Shell Polynomials and Dual Birth-Death Processes

Shell Polynomials and Dual Birth-Death Processes\star\star$\star$This paper is
a contribution to the Special Issue on Orthogonal Polynomials, Special
Functions and Applications. The full collection is available at
http://www.emis.de/journals/SIGMA/OPSFA2015.html

Erik A. VAN DOORN

E.A. van Doorn

Department of Applied Mathematics, University of Twente,
P.O. Box 217, 7500 AE Enschede, The Netherlands

<EMAIL_ADDRESS>http://wwwhome.math.utwente.nl/~doornea/

Received January 02, 2016, in final form May 14, 2016; Published online May
18, 2016

This paper aims to clarify certain aspects of the relations between birth-
death processes, measures solving a Stieltjes moment problem, and sets of
parameters defining polynomial sequences that are orthogonal with respect to
such a measure. Besides giving an overview of the basic features of these
relations, revealed to a large extent by Karlin and McGregor, we investigate a
duality concept for birth-death processes introduced by Karlin and McGregor
and its interpretation in the context of shell polynomials and the
corresponding orthogonal polynomials. This interpretation leads to increased
insight in duality, while it suggests a modification of the concept of
similarity for birth-death processes.

orthogonal polynomials; birth-death processes; Stieltjes moment problem; shell
polynomials; dual birth-death processes; similar birth-death processes

42C05; 60J80; 44A60

##  1 Introduction

In what follows a measure will always be a finite positive Borel measure on
the real axis with infinite support and finite moments of all (positive)
orders. It will be convenient to assume throughout that the measure is
normalized so that it becomes a probability measure. The Hamburger moment
problem associated with a measure $\psi$ is said to be determined ($\psi$ is
det(H), for short) if $\psi$ is uniquely determined by its moments; otherwise,
it is said to be indeterminate ($\psi$ is indet(H)). Similar terminology will
be used for the Stieltjes moment problem associated with $\psi$, in which we
limit our scope to measures with support on the nonnegative real axis, with
det(S) (indet(S)) replacing det(H) (indet(H)).

Chihara [9] showed that when a measure is indet(H) and has left-bounded
support, there is a unique solution of the associated moment problem with the
property that the minimum of its support is maximal. We will refer to this
solution as the natural solution. It will be convenient to qualify a measure
as natural also if it is the solution of a determined moment problem. Note
that the natural solution of an indeterminate Hamburger moment problem may be
the unique solution of a Stieltjes moment problem.

Our point of departure is a measure $\psi$ on the nonnegative real axis with
moments

$\displaystyle m_{n}(\psi):=\int_{[0,\infty)}x^{n}\psi(dx),\qquad n\geq 0.$

By assumption $m_{0}(\psi)=1$. In what follows we allow $\psi$ to be indet(S)
(and hence indet(H)) but assume in this case that $\psi$ is the natural
solution of the associated moment problem. The (monic) polynomials that are
orthogonal with respect to $\psi$ will be denoted by $P_{n}$.

As is well known there exist unique constants $c_{n}\in\mathbb{R}$ and
$d_{n+1}>0$, $n\geq 1$, such that the polynomials $P_{n}$ satisfy the three-
terms recurrence relation

$\displaystyle P_{n}(x)=(x-c_{n})P_{n-1}(x)-d_{n}P_{n-2}(x),\qquad n>1,$
$\displaystyle P_{1}(x)=x-c_{1},\qquad P_{0}(x)=1.$ (1.1)

Since the support of $\psi$ is a subset of the nonnegative real axis we
actually have $c_{n}>0$, and there is a more refined result (see, for
instance, Chihara [10, Corollary to Theorem I.9.1]) to the effect that there
are numbers $\mu_{0}\geq 0$ and $\lambda_{n}>0$, $\mu_{n+1}>0$ for $n\geq 0$,
such that

$\displaystyle c_{n}=\lambda_{n-1}+\mu_{n-1}\qquad\mbox{and}\qquad
d_{n+1}=\lambda_{n-1}\mu_{n},\qquad n\geq 1.$ (1.2)

(Further results in this vein can be found in [14].) Since $\lambda_{n}$ and
$\mu_{n}$ may be interpreted as the birth rates and death rates, respectively,
of a birth-death process on the nonnegative integers, we will refer to a
collection of such constants as a set of birth and death rates (or a rate set,
for short). More information on birth-death processes and their rates will be
given in later sections, but at this stage we note that, by Karlin and
McGregor [16, Lemma 1 and Lemma 6 on p. 527] (see also [14, Theorem 1.3]), we
must have $\mu_{0}=0$ unless $\psi$ has a finite moment of order $-1$, that
is,

$\displaystyle m_{-1}(\psi):=\int_{[0,\infty)}x^{-1}\psi(dx)<\infty,$ (1.3)

in which case $\mu_{0}$ may be any number in the interval
$[0,1/m_{-1}(\psi)]$. Evidently, once $\mu_{0}$ has been chosen, the other
rates are fixed.

In the remainder of this section we will assume that (1.3) is satisfied, so
that, in particular, $\psi(\\{0\\})=0$. Defining the measure $\phi^{(0)}$ by

$\displaystyle\phi^{(0)}([0,x]):=\frac{1}{m_{-1}(\psi)}\int_{[0,x]}y^{-1}\psi(dy),\qquad
x\geq 0,$ (1.4)

and letting, for $a>0$,

$\displaystyle\phi^{(a)}:=\frac{1}{a+1}\big{(}a\delta_{0}+\phi^{(0)}\big{)},$
(1.5)

where $\delta_{0}$ is the Dirac measure with mass $1$ at $0$, we observe that,
for any $a\geq 0$, $\phi^{(a)}$ is a probability measure on the nonnegative
real axis. As a consequence there exists a sequence of (monic) polynomials
$\big{\\{}S^{(a)}_{n}\big{\\}}$ that are orthogonal with respect to
$\phi^{(a)}$. Since, for all $a\geq 0$,

$\displaystyle\psi([0,x])=(a+1)m_{-1}(\psi)\int_{[0,x]}y\phi^{(a)}(dy),\qquad
x\geq 0,$

we will, following Chihara [13], refer to the polynomials $S^{(a)}_{n}$ as
shell polynomials corresponding to the orthogonal polynomial sequence
$\\{P_{n}\\}$. In the terminology of [10, Section I.7] the polynomials $P_{n}$
are, for any $a\geq 0$, the kernel polynomials with $K$-parameter $0$
corresponding to $\big{\\{}S^{(a)}_{n}\big{\\}}$.

Information on the status of the moment problem associated with $\phi^{(a)}$
is given in the next theorem.

######  Theorem 1.1 (Chihara [8, Theorem 2]).

  1. $(i)$

The measure $\phi^{(0)}$ is det(H).

  2. $(ii)$

The measure $\phi^{(a)}$, $a>0$, is det(S) if and only if the measure $\psi$
is det(S).

Evidently, $\phi^{(a)}$ cannot be a natural measure if it is indet(S), but we
will see that $\phi^{(a)}$ nevertheless has a certain significance in this
case, in particular in the context of birth-death processes. Before
introducing our findings it will be useful to state already some facts about
the relations between birth-death processes, rate sets, and measures that will
be expounded in Section 3.

