# On semi-continuity and continuity of the smallest and largest minimizing
point of real convex functions with applications in probability and statistics

Dietmar Ferger<EMAIL_ADDRESS>Fakultät Mathematik, Technische
Universität Dresden, Zellescher Weg 12-14, Dresden, 01069, Germany

###### Abstract

We prove that the smallest minimizer $\sigma(f)$ of a real convex function $f$
is less than or equal to a real point $x$ if and only if the right derivative
of $f$ at $x$ is non-negative. Similarly, the largest minimizer $\tau(f)$ is
greater or equal to $x$ if and only if the left derivative of $f$ at $x$ is
non-positive. From this simple result we deduce measurability and semi-
continuity of the functionals $\sigma$ and $\tau$. Furthermore, if $f$ has a
unique minimizing point, so that $\sigma(f)=\tau(f)$, then the functional is
continuous at $f$. With these analytical preparations we can apply Continuous
Mapping Theorems to obtain several Argmin theorems for convex stochastic
processes. The novelty here are statements about classical distributional
convergence and almost sure convergence, if the limit process does not have a
unique minimum point. This is possible by replacing the natural topology on
$\mathbb{R}$ with the order topologies. Another new feature is that not only
sequences but more generally nets of convex stochastic processes are allowed.

###### keywords:

order topologies, distributional convergence in topological spaces, Argmin
theorems, functional limit theorem for convex processes

##  1 Introduction

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a convex function with pertaining
set

$A(f):=\\{t\in\mathbb{R}:f(t)=\inf_{s\in\mathbb{R}}f(s)\\}$ (1)

of all minimizing points. This _minimum set_ can equivalently be described in
terms of the right and left derivative $D^{+}f$ and $D^{-}f$ of $f$. Indeed,
it follows from Theorem 23.2 of Rockafellar [21] that

$A(f)=\\{t\in\mathbb{R}:D^{-}f(t)\leq 0\leq D^{+}f(t)\\}.$ (2)

Of course, it can happen that $A(f)$ is empty. However, if this is not the
case, it is well-known (and actually easy to see) that then $A(f)$ is closed
and convex and hence is a closed interval. The extreme case $A(f)=\mathbb{R}$
occurs if and only if $f$ is a constant function. So, as long as $f$ is not a
constant function there are three possibilities: (i) $A(f)=[a,b],a\leq
b\in\mathbb{R}$. (ii) $A(f)=[a,\infty),a\in\mathbb{R}$. (iii)
$A(f)=(-\infty,b],b\in\mathbb{R}$.

Let $C:=\\{f:\mathbb{R}\rightarrow\mathbb{R};f\text{ convex}\\}$ be the class
of all convex functions. Introduce

$S:=\\{f\in C:\inf A(f)>-\infty\\}$ $S^{\prime}:=\\{f\in C:\sup
A(f)<\infty\\}.$

Thus $S$ and $S^{\prime}$ consist exactly of those functions for which the
smallest and largest minimizer, respectively, exist. Consequently the
functionals $\sigma:S\rightarrow\mathbb{R}$ and
$\tau:S^{\prime}\rightarrow\mathbb{R}$ given by $\sigma(f)=\min A(f)$ and
$\tau(f)=\max A(f)$ are well defined on their domains.

Our first main result provides a necessary and sufficient condition for the
location of the smallest and the largest minimizer, respectively. It is rather
simple with an elementary proof, but with it we will draw a whole series of
useful conclusions. In section 2 it is shown that $\sigma$ and $\tau$ are
measurable and semi-continuous. On the set $S_{u}$ of all convex functions
with exactly one minimizer the functionals $\sigma$ and $\tau$ coincide and
are furthermore continuous there. In section 3 this is used in combination
with Continuous Mapping Theorems to derive several Argmin theorems for convex
stochastic processes. For a further discussion of our findings we refer to the
concluding remarks at the end of section 3.

######  Theorem 1.

Let $\sigma(f)=\min A(f)$ and $\tau(f)=\max A(f)$ be the smallest and largest
minimizing point of the convex function $f$. Then the following equivalent
relations hold for every $x\in\mathbb{R}$:

$\sigma(f)\leq x\quad\Longleftrightarrow\quad D^{+}f(x)\geq 0$ (3)
$\tau(f)\geq x\quad\Longleftrightarrow\quad D^{-}f(x)\leq 0.$ (4)

