versus Extended Bodies
Particles are bodies where all the mass is concentrated at a single point in space. Particle analysis will only have to take into account translational motion and the forces acting on the body, because rotation is not considered for particles. Extended bodies, on the other hand, have mass that is distributed throughout a finite volume. Often in engineering statics, we will take a shortcut and say rigid bodies to describe extended bodies that also happen to be rigid. This is because particles, as a single point, cannot deform. Extended body analysis is more complex and also has to take into account moments and rotational motions. In actuality, no bodies are truly particles, but some bodies can be approximated as particles to simplify analysis. Bodies are often assumed to be particles if the rotational motions are negligible when compared to the translational motions, or in systems where there is no moment exerted on the body, such as a concurrent force system.


6.10: Outcome- Customer Relationship Management (CRM) Systems
What you’ll learn to do: explain how customer relationship management (CRM) systems can help organizations manage and gain customer insights from marketing information
To round out our discussion of marketing information and research, we need to add one more important tool to the mix: customer relationship management (CRM) systems. These increasingly prevalent systems are the centerpiece in how many organizations make sense of and manage marketing data about current and prospective customers. A basic understanding of CRM systems can help you recognize their potential for helping organizations use marketing information more effectively.
The specific things you’ll learn in this section include:
- Define CRM systems and explain their purpose
- Describe the types of marketing information CRM systems can capture and why it is valuable for generating customer insights
Learning Activities
The learning activities for this section include the following:
- Reading: Customer Relationship Management Systems
- Self Check: Customer Relationship Management (CRM) Systems
Contributors and Attributions
- Outcome: Customer Relationship Management (CRM) Systems. Provided by : Lumen Learning. License : CC BY: Attribution


2.E: Chapter Two (Exercises)
Is each inference an illustration of vagueness or ambiguity?
A.  It’s tricky.
B. I don’t think I lied, I said “I’ll be there to help you if you need it” and you didn’t really need help this time since you clearly survived without my help.
C. Teacher, you said to do the test “without help” and Jesus is using a pencil to complete the test. That’s helping him do it!
D. You said not to tattle, so when I saw the murderers running away from their victim I remained tight-lipped as ever.
E. Uggh. I said I wanted regular blonde hair, not platinum blonde hair! You messed my hair up.
F. The physics teacher said that on a level plane it’s indeterminate which way a stationary object will start to roll, but I’ve been setting this ball down carefully on the floor of this airplane and it has been rolling backwards every time!
G. Of course humans can fly. I just flew to Los Angeles last week!
H. She said “I’ll meet you inside,” so I went deep inside my soul in search of her, but never found her.
I. "As I was leaving this morning, I said to myself, 'The last thing you must do is forget your speech.' And, sure enough, as I left the house this morning, the last thing I did was to forget my speech."– Rowan Atkinson
J. You said we’d meet early today, but I’ve been waiting here since 5 am and you just now showed up at 7 am!
K. “I didn’t go to the party because Shamik was there.”
“Oh, so you went to sample the appetizers?”
L. I’m not a very good student, I only have a 3.87 GPA.
M. You’re so rich! You have a working refrigerator.
N. It’s so cold outside today, it’s like 50 degrees.
O. “Call me a taxi, please.”
“Okay, you’re a taxi!”
P. Union demands increased unemployment. -newspaper headline
Q.  Well I do listen to her, I just don’t do what she tells me to do!
Is each of the following series, determine whether the extension is increasing or decreasing and then determine whether it has more intensional information (it’s getting more specific) or decreasing intensional information (it’s getting less specific). “No change” is also an option for both dimensions (intension and extension).
A. Indian food, North Indian food, Masala dishes, Paneer Masala.
B. My rude neighbor guy, rude guys, rude people, people.
C. The Moon, moons, moons in our solar system, natural satellites (any natural object orbiting a planet) in our solar system, natural satellites.
D. Jimmy Carr (an English comedian), English comedians, British comedians, comedians.
E. Earth, planets in our solar system with intelligent life, the planet on which Captain America: Civil War takes place, the planet that is home to almost all living human beings.
F. Pink Floyd, rock groups from the 70’s and 80’s, English-speaking rock groups, rock groups
For each definition, determine the answer to each of the following questions: a) Is it Stipulative or Descriptive? b) Is it too broad or too narrow, or is it apt? c) Is it argumentative or Neutral?
A.
B. An immigrant is someone like Natalie Portman, Gal Gadot, John Oliver, or James McAvoy.
C. Furniture is a human-made object used to store or hold things; or sit or lay on.
D. A ruler is a straight and rigid device used to measure lengths.
E. A criminal is someone who has committed a crime.
F. A Russian is someone who is not to be trusted.
G.  A wristwatch is a time piece made to be worn around one’s wrist.
I. A smart device is any device with the ability to connect to the internet.
Explain how this is equivocation...
A. You said you would meet me at the park, but we’ve already met—years ago in fact—so I don’t think we should plan to meet in the future.
B. That bumper sticker says “the cost of freedom is great,” which is weird because I never think of war as all that great. It’s pretty awful.
C. Peter: Let’s go to the gym to do a bit of working out.
Tamil: Okay, but you won’t need your running shoes at the climbing gym, you should grab your climbing shoes instead and just wear street shoes on the way there.
Fill in each blank with either Necessary and Sufficient, Necessary but not Sufficient, Sufficient but not Necessary, or Neither Necessary nor Sufficient.
A. Giving birth a baby is a _______ condition to becoming a mother.
B. Eating spicy foods is, for many people, a _______ condition for having gastro-intestinal problems.
C. Drinking milk is, for lactose-intolerant people, a ________ condition for having gastro-intestinal problems.
D. Flying in a plane is a ________ condition for getting from California to Hawaii in under 24 hours.
E.
F. Being alive is a ________ condition for running for President.
G. Eating at least every three weeks is a ___________ condition for staying alive (for humans).
H. Having an ear of corn in your possession is a _______________ condition for making corn bread.
I. Selling drugs in front of a police officer is, in normal circumstances, a ___________ condition for being arrested
J. Being a set of bound pages containing a more or less unified piece of writing or collection of writings is a _____________ condition for being a book.
K. Putting fruit, yogurt, and milk in a blender and blending until smooth is a ________ condition for making a smoothie.