######  Fact 1.2.

A birth-death process uniquely defines a rate set.

######  Fact 1.3.

A birth-death process uniquely defines a measure on $[0,\infty)$ through
Karlin and McGregor’s representation formula (3.3) for the transition
functions of a birth-death process. This measure is natural or of a type known
as Nevanlinna extremal.

######  Fact 1.4.

A rate set $\\{\lambda_{n},\mu_{n}\\}$ uniquely defines, through (1.2), a
natural measure on $[0,\infty)$ with respect to which the polynomials $P_{n}$
of (1.1) are orthogonal. Conversely, a natural measure on $[0,\infty)$, with
monic orthogonal polynomials $P_{n}$ and moment $m_{-1}$ of order $-1$,
defines, through (1.1) and (1.2), a rate set $\\{\lambda_{n},\mu_{n}\\}$,
which, if $m_{-1}=\infty$, is unique and satisfies $\mu_{0}=0$. If
$m_{-1}<\infty$ the measure defines an infinite family of rate sets indexed by
the value of $\mu_{0}$, which can be any number in the interval
$[0,1/m_{-1}]$.

######  Fact 1.5.

A rate set $\\{\lambda_{n},\mu_{n}\\}$ uniquely defines a birth-death process
if and only if at least one of the following conditions prevails:

  1. (i)

the natural measure defined by the rate set is det(S);

  2. (ii)

$\mu_{0}>0$ and the natural measure defined by the rate set has
$m_{-1}=\mu_{0}^{-1}$.

Otherwise, there is an infinite, one-parameter family of birth-death processes
with the given rates. Two members of this family may be identified as extreme,
and are known as the minimal process (associated with the natural measure) and
the maximal process.

Fact 1.2 is a trivial consequence of the definition of a birth-death process
in Section 3.1; Facts 1.3 and 1.4 follow from the seminal work of Karlin and
McGregor [16] (Fact 1.4 also summarizes earlier observations in this section);
for Fact 1.5 we refer to [16] again and, for the second part, to [22]. Note
that, by Fact 1.4, our measure $\psi$ defines infinitely many rate sets, since
$m_{-1}(\psi)<\infty$ by assumption.

Our main goal in this paper is to interpret and characterize the measures
$\phi^{(a)}$, $a\geq 0$, and the relation between the measures $\psi$ and
$\phi^{(a)}$, in the context of birth-death processes. Concretely, we will
display a one-to-one correspondence between the measures $\phi^{(a)}$, $a\geq
0$, and the rate sets with $\mu_{0}>0$ defined by $\psi$ (to which we will
refer as $\psi$-rate sets for short). Moreover, if $\phi^{(a)}$ is det(S),
then the birth-death process (uniquely) defined by $\phi^{(a)}$ and the
(unique) birth-death process whose rate set is the $\psi$-rate set
corresponding to $\phi^{(a)}$ are shown to be dual to each other in sense of
Karlin and McGregor [17, Section 6]. As a result the duality concept for
birth-death processes can be extended to families of birth-death processes
that are similar in the sense of [18], after a slight modification of the
definition of similarity.

When $\phi^{(a)}$ is indet(S) – and hence, by Theorem 1.1, $a>0$ – the
situation is more complicated since the duality concept for birth-death
processes can be applied only to minimal and maximal processes when a rate set
does not define a birth-death process uniquely (see [22]). However, we will
see that in this case $\phi^{(a)}$ is Nevanlinna extremal (as stated already
by Berg and Christiansen [3]) and corresponds to a maximal birth-death
process, which happens to be dual to the minimal process whose rate set is the
$\psi$-rate set corresponding to $\phi^{(a)}$. We will also present a
counterpart of this result.

In the next section we will collect some further notation, terminology and
preliminary results about shell polynomials, rate sets, and measures, while in
Section 3 the relevant properties of birth-death processes are set forth and
put in proper perspective. Our findings are detailed in Section 4.

##  2 Preliminaries

###  2.1 Shell polynomials and rate sets

Applying [10, Corollary to Theorem I.9.1] to the polynomials $S^{(a)}_{n}$, we
conclude that, for any $a\geq 0$, there exist constants $\mu^{(a)}_{0}\geq 0$
and $\lambda^{(a)}_{n}>0$, $\mu^{(a)}_{n+1}>0$ for $n\geq 0$, such that

$\displaystyle
S^{(a)}_{n}(x)=\big{(}x-\lambda^{(a)}_{n-1}-\mu^{(a)}_{n-1}\big{)}S^{(a)}_{n-1}(x)-\lambda^{(a)}_{n-2}\mu^{(a)}_{n-1}S^{(a)}_{n-2}(x),\qquad
n>1,$ $\displaystyle S^{(a)}_{1}(x)=x-\lambda^{(a)}_{0}-\mu^{(a)}_{0},\qquad
S^{(a)}_{0}(x)=1.$ (2.1)

If $\phi^{(a)}$ is natural and $m_{-1}(\phi^{(a)})=\infty$ (so in particular
if $\phi^{(a)}$ is det(S) and $a>0$) then, by [16, Lemma 1 and Lemma 6 on p.
527)] again, we must have $\mu_{0}^{(a)}=0$. In other circumstances
$\mu^{(a)}_{0}$ may also be chosen positive, but it will be convenient to set
$\mu_{0}^{(a)}=0$ by definition in what follows.

We can now relate the parameters in the recurrence relation (2.1) for the
polynomials $S_{n}^{(a)}$ to the parameters $c_{n}$ and $d_{n}$ in the
recurrence relation (1.1). Indeed, since the polynomials $P_{n}$ are the
kernel polynomials with $K$-parameter $0$ corresponding to
$\big{\\{}S_{n}^{(a)}\big{\\}}$, we have, by [10, Theorem I.9.1],

$\displaystyle c_{n}=\lambda_{n-1}^{(a)}+\mu_{n}^{(a)}\qquad\mbox{and}\qquad
d_{n+1}=\lambda_{n}^{(a)}\mu_{n}^{(a)},\qquad n\geq 1.$

Subsequently defining birth rates $\lambda_{n}$ and death rates $\mu_{n}$ by

$\displaystyle\lambda_{n}=\mu_{n+1}^{(a)}\qquad\mbox{and}\qquad\mu_{n}=\lambda_{n}^{(a)},\qquad
n\geq 0,$ (2.2)

it follows that we have regained (1.2). So we see that we can parametrize the
birth and death rates in the representation (1.2) by the value of $\mu_{0}$,
but also by the size $a$ of the atom at $0$ of the measure $\phi^{(a)}$ of
(1.4) and (1.5), since the value of $a$ uniquely identifies the shell
polynomials $S_{n}^{(a)}$ corresponding to $\\{P_{n}\\}$, and hence, through
(2.1) (where $\mu_{0}^{(a)}=0$) and (2.2), the birth and death rates.

The next theorem gives an explicit one-to-one relation between $\mu_{0}$ and
$a$, and shows that the question of whether the alternative representation
yields all possibilities can be answered in the affirmative, provided we allow
$0\leq a\leq\infty$ and interpret $\lambda_{n}^{(\infty)}$ and
$\mu_{n}^{(\infty)}$ as limits as $a\to\infty$ of the corresponding quantities
with superindex (a). We will see in Section 2.2 that the theorem is an
immediate corollary of a theorem of Chihara [8].