###### Proof.

For the proof of (3) we briefly write $\sigma$ for $\sigma(f)$. Recall that
$D^{+}f$ and $D^{-}f$ are non-decreasing, confer, e.g., Theorem 1.3.3 in
Niculescu and Persson [20]. Thus $\sigma\leq x$ entails $D^{+}f(x)\geq
D^{+}f(\sigma)\geq 0$, where the last inequality follows from (2), because
$\sigma\in A(f)$. To see the reverse conclusion in (3) we use that by Theorem
1.3.1 in Niculescu and Persson [20] the difference quotients
$\frac{f(t)-f(x)}{t-x}$ are non-increasing as $t\downarrow x$. Therefore we
obtain

$0\leq
D^{+}f(x)=\inf_{t>x}\frac{f(t)-f(x)}{t-x}\leq\frac{f(t)-f(x)}{t-x}\quad\forall\;t>x.$
(5)

Multiplication with the positive difference $t-x$ yields that $f(t)\geq f(x)$
for all $t>x$ and so

$\inf_{t>x}f(t)\geq f(x)\geq\inf_{t\leq x}f(t),$

where the second inequality is trivial. To sum up we arrive at

$\inf_{t\in\mathbb{R}}f(t)=\inf_{t\leq x}f(t).$ (6)

Next we consider a point $t<\sigma$. Infer from the monotonicity of $D^{-}f$
that $D^{-}f(t)\leq D^{-}f(\sigma)\leq 0$, where the last inequality is
ensured by (2). Just as for $D^{+}f$, we see that
$D^{-}f(t)=\sup_{s<t}\frac{f(s)-f(t)}{s-t}$, whence in view of $D^{-}f(t)\leq
0$ we have that

$\frac{f(s)-f(t)}{s-t}\leq 0\quad\forall\;s<t.$

Multiplication with the negative difference $s-t$ gives $f(s)\geq f(t)$ for
all $s<t<\sigma$. Since $f$ is continuous, taking the limit $t\uparrow\sigma$
finally shows that $f$ is non-increasing on the closed half-line
$(-\infty,\sigma]$. In particular, $f(t)\geq f(\sigma)$ for all $t\leq\sigma$.
Actually, on the open interval $(-\infty,\sigma)$ the inequality is strict:

$f(t)>f(\sigma)\quad\forall\;t<\sigma.$ (7)

This is because otherwise there exists some $t_{0}<\sigma$ such that
$f(t_{0})\leq f(\sigma)$. Since $\sigma$ is a minimizing point, $t_{0}$ must
be a minimizing point as well. This is a contradiction to minimality of
$\sigma$.

Now, assume that $x<\sigma$. Deduce from $f$ is non-increasing on
$(-\infty,x]\subseteq(-\infty,\sigma]$ that $f(t)\geq f(x)$ for all $t\leq x$.
Consequently, $\inf_{t\leq x}f(t)\geq f(x)$ and therefore by (6) and (7):

$f(\sigma)=\inf_{t\in\mathbb{R}}f(t)=\inf_{t\leq x}f(t)\geq f(x)>f(\sigma).$

This is a contradiction and thus $x\geq\sigma$ is true as desired.

For the proof of (4) we use a time-reversing argument. Introduce the function
$f_{-}$ defined by $f_{-}(t):=f(-t),t\in\mathbb{R}$. One easily verifies that
$f_{-}$ is convex and that $\tau(f)=-\sigma(f_{-})$. We obtain by (3):

$\tau(f)\geq x\quad\Longleftrightarrow\quad-\sigma(f_{-})\geq
x\quad\Longleftrightarrow\quad\sigma(f_{-})\leq-x\quad\Longleftrightarrow\quad
D^{+}f_{-}(-x)\geq 0$

and the assertion follows upon noticing that $D^{+}f_{-}(-x)=-D^{-}f(x)$. ∎

######  Remark 1.

Under the existence of the minimizers one has that:

$(a)\quad\sigma(f)\leq x,\tau(f)\geq x\quad\Longleftrightarrow\quad(b)\quad
D^{+}f(x)\geq 0,D^{-}f(x)\leq 0.$ (8)

Indeed, recall that $D^{+}f$ and $D^{-}f$ are non-decreasing, which by (2)
shows necessity of (b). By another application of (2) the point x in (b) is a
minimizer, $x\in A(f)$, and (a) follows using the minimum and maximum property
of $\sigma(f)=\min A(f)$ and $\tau(f)=\max A(f)$, respectively.

Note that we cannot readily infer Proposition 1 from (8). For example, if we
only know that $D^{+}f(x)\geq 0$ holds, then $x$ need not be a minimizing
point in general, and is not, as can be seen from simple examples as for
instance $f(x)=x^{2}$. Consequently, the argument via the minimum property of
$\sigma(f)$ fails.

For every non-decreasing function $F:\mathbb{R}\rightarrow\mathbb{R}$ we
introduce the _generalized inverses_ :

$F^{\wedge}(y):=\inf\\{x\in\mathbb{R}:F(x)\geq y\\}\quad\text{and}\quad
F^{\vee}(y):=\sup\\{x\in\mathbb{R}:F(x)\leq y\\},\;y\in\mathbb{R}.$

For properties of these inverses, confer Embrechts and Hofert [5], Feng et.al.
[6] or Fortelle [10]. Notice that (3) is the same as
$[\sigma(f),\infty)=\\{x\in\mathbb{R}:D^{+}f(x)\geq 0\\}$ and hence

$\sigma(f)=(D^{+}f)^{\wedge}(0).$

Similarly, $(-\infty,\tau(f)]=\\{x\in\mathbb{R}:D^{-}f(x)\leq 0\\}$ by (4),
whence

$\tau(f)=(D^{-}f)^{\vee}(0).$

##  2 Measurability, Semi-continuity and Continuity of the argmin-functionals

Recall that $C$ is the class of all convex functions on $\mathbb{R}$. For each
$t\in\mathbb{R}$ let $\pi_{t}:C\rightarrow\mathbb{R}$ denote the projection
(evaluation map) at $t$, that is $\pi_{t}(f)=f(t)$. We endow the function
space $C$ with the $\sigma-$algebra generated by the projections:
$\mathcal{C}:=\sigma(\pi_{t}:t\in\mathbb{R})$. Furthermore, $C$ is equipped
with the topology $\mathcal{T}$ of pointwise convergence, which is known to be
generated by the projections: $\mathcal{T}:=\tau(\pi_{t}:t\in\mathbb{R})$.
Recall that $\mathcal{T}$ is the smallest topology on $C$ for which all
projections are continuous. Note that the trace
$\mathcal{C}_{S}:=S\cap\mathcal{C}$ in $S$ is generated by the restrictions of
$\pi_{t}$ to $S$. Analogously, the subspace topology
$\mathcal{T}_{S}=S\cap\mathcal{T}$ on $S$ is generated by these restrictions.
The corresponding statements hold for $S^{\prime}$ endowed with the trace
$\mathcal{C}_{S^{\prime}}=S^{\prime}\cap\mathcal{C}$ and the subspace topology
$\mathcal{T}_{S^{\prime}}=S^{\prime}\cap\mathcal{T}$.

######  Proposition 2.

  * (i)

$\sigma:S\rightarrow\mathbb{R}$ is $\mathcal{C}_{S}$-Borel measurable and
$\mathcal{T}_{S}-$lower semicontinuous.

  * (ii)

$\tau:S^{\prime}\rightarrow\mathbb{R}$ is $\mathcal{C}_{S^{\prime}}$-Borel
measurable and $\mathcal{T}_{S^{\prime}}-$upper semicontinuous.

###### Proof.

For each $x\in\mathbb{R}$ we have that:

$\displaystyle\\{f\in S:\sigma(f)\leq x\\}$ $\displaystyle=$
$\displaystyle\\{f\in S:D^{+}f(x)\geq 0\\}\hskip 56.9055pt\text{ by Theorem
\ref{location}}$ $\displaystyle=$ $\displaystyle\\{f\in
S:\inf_{x<t}\frac{f(t)-f(x)}{t-x}\geq 0\\}\hskip 28.45274pt\text{by the first
equality in (\ref{D+})}$ $\displaystyle=$ $\displaystyle\\{f\in
S:\inf_{x<t\in\mathbb{Q}}\frac{f(t)-f(x)}{t-x}\geq 0\\}\hskip
18.20973pt\text{by continuity of }f$ $\displaystyle=$
$\displaystyle\bigcap_{x<t\in\mathbb{Q}}\\{f\in S:\frac{f(t)-f(x)}{t-x}\geq
0\\}$ $\displaystyle=$ $\displaystyle\bigcap_{x<t\in\mathbb{Q}}\\{f\in
S:f(t)-f(x)\geq 0\\}$ $\displaystyle=$
$\displaystyle\bigcap_{x<t\in\mathbb{Q}}\big{(}S\cap(\pi_{t}-\pi_{x})^{-1}([0,\infty))\big{)}$
$\displaystyle=$ $\displaystyle
S\cap\bigcap_{x<t\in\mathbb{Q}}(\pi_{t}-\pi_{x})^{-1}([0,\infty))$

By construction of $\mathcal{C}$ every projection is $\mathcal{C}$-measurable,
whence the differences $\pi_{t}-\pi_{x}$ are $\mathcal{C}$-measurable as well
and therefore $(\pi_{t}-\pi_{x})^{-1}([0,\infty))\in\mathcal{C}$ for all
rationals $t>x$. Since $\mathcal{C}$ is closed under denumerable intersections
we arrive at $\\{f\in S:\sigma(f)\leq x\\}\in
S\cap\mathcal{C}=\mathcal{C}_{S}$ for all $x\in\mathbb{R}$, which by Lemma 1.4
in Kallenberg [17] shows measurability of $\sigma$.