######  Theorem 2.1.

Let $\psi$ be a natural measure satisfying (1.3) and let $\mu_{0}$ be
determined by $a$ via (1.4), (1.5), (2.1) $($with $\mu_{0}^{(a)}=0)$ and
(2.2). Then, for $0\leq a\leq\infty$,

$\displaystyle\mu_{0}=\frac{1}{(a+1)m_{-1}(\psi)},$

whence $\mu_{0}$ can have any value in the interval $[0,1/m_{-1}(\psi)]$.

Note that, as $a\to\infty$, $\phi^{(a)}$ converges strongly to $\delta_{0}$,
so we cannot (and need not) extend the definition of $S_{n}^{(a)}$ to include
the case $a=\infty$. An interpretation of $\lambda_{n}^{(\infty)}$ and
$\mu_{n}^{(\infty)}$ as birth and death rates of a birth-death process on the
nonnegative integers is possible, but does not fit in the setting described
around (1.2) since $\lambda_{0}^{(\infty)}=\mu_{0}^{(\infty)}=0$.

Let us mention at this point that (2.2) displays the duality concept for
birth-death processes that will be further discussed in Section 3.2 and plays
a crucial role in Section 4.

###  2.2 Chain sequences and rate sets

We first recall some definitions and basic results (see Chihara [10, Section
III.5] and [12] for more information). A sequence $\\{a_{n}\\}_{n=1}^{\infty}$
is a chain sequence if there exists a second sequence
$\\{g_{n}\\}_{n=0}^{\infty}$ such that

$\displaystyle(i)\quad$ $\displaystyle 0\leq g_{0}<1,\qquad 0<g_{n}<1,\qquad
n\geq 1,$ $\displaystyle(ii)\quad$ $\displaystyle
a_{n}=(1-g_{n-1})g_{n},\qquad n\geq 1.$

The sequence $\\{g_{n}\\}$ is called a parameter sequence for $\\{a_{n}\\}$.
If both $\\{g_{n}\\}$ and $\\{h_{n}\\}$ are parameter sequences for
$\\{a_{n}\\}$, then

$\displaystyle g_{n}<h_{n},\qquad n\geq 0\quad\Longleftrightarrow\quad
g_{0}<h_{0}.$

Every chain sequence $\\{a_{n}\\}$ has a minimal parameter sequence, uniquely
determined by the condition $g_{0}=0$, and a maximal parameter sequence
$\\{M_{n}\\}$, characterized by the fact that $M_{0}>g_{0}$ for any other
parameter sequence $\\{g_{n}\\}$. For every $x$, $0\leq x\leq M_{0},$ there is
a unique parameter sequence $\\{g_{n}\\}$ for $\\{a_{n}\\}$ such that
$g_{0}=x.$

Linking the parameters in the three-terms recurrence relation (1.1) to birth
and death rates is an alternative for the approach involving chain sequences
chosen by Chihara in, for instance, [8, 10]. Indeed, letting

$\displaystyle a_{n}=\frac{d_{n+1}}{c_{n}c_{n+1}},\qquad n\geq 1,$

we see that the sequence $\\{a_{n}\\}_{n=1}^{\infty}$ is a chain sequence,
since $a_{n}=(1-g_{n-1})g_{n}$ if we choose

$\displaystyle g_{n}=\frac{\mu_{n}}{\lambda_{n}+\mu_{n}},\qquad n\geq 0,$
(2.3)

for any set of birth rates $\lambda_{n}$ and death rates $\mu_{n}$ satisfying
(1.2). So (2.3) gives a one-to-one correspondence between a parameter sequence
for the chain sequence $\\{a_{n}\\}$ and a rate set satisfying (1.2). Since
$0\leq\mu_{0}\leq 1/m_{-1}(\psi)$, we can also characterize the maximal
parameter sequence for $\\{a_{n}\\}$ by

$\displaystyle M_{0}=\frac{1}{c_{1}m_{-1}(\psi)}.$

Invoking [8, Theorem 2] we can now conclude that it implies Theorem 2.1, for
on comparing our (1.5) with [8, equation (3.3)] and making appropriate
identifications, we find that $a=(\mu_{0}m_{-1}(\psi))^{-1}-1$, as required.

###  2.3 Spectral properties and Nevanlinna extremal measures

In this subsection we will introduce some notation and terminology concerning
the natural measure $\psi$ introduced in Section 1 and, if $\psi$ is indet(S),
related measures called Nevanlinna extremal.

Of interest to us will be the quantities $\xi_{i},$ recurrently defined by

$\displaystyle\xi_{1}:=\inf\operatorname{supp}(\psi),$ (2.4)

and

$\displaystyle\xi_{i+1}:=\inf\\{\operatorname{supp}(\psi)\cap(\xi_{i},\infty)\\},\qquad
i\geq 1.$ (2.5)

where supp($\psi$) denotes the support (or spectrum) of the measure $\psi$. We
further define

$\displaystyle\sigma:=\lim_{i\to\infty}\xi_{i},$ (2.6)

the first accumulation point of $\operatorname{supp}(\psi)$ if it exists, and
infinity otherwise. So $(\operatorname{supp}\psi)$ is discrete with no finite
limit point if and only if $\sigma=\infty$. It is clear from the definition of
$\xi_{i}$ that, for all $i\geq 1$,

$\displaystyle\xi_{i+1}\geq\xi_{i}\geq 0,$

and

$\displaystyle\xi_{i}=\xi_{i+1}\quad\Longleftrightarrow\quad\xi_{i}=\sigma.$

Note that we must have $\sigma=0$ if $\xi_{1}=0$ and $\psi(\\{0\\})=0$. Also,
$\psi$ must be det(S) if $\xi_{1}=0$.

From Karlin and McGregor [16] (see also Chihara [11]) we know that

$\displaystyle\psi\quad\mbox{is~{}indet(S)}\quad\Longleftrightarrow\quad\sum_{n=0}^{\infty}\left(\pi_{n}+\frac{1}{\lambda_{n}\pi_{n}}\right)<\infty,$
(2.7)

where

$\displaystyle\pi_{0}:=1\qquad\mbox{and}\qquad\pi_{n}:=\frac{\lambda_{0}\lambda_{1}\cdots\lambda_{n-1}}{\mu_{1}\mu_{2}\cdots\mu_{n}},\qquad
n\geq 1,$ (2.8)

and $\\{\lambda_{n},\mu_{n}\\}$ is the rate set with $\mu_{0}=0$ satisfying
(1.2). We note, parenthetically, that for a rate set with $\mu_{0}>0$ the
right-hand side of (2.7) is sufficient, but not necessary for the
corresponding natural measure to be indet(S) (see [11]).