As to semicontinuity recall that by construction of $\mathcal{T}$ every
projection $\pi_{t}$ is $\mathcal{T}$-continuous, whence the differences
$\pi_{t}-\pi_{x}$ are $\mathcal{T}$-continuous as well and therefore
$(\pi_{t}-\pi_{x})^{-1}([0,\infty))$ are $\mathcal{T}-$closed for all $t>x$.
Since $\mathcal{T}$ is closed under every kind of intersections we arrive at
$\\{f\in S:\sigma(f)\leq x\\}$ is $\mathcal{T}_{S}-$closed for all
$x\in\mathbb{R}$, which shows $\mathcal{T}_{S}-$semicontinuity of
$\mathcal{\sigma}$.

The second part follows in the same way. Indeed, since
$D^{-}f(x)=\sup_{t<x}\frac{f(t)-f(x)}{t-x}$ it follows analogously that

$\\{f\in S^{\prime}:\tau(f)\geq
x\\}=S^{\prime}\cap\bigcap_{x<t\in\mathbb{Q}}(\pi_{t}-\pi_{x})^{-1}([0,\infty)).$

∎

Next we give further equivalent characterizations of semi-continuity. The
first one is an immediate consequence of Proposition 2 and the definition of
continuity (pre-images of open sets are open).

######  Corollary 1.

Let
$\mathcal{O}_{<}:=\\{(-\infty,x):x\in\mathbb{R}\\}\cup\\{\emptyset,\mathbb{R}\\}$
and
$\mathcal{O}_{>}:=\\{(x,\infty):x\in\mathbb{R}\\}\cup\\{\emptyset,\mathbb{R}\\}$
be the left-order topology and the right-order topology. Then:

  * (1)

$\sigma:(S,\mathcal{T}_{S})\rightarrow(\mathbb{R},\mathcal{O}_{>})$ is
continuous.

  * (2)

$\tau:(S^{\prime},\mathcal{T}_{S^{\prime}})\rightarrow(\mathbb{R},\mathcal{O}_{<})$
is continuous.

Sometimes it is advantageous to consider the restrictions of $\sigma$ and
$\tau$ on subspaces.

######  Remark 2.

Let $\emptyset\neq U\subseteq S$ be endowed with
$\mathcal{C}_{U}:=U\cap\mathcal{C}$ and $\mathcal{T}_{U}:=U\cap\mathcal{T}$.
Then $\sigma:U\rightarrow\mathbb{R}$ is $\mathcal{C}_{U}-$Borel measurable and
$\sigma:(U,\mathcal{T}_{U})\rightarrow(\mathbb{R},\mathcal{O}_{>})$ is
continuous. Similarly, if $U\subseteq S^{\prime}$, then
$\tau:U\rightarrow\mathbb{R}$ is $\mathcal{C}_{U}-$Borel measurable and
$\tau:(U,\mathcal{T}_{U})\rightarrow(\mathbb{R},\mathcal{O}_{<})$ is
continuous.

Corollary 1 in turn yields a second equivalent description of semi-continuity
via net-convergence. For this purpose, let $(I,\leq)$ be here and in the
following a directed set. Also recall the definition

$S_{u}:=\\{f\in C:f\text{ has a unique minimizing point}\\}.$ (9)

######  Corollary 2.

Assume that $(f_{\alpha})_{\alpha\in I}\subseteq C$ converges pointwise to $f$
on $\mathbb{R}$. Then the following statements apply:

  * (1)

If $(f_{\alpha})_{\alpha\in I}\subseteq S$ and $f\in S$, then
$\liminf_{\alpha}\sigma(f_{\alpha})\geq\sigma(f)$.

  * (2)

If $(f_{\alpha})_{\alpha\in I}\subseteq S^{\prime}$ and $f\in S^{\prime}$,
then $\limsup_{\alpha}\tau(f_{n})\leq\tau(f)$.

  * (3)

If $(f_{\alpha})_{\alpha\in I}\subseteq S\cap S^{\prime}$ and $f\in S_{u}$,
then $\lim_{\alpha}\sigma(f_{\alpha})=\sigma(f)$ and
$\lim_{\alpha}\tau(f_{\alpha})=\tau(f)$. Note that $\sigma(f)=\tau(f)$,
because $f\in S_{u}$. Thus the smallest minimizer and the largest minimizer
converge to the same limit.