It is well known that $\sigma=\infty$ if $\psi$ is indet(S). (A necessary and
sufficient condition for $\sigma=\infty$ in terms of any rate set satisfying
(1.2) has recently been revealed in [24].) Moreover, if $\psi$ is indet(S)
there are infinitely many solutions of the Stieltjes moment problem associated
with $\psi$. We shall be interested in particular in solutions known as
Nevanlinna extremal (or $N$-extremal, for short), which may be defined as
follows (see, for example, Berg and Valent [7, Section 1] and, for more
background information, Shohat and Tamarkin [21, pp. 51–60] and Berg and
Valent [6, Section 2]). Let

$\displaystyle\rho(x):=\left\\{\sum_{n=0}^{\infty}p_{n}^{2}(x)\right\\}^{-1},\qquad
x\in\mathbb{R},$

where $p_{n}(x)$ are the orthonormal polynomials corresponding to $\psi$. Then
$\rho(x)$ is positive for all real $x$ and equals, if $x\geq 0$, the maximal
mass any solution can concentrate at $x$. Supposing that a solution of the
Stieltjes moment problem locates positive mass at the point $x$, then that
solution is an $N$-extremal solution if and only if the point mass at $x$
equals $\rho(x)$.

Some pertinent properties of $N$-extremal solutions are the following. There
is a one-to-one correspondence between the real numbers in the interval
$[0,\xi_{1}]$ and the $N$-extremal solutions of the Stieltjes moment problem
associated with $\psi$. For $\xi\in[0,\xi_{1}]$ we denote the corresponding
$N$-extremal solution by $\psi_{\xi}$. The spectrum of $\psi_{\xi}$ is
discrete and consists of the point $\xi$ and exactly one point in each of the
intervals $(\xi_{i},\xi_{i+1}]$, $i\geq 1$. Evidently, we have
$\psi_{\xi_{1}}=\psi$ and supp($\psi_{\xi_{1}}$)
$=\\{\xi_{1},\xi_{2},\dots\\}$. The spectral points of two different
$N$-extremal solutions strictly separate each other.

We finally note that the natural measure $\psi$ can be identified with the
solution of the associated Stieltjes moment problem that is related to the
Friedrichs extension of some semi-bounded operator (see Pedersen [19] for
details). The parametrization in [19] (and in [6, 7]) of the $N$-extremal
solutions of the indeterminate Stieltjes moment problem associated with $\psi$
differs from ours and is effectuated by a number in an interval $[\alpha,0]$;
the solution corresponding to the parameter value $\alpha$ ($<0$) is our
natural solution $\psi$ ($=\psi_{\xi_{1}}$).

##  3 Birth-death processes

###  3.1 Basic properties

In this paper a birth-death process $\mathcal{X}\equiv\\{X(t),\,t\geq 0\\}$,
say, will always be a continuous-time Markov chain taking values in
$\mathcal{N}:=\\{0,1,\ldots\\}$ with the property that only transitions to
neighbouring states are permitted. The process has upward transition (or
birth) rates $\lambda_{n}$, $n\in\mathcal{N}$, and downward transition (or
death) rates, $\mu_{n}$, $n\in\mathcal{N}$, all strictly positive except
$\mu_{0}$, which might be equal to 0. When $\mu_{0}=0$ the process is
irreducible, but when $\mu_{0}>0$ the process may escape from $\mathcal{N}$,
via $0$, to an absorbing state $-1$. The $q$-matrix of transition rates of
$\mathcal{X}$, restricted to the states in $\mathcal{N}$, will be denoted by
$Q$, that is,

$\displaystyle
Q=\begin{pmatrix}-(\lambda_{0}+\mu_{0})&\lambda_{0}&0&0&0&\ldots\cr\mu_{1}&-(\lambda_{1}+\mu_{1})&\lambda_{1}&0&0&\ldots\cr
0&\mu_{2}&-(\lambda_{2}+\mu_{2})&\lambda_{2}&0&\ldots\cr\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\cr\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\cr\end{pmatrix},$
(3.1)

and $\mathcal{X}$ will be referred to as a $Q$-process. The process
$\mathcal{X}$ will be identified with its transition functions

$\displaystyle p_{ij}(t):=\Pr\\{X(t)=j\,|\,X(0)=i\\},\qquad
i,j\in\mathcal{N},\qquad t\geq 0,$

and we write $P(\cdot):=(p_{ij}(\cdot)$, $i,j\in\mathcal{N})$. Besides the
usual probabilistic requirements and the Chapman–Kolmogorov equations

$\displaystyle P(s+t)=P(s)P(t),\qquad s\geq 0,\qquad t\geq 0,$

imposed by the Markov property, the transition functions of $\mathcal{X}$ will
be assumed to satisfy both the Kolmogorov backward equations

$\displaystyle P^{\prime}(t)=QP(t),\qquad t\geq 0,$

and forward equations

$\displaystyle P^{\prime}(t)=P(t)Q,\qquad t\geq 0,$

with initial condition $P(0)=I$, the identity matrix. It follows in particular
that $P^{\prime}(0)=Q$, establishing Fact 1.2. We refer to Anderson [1] for
more information on continuous-time Markov chains in general and birth-death
processes in particular.

A matrix of the type (3.1) is always the $q$-matrix of a birth-death process,
but not necessarily of a unique process. Karlin and McGregor [16] have shown
that the $Q$-process $\mathcal{X}$ is uniquely determined by $Q$ – that is, by
its rates – if and only if the series

$\displaystyle\sum_{n=0}^{\infty}\left(\pi_{n}+\frac{1}{\lambda_{n}\pi_{n}}\right),$
(3.2)

where $\pi_{n}$ is given by (2.8), diverges. If $\mu_{0}=0$ then, in view of
(2.7) (where $\mu_{0}=0$ is assumed), the series diverges if and only if
$\psi$ is det(S), where $\psi$ denotes the (natural) measure defined by the
rate set $\\{\lambda_{n},\mu_{n}\\}$. If $\mu_{0}>0$ then, by [16, Theorem
15], the series (3.2) diverges if and only if $\psi$ is det(S) or
$m_{-1}(\psi)=1/\mu_{0}$.

If the series (3.2) converges there is an infinite, one-parameter family of
$Q$-processes, which includes two members – the minimal and the maximal
$Q$-process – with matrices of transition functions $P^{\min}(\cdot)$ and
$P^{\max}(\cdot)$ that are uniquely defined by the requirement that any
$Q$-process with matrix of transition functions $P(\cdot)$ satisfies

$\displaystyle P^{\min}(t)\leq P(t)\leq P^{\max}(t),\qquad
i,j\in\mathcal{N},\qquad t\geq 0,$

where $\leq$ denotes componentwise inequality. After introducing duality for
birth-death processes in the next subsection we will able to identify the
parameter characterizing the individual $Q$-processes.

Given the birth rates $\lambda_{n}$ and death rates $\mu_{n}$ of $\mathcal{X}$
we can define positive numbers $c_{n}$ and $d_{n}$ by (1.2) and, subsequently,
polynomials $P_{n}$ by the recurrence relation (1.1). By Favard’s theorem the
polynomials $P_{n}$ are orthogonal with respect to a finite positive Borel
measure on the real axis (with finite moments of all positive orders), and it
is shown in [16] and [8] that, in fact, there is such a measure with support
on the nonnegative real axis. As before we will assume that the measure is
normalized to be a probability measure. So we conclude that a set of birth and
death rates uniquely defines a natural measure on the nonnegative real axis,
thus confirming the first part of Fact 1.4.