###### Proof.

By assumption, $f_{\alpha}\rightarrow f$ in $(C,\mathcal{T})$, which by the
requirement in (1) is the same as $f_{\alpha}\rightarrow f$ in
$(S,\mathcal{T}_{S})$. According to Corollary 1
$\sigma:(S,\mathcal{T}_{S})\rightarrow(\mathbb{R},\mathcal{O}_{>})$ is
continuous at every point $f\in S$. Consequently
$\sigma(f_{\alpha})\rightarrow\sigma(f)$ in $(\mathbb{R},\mathcal{O}_{>})$.
Now, a net $(y_{\alpha})$ converges in $(\mathbb{R},\mathcal{O}_{>})$ to $y$
if and only if $\liminf_{\alpha}y_{\alpha}\geq y$, which gives (1). In the
same way one obtains (2) upon noticing that $y_{\alpha}\rightarrow y$ in
$(\mathbb{R},\mathcal{O}_{<})$ if and only if $\limsup_{\alpha}y_{\alpha}\leq
y$. Finally, (3) follows from (1) and (2), because $\sigma\leq\tau$ and
therefore

$\sigma(f)\leq\liminf_{\alpha}\sigma(f_{\alpha})\leq\limsup_{\alpha}\sigma(f_{\alpha})\leq\limsup_{\alpha}\tau(f_{\alpha})\leq\tau(f)=\sigma(f).$

This shows that $\sigma(f_{\alpha})\rightarrow\sigma(f)$. Similarly

$\sigma(f)\leq\liminf_{\alpha}\sigma(f_{\alpha})\leq\liminf_{\alpha}\tau(f_{\alpha})\leq\limsup_{\alpha}\tau(f_{\alpha})\leq\tau(f)=\sigma(f)$

resulting in $\tau(f_{\alpha})\rightarrow\tau(f)$. ∎

Semi-continuity of $\sigma$ and $\tau$ as stated in Proposition 2 and its
reformulations in Corollaries 1 and 2 turn out to be a very strong tool for
proving so-called Argmin theorems in probability and statistics.

Occasionally it is stated in the literature that $\sigma$ or $\tau$ are
actually continuous with respect to the natural topology $\mathcal{O}_{n}$ on
$\mathbb{R}$. But the following examples shows that this is not true.

######  Example 1.

Consider

$f(t)=\left\\{\begin{array}[]{l@{\quad,\quad}l}0&|t|\leq 1\\\
|t|-1&|t|>1\end{array}\right.$

and for every $n\in\mathbb{N}$ let

$f_{n}(t)=\left\\{\begin{array}[]{l@{\quad,\quad}l}f(t)&t<0\text{ or
}t>1+\frac{1}{n}\\\ \frac{1}{n+1}t&t\in[0,1+\frac{1}{n}].\end{array}\right.$

Obviously, $f$ and $f_{n},n\in\mathbb{N}$ are convex and $f_{n}$ converges at
every point (actually uniformly on $\mathbb{R}$) to $f$. However,
$\tau(f_{n})=0$ for all $n\in\mathbb{N}$, whereas $\tau(f)=1$ and consequently
$\tau(f_{n})\not\rightarrow\tau(f)$. Thus $\tau$ is not continuous at $f$ and
from $\sigma(f)=-\tau(f_{-})$ we infer that $\sigma$ is not continuous at
$f_{-}$.

Note that the limit function in our example has no unique minimizing point. So
let us consider the family $S_{u}$ in (9) of all functions $f$ with a unique
minimizer. Clearly it holds that $S_{u}\subseteq S\cap S^{\prime}$ and that
the functionals $\sigma$ and $\tau$ coincide on $S_{u}$. Therefore, from
Remark 2 we can infer that $\sigma$ is lower- and upper-semicontinuous on the
subspace $(S_{u},\mathcal{T}_{S_{u}})$, whence $\sigma$ is continuous on
$(S_{u},\mathcal{T}_{S_{u}})$ with respect to the natural topology
$\mathcal{O}_{n}$ on $\mathbb{R}$. We note this in the following

######  Corollary 3.

$\sigma=\tau$ on $S_{u}$ and

$\sigma:(S_{u},\mathcal{T}_{S_{u}})\rightarrow(\mathbb{R},\mathcal{O}_{n})$

is continuous.