Actually, the natural measure that is defined by the rates $\lambda_{n}$ and
$\mu_{n}$ – and hence by the matrix $Q$ – is precisely the measure $\psi$
appearing in Karlin and McGregor’s [16] spectral representation for the
transition functions of the unique (if the series (3.2) diverges) or minimal
(if the series (3.2) converges) $Q$-process, namely,

$\displaystyle
p_{ij}(t)=(-1)^{i+j}\prod_{k=1}^{j}\frac{1}{\lambda_{k-1}\mu_{k}}\int_{0}^{\infty}e^{-xt}P_{i}(x)P_{j}(x)\psi(dx),\qquad
i,j\in\mathcal{N},\qquad t\geq 0,$ (3.3)

where an empty product is defined to be $1$. If the series (3.2) converges the
representation (3.3) still holds for any $Q$-process, provided $\psi$ is
replaced by the appropriate $N$-extremal solution of the Stieltjes moment
problem associated with the rate set. If $\mu_{0}=0$ every $N$-extremal
measure $\psi_{\xi}$, $0\leq\xi\leq\xi_{1}$, corresponds to a birth-death
process. The $N$-extremal measures corresponding to a birth-death process with
$\mu_{0}>0$ will be identified in the next subsection. In any case, the
preceding remarks confirm Fact 1.3.

For completeness’ sake we recall from Section 1 that a natural measure on the
nonnegative real axis corresponds to an infinite family of rate sets, indexed
by the value of $\mu_{0}$, if (and only if) the measure has a finite moment of
order $-1$.

###  3.2 Dual birth-death processes

Our point of departure in this subsection is a birth-death process
$\mathcal{X}$ that is uniquely defined by its birth rates $\lambda_{n}$ and
death rates $\mu_{n}$, where $\mu_{0}>0$. Following Karlin and McGregor [17,
Section 6], we define the process $\mathcal{X}^{d}$ to be a birth-death
process on $\mathcal{N}$ with birth rates $\lambda_{n}^{d}$ and death rates
$\mu_{n}^{d}$ given by $\mu_{0}^{d}:=0$ and

$\displaystyle\lambda_{n}^{d}:=\mu_{n},\qquad\mu_{n+1}^{d}:=\lambda_{n},\qquad
n\geq 0.$ (3.4)

Accordingly, we let

$\displaystyle\pi_{0}^{d}:=1\qquad\mbox{and}\qquad\pi_{n}^{d}:=\frac{\lambda^{d}_{0}\lambda^{d}_{1}\cdots\lambda^{d}_{n-1}}{\mu^{d}_{1}\mu^{d}_{2}\cdots\mu^{d}_{n}}=\frac{\mu_{0}\mu_{1}\cdots\mu_{n-1}}{\lambda_{0}\lambda_{1}\cdots\lambda_{n-1}},\qquad
n\geq 1,$

and note that

$\displaystyle\pi_{n+1}^{d}=\mu_{0}(\lambda_{n}\pi_{n})^{-1}\qquad\mbox{and}\qquad\big{(}\lambda^{d}_{n}\pi^{d}_{n}\big{)}^{-1}=\mu_{0}^{-1}\pi_{n},\qquad
n\geq 0.$

Hence divergence of the series (3.2) is equivalent to divergence of the series

$\displaystyle\sum_{n=0}^{\infty}\left(\pi^{d}_{n}+\frac{1}{\lambda^{d}_{n}\pi^{d}_{n}}\right),$
(3.5)

so that $\mathcal{X}^{d}$ is uniquely defined by its rates if and only if
$\mathcal{X}$ is uniquely defined by its rates. So within the setting of
birth-death processes that are uniquely defined by their rates, the mapping
(3.4) establishes a one-to-one correspondence between processes with
$\mu_{0}=0$ and those with $\mu_{0}>0$. The processes $\mathcal{X}$ and
$\mathcal{X}^{d}$ are therefore called each other’s dual.

The transition functions of $\mathcal{X}^{d}$ satisfy a representation formula
analogous to (3.3), involving birth-death polynomials $P_{n}^{d}$ and a unique
natural probability measure $\psi^{d}$ on the nonnegative real axis with
respect to which the polynomials $P_{n}^{d}$ are orthogonal. Still assuming
divergence of (3.2) (and hence of (3.5)), we have, by [16, Lemma 3],

$\displaystyle\psi^{d}([0,x])=1-\mu_{0}m_{-1}(\psi)+\mu_{0}\int_{[0,x]}y^{-1}\psi(dy),\qquad
x\geq 0,$ (3.6)

where $\psi$ is the (natural) measure defined by $\mathcal{X}$ (which must
have $m_{-1}(\psi)<\infty$ since $\mu_{0}>0$). With $\xi^{d}_{i}$ and
$\sigma^{d}$ denoting the quantities defined by (2.4), (2.5) and (2.6) if we
replace $\psi$ by $\psi^{d}$, we thus have $\sigma^{d}=\sigma$,

$\displaystyle\xi^{d}_{1}=0\qquad\mbox{and}\qquad\xi^{d}_{i+1}=\xi_{i},~{}i>1,\qquad\mbox{if}\qquad\mu_{0}m_{-1}(\psi)<1,$

and

$\displaystyle\xi^{d}_{i}=\xi_{i},\qquad i\geq
1,\qquad\mbox{if}\qquad\mu_{0}m_{-1}(\psi)=1.$

Interestingly, with $p_{ij}^{d}(t)$ denoting the transition functions of the
dual process, we also have

$\displaystyle\sum_{j\geq k}p_{ij}^{d}(t)=\sum_{j<i}p_{k-1,j}(t),\qquad
i,k\in\mathcal{N},\qquad t\geq 0.$ (3.7)

provided the summations are interpreted to include probability mass, if any,
having escaped from $\mathcal{N}$ to the absorbing state $-1$ or to infinity
(see [22] and the references there for details). This property makes duality a
useful tool in the analysis of birth-death processes (see, for example, [23]).

If the series (3.2) (and hence the series (3.5)) converges, the situation is
more complicated since the rate sets $\\{\lambda_{n},\mu_{n}\\}$ and
$\\{\lambda_{n}^{d},\mu_{n}^{d}\\}$ are associated with infinite families of
birth-death processes. The following facts have been established in [22].
First, there is the separation result

$\displaystyle 0<\xi_{i}^{d}<\xi_{i}<\xi_{i+1}^{d},\qquad i\geq 1,$

where $\xi_{i}$ and $\xi^{d}_{i}$ now represent the spectral points of the
natural measures $\psi=\psi_{\xi_{1}}$ and $\psi^{d}=\psi^{d}_{\xi_{1}^{d}}$
that are uniquely defined by the rate sets $\\{\lambda_{n},\mu_{n}\\}$ and
$\\{\lambda_{n}^{d},\mu_{n}^{d}\\}$, respectively. (Recall that
$\sigma=\sigma^{d}=\infty$.)

Secondly, the $N$-extremal measure $\psi_{\xi}$ is associated with a birth-
death process (in the sense of Section 3.1) if and only if
$\xi_{1}^{d}\leq\xi\leq\xi_{1}$, while the $N$-extremal solution
$\psi_{\xi}^{d}$ is associated with a birth-death process for all $\xi$
satisfying $0\leq\xi\leq\xi_{1}^{d}$. The birth-death processes associated
with the $N$-extremal solutions $\psi_{\xi_{1}}=\psi$ and
$\psi_{\xi_{1}^{d}}^{d}=\psi^{d}$ are minimal processes, whereas the birth-
death processes corresponding to the $N$-extremal solutions
$\psi_{\xi_{1}^{d}}$ and $\psi_{0}^{d}$ are maximal processes.

Thirdly, (3.6) (and (3.7)) remain valid if (and only if) either $\psi^{d}$ is
replaced by $\psi^{d}_{0}$ or $\psi$ by $\psi_{\xi_{1}^{d}}$, that is, we have

$\displaystyle\psi^{d}_{0}([0,x])=1-\mu_{0}m_{-1}(\psi)+\mu_{0}\int_{[0,x]}y^{-1}\psi(dy),\qquad
x\geq 0,$ (3.8)

and

$\displaystyle\psi^{d}([0,x])=1-\mu_{0}m_{-1}(\psi_{\xi_{1}^{d}})+\mu_{0}\int_{[0,x]}y^{-1}\psi_{\xi_{1}^{d}}(dy),\qquad
x\geq 0.$

It follows that the duality concept for rate sets can be extended to birth-
death processes also if they are not uniquely defined by their rates, provided
one restricts oneself to minimal and maximal processes, and links a minimal
process to a maximal process.

We finally remark that a probabilistic interpretation of minimal and maximal
processes involves the character of the boundary at infinity (which is not
specified by the rates). This boundary may be completely absorbing (the
minimal process), completely reflecting (the maximal process), or something in
between. Evidently, the distinction is relevant only if the process can
explode, that is, reach infinity in finite time.

###  3.3 Similar birth-death processes

Consider, besides the birth-death process $\mathcal{X}$ of Section 3.1,
another birth-death process $\tilde{\mathcal{X}}$, with birth rates
$\tilde{\lambda}_{n}$ and death rates $\tilde{\mu}_{n}$, coefficients
$\tilde{\pi}_{n}$ and transition functions $\tilde{p}_{ij}(\cdot)$. The
processes $\mathcal{X}$ and $\tilde{\mathcal{X}}$ are said to be similar if
there are constants $c_{ij}$, $i,j\in\mathcal{N},$ such that

$\displaystyle{\tilde{p}}_{ij}(t)=c_{ij}p_{ij}(t),\qquad
i,j\in\mathcal{N},\qquad t\geq 0.$

The next theorem shows that, under certain regularity conditions, similarity
imposes strong restrictions on the birth and death rates.

######  Theorem 3.1.

Let the birth-death processes $\mathcal{X}$ and $\tilde{\mathcal{X}}$ be
either uniquely determined by their rates or minimal. If $\mathcal{X}$ and
$\tilde{\mathcal{X}}$ are similar, then their birth and death rates are
related as

$\displaystyle\tilde{\lambda}_{n}+\tilde{\mu}_{n}=\lambda_{n}+\mu_{n},\qquad\tilde{\lambda}_{n}\tilde{\mu}_{n+1}=\lambda_{n}\mu_{n+1},\qquad
n\in\mathcal{N},$ (3.9)

while their transition functions satisfy

$\displaystyle\tilde{p}_{ij}(t)=\sqrt{\frac{\pi_{i}\tilde{\pi}_{j}}{\tilde{\pi}_{i}\pi_{j}}}p_{ij}(t),\qquad
i,j\in\mathcal{N},\qquad t\geq 0.$

Conversely, if $\mathcal{X}$ and $\tilde{\mathcal{X}}$ are birth-death
processes with rates related as in (3.9) then $\mathcal{X}$ and
$\tilde{\mathcal{X}}$ are similar.

In the more restricted setting in which $\mathcal{X}$ and
$\tilde{\mathcal{X}}$ are uniquely determined by their rates the statements of
this theorem were given in [18, Theorems 1 and 2]. Since the additional
restrictions are not used in the proof of the necessity of (3.9) for
similarity of $\mathcal{X}$ and $\tilde{\mathcal{X}}$, the question remains
whether (3.9) is sufficient for similarity of $\mathcal{X}$ and
$\tilde{\mathcal{X}}$ when the processes are not uniquely defined by their
rates (but minimal). This, however, follows immediately from Karlin and
McGregor’s representation formula (3.3), since, considering the remarks
preceding (3.3), the polynomials and natural measure associated with
$\mathcal{X}$ must be identical to those of $\tilde{\mathcal{X}}$ if (3.9)
prevails. Interestingly, Fralix [15] recently established a sufficient
condition for similarity in the setting of continuous-time Markov chains
(conjectured earlier by Pollett [20]), which amounts to (3.9) when applied to
birth-death processes.

On relating the results of Theorem 3.1 to (1.1) and (1.2) we see that a family
of similar birth-death processes is characterized by the fact that all members
are associated with the same orthogonal polynomial sequence $\\{P_{n}\\}$ –
and hence with the same (natural) orthogonalizing measure $\psi$ – while each
individual member may be characterized by the value of $\mu_{0}$, which can be
any number in $[0,1/m_{-1}(\psi)]$. So a family of similar birth-death
processes has either one member (if $m_{-1}(\psi)=\infty$) or infinitely many
members (if $m_{-1}(\psi)<\infty$). Note that there is always a member in the
family with $\mu_{0}=0$, the representative of the family. By [16, equation
(2.4), Lemma 6 on p. 527] we have

$\displaystyle\mu_{0}=0\quad\Longrightarrow\quad
m_{-1}(\psi)=\sum_{n=0}^{\infty}\frac{1}{\lambda_{n}\pi_{n}},$

so to decide whether the representative of a family is the only member of the
family is, given the birth and death rates of the representative, a trivial
task.

##  4 Results

Having collected all we need, we are ready to draw conclusions. To start with,
consider a rate set $\\{\lambda_{n},\mu_{n}\\}$ with $\mu_{0}>0$ and the
natural measure $\psi$ that, by Fact 1.4, is defined by this set. Since, by
Fact 1.4 again, $m_{-1}(\psi)<\infty$ and $\mu_{0}\leq 1/m_{-1}(\psi)$, we can
choose $a=(\mu_{0}m_{-1}(\psi))^{-1}-1\geq 0$ and thus link the rate set
$\\{\lambda_{n},\mu_{n}\\}$ to the measure $\phi^{(a)}$ defined in (1.4) and
(1.5).