Let $S^{*}:=S\cap S^{\prime}=\\{f:\mathbb{R}\rightarrow\mathbb{R};f\text{
convex with }A(f)\text{ is a compact interval}\\}$ and let
$\xi:S^{*}\rightarrow\mathbb{R}$ be any measurable selection of $A$, i.e.,
$\xi(f)\in A(f)$ for every $f\in S^{*}$ and measurability refers to the trace
$\mathcal{C}_{S^{*}}=S^{*}\cap\mathcal{C}$. Assume that $(f_{\alpha})\subseteq
S^{*}$ converges pointwise to $f\in S^{*}$. Since
$\sigma(f_{\alpha})\leq\xi(f_{\alpha})\leq\tau(f_{\alpha})$ for all $\alpha\in
I$, it follows from the above Corollary 2 (and the characterization of net-
convergence in the order topologies, confer the proof of Corollary 2) that

$\xi(f_{\alpha})\rightarrow\sigma(f)\;\text{ in
}\;(\mathbb{R},\mathcal{O}_{>})\quad\text{and
}\;\xi(f_{\alpha})\rightarrow\tau(f)\;\text{ in
}\;(\mathbb{R},\mathcal{O}_{<}).$

If $f\in S_{u}$, then $\sigma(f)=\tau(f)$, whence
$\xi(f_{\alpha})\rightarrow\sigma(f)$ in $(\mathbb{R},\mathcal{O}_{>})$ and in
$(\mathbb{R},\mathcal{O}_{<})$, which, as we know, is the same as
$\xi(f_{\alpha})\rightarrow\sigma(f)=\tau(f)$ in the natural topology
$\mathcal{O}_{n}$. In particular, every measurable selection of $A$ is
continuous on the subspace $S_{u}$ with limit $\sigma=\tau$. We see here, and
will see it another time later, that the class $S_{u}$ of convex functions
with unique minimization point plays a special role.

######  Lemma 1.

$S_{u}\in\mathcal{C}_{S^{*}}$.

###### Proof.

Remark 2 says that $\sigma:(S^{*},\mathcal{C}_{S^{*}})\rightarrow\mathbb{R}$
and $\tau:(S^{*},\mathcal{C}_{S^{*}})\rightarrow\mathbb{R}$ are Borel
measurable. Infer from $S_{u}\subseteq S^{*}$ that $S_{u}=\\{f\in
S^{*}:\sigma(f)=\tau(f)\\}=(\sigma-\tau)^{-1}(\\{0\\})\in\mathcal{C}_{S^{*}}$.
∎

In addition to the topology $\mathcal{T}$ of pointwise convergence, let $C$
also be endowed with the topology $\mathcal{T}_{uc}$ of uniform convergence on
compacta. It is well-known that $\mathcal{T}\subseteq\mathcal{T}_{uc}$,
because uniform convergence on compacta implies pointwise convergence. From
Theorem 10.8 of Rockafellar [21] we know that on $C$ the reverse is true.
Notice that this is valid only for sequences. Thus the identity
$i:(C,\mathcal{T})\rightarrow(C,\mathcal{T}_{uc})$ is sequentially continuous
at every $f\in C$. Unfortunately, in general topological spaces sequential
continuity does not imply continuity, which in turn would give
$\mathcal{T}_{uc}=i^{-1}(\mathcal{T}_{uc})\subseteq\mathcal{T}$ as desired. In
fact the implication is true, if the space is first countable, confer Theorem
7.1.3 in Singh [22]. At this stage, however, we do not know whether first
countability holds for $(C,\mathcal{T})$. So, we will prove continuity of
$i:(C,\mathcal{T})\rightarrow(C,\mathcal{T}_{uc})$ traditionally via net-
convergence, confer Theorem 4.2.6 in Singh [22]. Theorem 3 below on net-
convergence is not only the key to success, but above all interesting in
itself when compared with the sequential convergence occurring in Theorem 10.8
of Rockafellar [21]. The proof of Theorem 3 is based on the following
inequality.

######  Lemma 2.

Let $D$ be a dense subset of $\mathbb{R}$. Then for every compact set
$K\subseteq\mathbb{R}$ there exist a constant $C$ and points
$d_{1},\ldots,d_{8}\in D$ such that for each convex function
$f:\mathbb{R}\rightarrow\mathbb{R}$ it follows:

$|f(s)-f(t)|\leq C\sum_{i=1}^{8}|f(d_{i})||s-t|\quad\forall\;s,t\in K.$ (10)

###### Proof.

First, find points $a$ and $b$ from $D$ such that $K\subseteq[a,b]$. By
Theorem 1.3.7 in Niculescu and Persson [20] we have that:

$|f(s)-f(t)|\leq L|s-t|\quad\forall\;s,t\in[a,b],$

where $L=\max\\{|D^{+}f(a)|,|D^{-}f(b)|\\}$. Now,

$D^{+}f(a)=\inf_{x>a}\frac{f(x)-f(a)}{x-a}\leq\frac{f(x)-f(a)}{x-a}\leq(x-a)^{-1}(|f(x)|+|f(a)|)\quad\forall\;x>a$
(11)

and since $D^{+}f(a)\geq D^{-}f(a)=\sup_{y<a}\frac{f(y)-f(a)}{y-a}$ it follows
that

$D^{+}f(a)\geq\frac{f(y)-f(a)}{y-a}\geq(y-a)^{-1}(|f(y)|+|f(a)|)\quad\forall\;y<a.$
(12)

Next, in (11) and in (12) we can choose the points $x$ and $y$ from $D$. Put
$C_{1}:=\max\\{(x-a)^{-1},(a-y)^{-1}\\}\in(0,\infty)$. Then $|D^{+}f(a)|\leq
C_{1}(|f(x)|+|f(a)|+|f(y)|+|f(a)|)$. Similarly one obtains: $|D^{-}f(b)|\leq
C_{2}(|f(u)|+|f(b)|+|f(v)|+|f(b)|)$ with positive constant $C_{2}$ and points
$u\in D,u<b$ and $v\in D,v>b$. Finally, if we put $C:=\max\\{C_{1},C_{2}\\}$,
then $L\leq C(|f(x)|+|f(a)|+|f(y)|+|f(a)|+|f(u)|+|f(b)|+|f(v)|+|f(b)|)$, which
shows (10). ∎

To state our next result recall that $(I,\leq)$ is a directed set.

######  Theorem 3.

Let $D$ be dense in $\mathbb{R}$ and let $(f_{\alpha})_{\alpha\in I}$ be a net
in $C$, which converges pointwise on $D$ to a function $f$, that is
$f_{\alpha}(t)\rightarrow f(t)$ for all $t\in D$. Then $f$ is convex and
$(f_{\alpha})$ converges uniformly to $f$ on every compact subset of
$\mathbb{R}$.

###### Proof.

Convexity of $f$ follows from the convexity of $f_{\alpha}$ by taking the
limit. As to the second assertion let $K\subseteq\mathbb{R}$ be compact. By
Lemma 2 there are a constant $C$ and $d_{1},\ldots,d_{8}\in\mathbb{R}$ such
that

$|f_{\alpha}(s)-f_{\alpha}(t)|\leq
C\sum_{i=1}^{8}|f_{\alpha}(d_{i})||s-t|\quad\forall\;s,t\in
K\quad\forall\;\alpha\in I.$ (13)

By pointwise convergence we find for every $1\leq i\leq 8$ an index
$\alpha_{i}\in I$ and a constant $c_{i}\in\mathbb{R}$ such that
$|f_{\alpha}(d_{i})|\leq c_{i}$ for all $\alpha\geq\alpha_{i}$. To
$\alpha_{1},\ldots,\alpha_{8}$ there exist a dominating index $\alpha^{*}\in
I$ with $\alpha^{*}\geq\alpha_{1},\ldots,\alpha_{8}$. It follows that
$C\sum_{i=1}^{8}|f_{\alpha}(d_{i})|\leq C\sum_{i=1}^{8}c_{i}=:L\in[0,\infty)$
for all $\alpha\geq\alpha^{*}$, whence by (13) the family
$\mathcal{F}:=\\{f_{\alpha}:\alpha^{*}\leq\alpha\in I\\}$ is equicontinuous.
Furthermore,

$|f_{\alpha}(t)|\leq|f_{\alpha}(t)-f_{\alpha}(d_{1})|+|f_{\alpha}(d_{1})|\leq
L|t-d_{1}|+c_{1}\quad\forall\alpha\geq\alpha^{*}\geq\alpha_{1},$

whence $\mathcal{F}$ is pointwise bounded. Thus by the Arzelà-Ascoli theorem,
confer, e.g., Heuser [16], the family $\mathcal{F}$ is compact. Therefore, if
$(f_{\alpha^{\prime}})$ is a subnet of $(f_{\alpha})_{\alpha^{*}\leq\alpha\in
I}$, then there exists a further subnet $(f_{\alpha^{\prime\prime}})$ of
$(f_{\alpha^{\prime}})$, which converges to a function $g$ uniformly on $K$.
In particular, $f_{\alpha^{\prime\prime}}(t)\rightarrow g(t)$ for all $t\in
K$. But by the assumption of pointwise convergence we also know that
$f_{\alpha^{\prime\prime}}(t)\rightarrow f(t)$ for all $t\in K$. Thus $g=f$ on
$K$ and by the subnet-criterion it follows that
$(f_{\alpha})_{\alpha^{*}\leq\alpha\in I}$ converges to $f$ uniformly on $K$,
which a fortiori holds for the entire net $(f_{\alpha})_{\alpha\in I}$. ∎