Let us first assume that the rate set $\\{\lambda_{n},\mu_{n}\\}$ is such that
the series (3.2) diverges, whence it corresponds to a unique birth-death
process $\mathcal{X}$. Then, by Fact 1.5, there are two possibilities. The
first is that $\psi$ is det(S), in which case, by Theorem 1.1, $\phi^{(a)}$ is
also det(S). The second possibility is that $\psi$ is indet(S) and
$m_{-1}(\psi)=1/\mu_{0}$. But then $a=0$, so that, by Theorem 1.1,
$\phi^{(a)}=\phi^{(0)}$ is det(S) again. So, in any case, $\phi^{(a)}$ is
det(S) and hence natural. If $a=0$ it is possible for $\phi^{(a)}$ to define
an infinite family of rate sets, but, as agreed upon in Section 2.1, we will
always associate with $\phi^{(a)}$ the unique rate set
$\\{\lambda_{n}^{(a)},\mu_{n}^{(a)}\\}$ with $\mu_{0}^{(a)}=0$ determined by
the parameters in the recurrence relation (2.1) for the shell polynomials
$S_{n}^{(a)}$. So we have now linked the rate set $\\{\lambda_{n},\mu_{n}\\}$
with $\mu_{0}>0$ to a rate set $\\{\lambda_{n}^{(a)},\mu_{n}^{(a)}\\}$ with
$\mu_{0}^{(a)}=0$, which, by Fact 1.5, uniquely defines a birth-death process
$\mathcal{X}^{(a)}$. But on comparing (2.2) and (3.4), we see that

$\displaystyle\lambda_{n}^{(a)}=\lambda_{n}^{d}\qquad\mbox{and}\qquad\mu_{n}^{(a)}=\mu_{n}^{d},\qquad
n\geq 0,$

so that, actually, $\mathcal{X}^{(a)}=\mathcal{X}^{d}$, the dual process of
$\mathcal{X}$. Indeed, on comparing (1.4) and (1.5) with (3.6), we also
observe that $\phi^{(a)}=\psi^{d}$, as required. We summarize our findings in
the next theorem.

######  Theorem 4.1.

Let $\\{\lambda_{n},\mu_{n}\\}$ with $\mu_{0}>0$ be a rate set for which the
series (3.2) diverges, $\mathcal{X}$ the birth-death process defined by this
set, and $\psi$ the corresponding measure. Then $\phi^{(a)}$, defined by (1.4)
and (1.5), is ${\rm det(S)}$ for all $a\geq 0$, while, for
$a=(\mu_{0}m_{-1}(\psi))^{-1}-1$, it is the measure corresponding to
$\mathcal{X}^{d}$, the dual process of $\mathcal{X}$.

Note that the (not so obvious) condition $\mu_{0}m_{-1}(\psi)\leq 1$, which a
rate set associated with the natural measure $\psi$ with $m_{-1}(\psi)<\infty$
should satisfy, has a very natural counterpart for the measure of the dual
process, namely $a\geq 0$.

Evidently, (3.4) establishes a one-to-one correspondence between rate sets
with $\mu_{0}=0$ and those with $\mu_{0}>0$. So, as long as we work in the
setting of rate sets that uniquely define a birth-death process, the above
procedure mapping the rate set $\\{\lambda_{n},\mu_{n}\\}$ with $\mu_{0}>0$ to
the rate set $\\{\lambda_{n}^{(a)},\mu_{n}^{(a)}\\}$ with $\mu_{0}^{(a)}=0$,
via the corresponding birth-death processes, must reflect this correspondence.
In other words, every measure $\phi$ that corresponds to a rate set with
$\mu_{0}=0$, must be of the form $\phi^{(a)}$ of (1.4) and (1.5) for some
$a\geq 0$, with $\psi$ being the measure of the dual process. Here are the
details of this correspondence.

Consider a rate set $\\{\tilde{\lambda}_{n},\tilde{\mu}_{n}\\}$ with
$\tilde{\mu}_{0}=0$ for which the analogue of the series (3.2) diverges. Let
$\tilde{\mathcal{X}}$ be the birth-death process uniquely defined by this rate
set, $\tilde{P}_{n}$ the corresponding polynomials and $\tilde{\phi}$ the
corresponding measure, which, in view of the analogue of (2.7), must be
det(S). Then, letting

$\displaystyle
a=\frac{\tilde{\phi}(\\{0\\})}{1-\tilde{\phi}(\\{0\\})}\qquad\mbox{and}\qquad\phi^{(0)}=\frac{\tilde{\phi}-\tilde{\phi}(\\{0\\})\delta_{0}}{1-\tilde{\phi}(\\{0\\})},$

$\tilde{\phi}$ can be represented as

$\displaystyle\tilde{\phi}=\frac{1}{a+1}\big{(}a\delta_{0}+\phi^{(0)}\big{)}.$

Defining the (probability) measure $\psi$ by

$\displaystyle\psi([0,x])=\frac{1}{m_{1}(\phi^{(0)})}\int_{[0,x]}y\phi^{(0)}(dy),\qquad
x\geq 0,$ (4.1)

we can apply some results of Berg and Thill [4, 5] to conclude the following.

######  Lemma 4.2.

The measure $\psi$ defined by (4.1) is the natural solution of the
corresponding moment problem.

###### Proof.

If $\tilde{\phi}(\\{0\\})>0$ then, by [5, Lemma 5.4], $\psi$ must be det(S),
and hence natural, since $\tilde{\phi}$ is det(S). If
$\tilde{\phi}(\\{0\\})=0$ (so that $\tilde{\phi}=\tilde{\phi}^{(0)}$) and
$\psi$ is indet(S), then, by [4, Theorem 2.4], the density index of
$\tilde{\phi}^{(0)}$ (the largest $n\in\mathbb{N}$ such that the polynomials
are dense in $x^{n}\tilde{\phi}^{(0)}(dx)$) equals 2, implying that $\psi$ has
density index 1. Hence, by [4, Theorem 2.1], $\psi$ must be natural. ∎

Evidently, $m_{-1}(\psi)<\infty$, so we see that $\psi$ has the properties
imposed on $\psi$ in Section 1. Since

$\displaystyle\int_{[0,\infty)}\tilde{P}_{1}(x)\tilde{\phi}(dx)=\int_{[0,\infty)}(x-\tilde{\lambda}_{0})\tilde{\phi}(dx)=0,$

we have $m_{1}(\tilde{\phi})=\tilde{\lambda}_{0}$, while

$\displaystyle\psi([0,x])=\frac{1}{m_{1}(\tilde{\phi})}\int_{[0,x]}y\tilde{\phi}(dy)=\frac{1}{\tilde{\lambda}_{0}}\int_{[0,x]}y\tilde{\phi}(dy),\qquad
x\geq 0,$

so that $m_{-1}(\psi)=(1-\tilde{\phi}(\\{0\\}))/\tilde{\lambda}_{0}$. We can
now associate a rate set $\\{\lambda_{n},\mu_{n}\\}$ with $\psi$ by letting

$\displaystyle\mu_{0}=\frac{1}{(a+1)m_{-1}(\psi)}=\tilde{\lambda}_{0},$

so that $0<\mu_{0}\leq 1/m_{-1}(\psi)$, and choosing $\lambda_{n}$ and
$\mu_{n}$ such that the polynomials $P_{n}$ defined by (1.1) and (1.2) are
orthogonal with respect to $\psi$. Next identifying $\tilde{\phi}$ with the
measure $\phi^{(a)}$ defined in (1.4) and (1.5), we can identify the rates
$\tilde{\lambda}_{n}$ and $\tilde{\mu}_{n}$ with the rates $\lambda_{n}^{(a)}$
and $\mu_{n}^{(a)}$, respectively, appearing in the recurrence relation (2.1)
for the shell polynomials corresponding to the sequence $\\{P_{n}\\}$. On
comparing (2.2) and (3.4), we thus find

$\displaystyle\tilde{\lambda}_{n}=\lambda_{n}^{d}\qquad\mbox{and}\qquad\tilde{\mu}_{n}=\mu_{n}^{d},\qquad
n\geq 0.$

It follows that the series (3.5), and hence the series (3.2), diverges, so
that the rate set $\\{\lambda_{n},\mu_{n}\\}$ defines a unique birth-death
process $\mathcal{X}$. Moreover, $\tilde{\mathcal{X}}=\mathcal{X}^{d}$, the
dual process of $\mathcal{X}$. In summary, we can state the converse of
Theorem 4.1 as follows.

######  Theorem 4.3.

Let $\\{\tilde{\lambda}_{n},\tilde{\mu}_{n}\\}$ with $\tilde{\mu}_{0}=0$ be a
rate set for which the analogue of the series (3.2) diverges,
$\tilde{\mathcal{X}}$ the birth-death process defined by this set, and
$\tilde{\phi}$ the corresponding measure. Then, letting
$a=\tilde{\phi}(\\{0\\})/(1-\tilde{\phi}(\\{0\\}))$, the measure
$\tilde{\phi}$ can be identified with $\phi^{(a)}$, defined by (1.4) and
(1.5), where $\psi$ is the natural measure corresponding to a birth-death
process $\mathcal{X}$ with $\mu_{0}=((a+1)m_{-1}(\psi))^{-1}>0$. Also,
$\tilde{\mathcal{X}}=\mathcal{X}^{d}$, the dual process of $\mathcal{X}$.

Still residing in the setting of birth-death processes that are uniquely
defined by their rates, we recall from Section 3.3 that a collection of birth-
death processes sharing the same natural measure $\psi$ with finite moment of
order $-1$ is called a family of similar processes. The individual members of
this family are identified by the value of $\mu_{0}$, which may be any value
in the interval $0\leq\mu_{0}\leq 1/m_{-1}(\psi)$. However, in view of the
preceding observations, it seems more appropriate to exclude the process with
$\mu_{0}=0$ from this family and view this process as a member of a new family
of birth-death processes, which all have $\mu_{0}=0$ and a measure of the type

$\displaystyle\frac{1}{a+1}(a\delta_{0}+\psi),$ (4.2)

where $a\geq 0$, the process at hand corresponding to $a=0$.

Defining families of birth-death processes in this way allows us to extend the
duality concept for individual birth-death processes to families of birth-
death processes. Indeed, if $\psi_{1}$ is a natural measure with
$m_{-1}(\psi_{1})<\infty$ then there is a one-to-one correspondence between
the family of similar processes with measure $\psi_{1}$ and $\mu_{0}>0$, and
the family of processes with $\mu_{0}=0$ and a measure of the type (4.2),
where $a\geq 0$ and $\psi\equiv\psi_{2}$ is given by

$\displaystyle\psi_{2}([0,x])=\frac{1}{m_{-1}(\psi_{1})}\int_{[0,x]}y^{-1}\psi_{1}(dy),\qquad
x\geq 0,$

in the sense that corresponding processes are each other’s dual. Note that
$\psi_{2}$ is det(S) by Theorem 1.1. If $m_{-1}(\psi_{2})<\infty$, or,
equivalently, $m_{-2}(\psi_{1})<\infty$, we can view $\psi_{2}$ as the
producer of a family of similar birth-death processes with $\mu_{0}>0$, which,
in turn, is dual to a family of processes with $\mu_{0}=0$ and a measure of
the type (4.2), etc.

Moving beyond the setting of birth-death processes that are uniquely defined
by their rates the situation becomes more complicated, but as noted in Section
3.2 the duality concept for rate sets can be extended, provided one restricts
oneself to minimal and maximal processes, and links a minimal process to a
maximal process. We next elaborate on how this affects the measures involved.

So consider again a rate set $\\{\lambda_{n},\mu_{n}\\}$ with $\mu_{0}>0$, and
the natural measure $\psi$ defined by this set. As before we can choose
$a=(\mu_{0}m_{-1}(\psi))^{-1}-1\geq 0$ and thus link the rate set
$\\{\lambda_{n},\mu_{n}\\}$ to the measure $\phi^{(a)}$ defined in (1.4) and
(1.5). We now assume that the rate set $\\{\lambda_{n},\mu_{n}\\}$ is such
that the series (3.2) converges, so that, by Fact 1.5, $\phi^{(a)}$ is
indet(S) and $a>0$. Subsequently comparing (1.5) and (3.8) we conclude that we
actually have $\phi^{(a)}=\psi_{0}^{d}$. It follows that $\phi^{(a)}$ is
$N$-extremal (as noted already in [3]) and corresponds to the maximal birth-
death process associated with the rate set $\\{\lambda_{n}^{d},\mu_{n}^{d}\\}$
with $\mu_{0}^{d}=0$, given by (3.4). Summarizing we can state the following.

######  Theorem 4.4.

Let $\\{\lambda_{n},\mu_{n}\\}$ with $\mu_{0}>0$ be a rate set for which the
series (3.2) converges, $\mathcal{X}$ the minimal birth-death process defined
by this set, and $\psi$ the corresponding measure. Then
$\mu_{0}<1/m_{-1}(\psi)$, and $\phi^{(a)}$, defined by (1.4) and (1.5), is
${\rm indet(S)}$ for $a>0$. Moreover, for $a=(\mu_{0}m_{-1}(\psi))^{-1}-1$,
$\phi^{(a)}$ is the measure corresponding to $\mathcal{X}^{d}$, the maximal
process that is dual to $\mathcal{X}$, and hence $N$-extremal.

######  Remark 4.5.

The argument given in [3] for the fact that $\phi^{(a)}$ is $N$-extremal is
not entirely clear, but, in any case, a reference to [2, Theorem 8] (besides
the reference to [5, Theorem 5.5] given in [3]) is sufficient to justify the
statement.

Of course there is a converse to Theorem 4.4 – the analogue of Theorem 4.3 –
which, however, we will not formulate explicitly. It may be more interesting
to look at the minimal process corresponding to the rate set
$\\{\tilde{\lambda}_{n},\tilde{\mu}_{n}\\}$ with $\tilde{\mu}_{0}=0$ which
does not uniquely define a birth-death process, since the associated measure
is natural. We give the result without proof and refrain again from
formulating its converse explicitly.

######  Theorem 4.6.

Let $\\{\tilde{\lambda}_{n},\tilde{\mu}_{n}\\}$ with $\tilde{\mu}_{0}=0$ be a
rate set for which the analogue of the series (3.2) converges,
$\tilde{\mathcal{X}}$ the minimal birth-death process defined by this set, and
$\tilde{\phi}$ the corresponding measure. Then, letting
$a=\tilde{\phi}(\\{0\\})/(1-\tilde{\phi}(\\{0\\}))$, the measure
$\tilde{\phi}$ can be identified with $\phi^{(a)}$, defined by (1.4) and
(1.5), where $\psi$ is the $($natural$)$ measure corresponding to a maximal
birth-death process $\mathcal{X}$ with $\mu_{0}=((a+1)m_{-1}(\psi))^{-1}>0$,
and hence $N$-extremal. Also, $\tilde{\mathcal{X}}=\mathcal{X}^{d}$, the dual
process of $\mathcal{X}$.

We finally remark that Theorems 4.1 and 4.4 augment the information given in
[16, Lemma 2], while Theorems 4.3 and 4.6 elaborate on [16, Lemma 3].

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