Infer from the above Theorem 3 that if a net $f_{\alpha}\rightarrow f$ in
$(C,\mathcal{T})$, then $f_{\alpha}\rightarrow f$ in $(C,\mathcal{T}_{uc})$.
Obviously, the reverse is true as well. As a consequence we obtain

######  Corollary 4.

The topology of pointwise convergence and the topology of uniform convergence
on compacta coincide on $C$:

$\mathcal{T}=\mathcal{T}_{uc}.$

######  Remark 3.

Let $\mathcal{T}(D)$ be the topology of pointwise convergence on $D$. It is
generated by the projections $\pi_{t},t\in D$. If $D$ is dense in
$\mathbb{R}$, then Theorem 3 actually yields that
$\mathcal{T}(D)=\mathcal{T}_{uc}=\mathcal{T}$. So all the topologies match.

The observation in Remark 3 leads to the following variant of the semi-
continuity.

######  Corollary 5.

Let $D$ be dense in $\mathbb{R}$. If in Corollary 2 the assumption is replaced
by $(f_{\alpha})_{\alpha\in I}\subseteq C$ converges pointwise to $f$ on D,
then all statements of Corollary 2 remain valid.

Let $D=\\{t_{i}:i\in\mathbb{N}\\}$ be a countable and dense subset of
$\mathbb{R}$. Introduce the special projection map
$H:C\rightarrow\mathbb{R}^{\mathbb{N}}$ by
$H(f):=(f(t_{i}))_{i\in\mathbb{N}}$. Note that $H$ depends on $D$, but we
suppress this in our notation. Equip $\mathbb{R}^{\mathbb{N}}$ with the
product topology $\Pi$. Denote the range $H(C)$ by $R$ and the relative
topology $R\cap\Pi$ by $\mathcal{R}$. With the following result we will prove
a functional limit theorem for convex stochastic processes.

######  Lemma 3.

The map $H$ is a bijection onto its range and its inverse
$H^{-1}:(R,\mathcal{R})\rightarrow(C,\mathcal{T})$ is continuous.

###### Proof.

If $H(f)=H(g)$, then $f=g$ on $D$ and by continuity and denseness of $D$ the
equality holds on the entire real line. Thus $H$ is injective, and it is
surjective by construction.

As to continuity of the inverse consider a sequence $(r_{n})$ with

$r_{n}\rightarrow r\;\text{ in }\;(R,\mathcal{R}).$ (14)

Since $(r_{n})\subseteq R$, we find to each $n\in\mathbb{N}$ a function
$f_{n}\in C$ such that $r_{n}=H(f_{n})=(f_{n}(t_{i}))_{i\in\mathbb{N}}$. For
the same reason there is some $f\in C$ with
$r=H(f)=(f(t_{i}))_{i\in\mathbb{N}}$. Recall that convergence in $\Pi$ or in
$\mathcal{R}$, respectively, is the same as coordinate-wise convergence. Thus
the convergence in (14) means that $f_{n}(t_{i})\rightarrow f(t_{i})$ for all
$i\in\mathbb{N}$. Since $D$ lies dense in $\mathbb{R}$ we can apply Theorem
10.8 in Rockafellar [21] (or our Theorem 3) to infer that actually
$f_{n}(t)\rightarrow f(t)$ for every $t\in\mathbb{R}$. Now, by definition
$f_{n}=H^{-1}(r_{n})$ and $f=H^{-1}(r)$, whence we arrive at
$H^{-1}(r_{n})\rightarrow H^{-1}(r)$ in $(C,\mathcal{T})$. Consequently,
$H^{-1}$ is continuous. ∎

##  3 Applications in probability and statistics

Let $(\Omega,\mathcal{A})$ be a measurable space. For a map
$Z:\Omega\rightarrow C$ we write $Z(\omega,t):=Z(\omega)(t)$ for the value of
the function $Z(\omega):\mathbb{R}\rightarrow\mathbb{R}$ (_trajectory_) at
point $t\in\mathbb{R}$. Very often it is more convenient to write $Z(t)$
instead of $Z(\omega,t)$ for this ambiguity in the notation explains in the
context. Let $\mathcal{B}:=\mathcal{B}(\mathbb{R})$ denote the Borel-$\sigma$
algebra on $\mathbb{R}$. If $Z(\cdot,t):\Omega\rightarrow\mathbb{R}$ is
$\mathcal{A}-\mathcal{B}$ measurable for each $t\in\mathbb{R}$, then $Z$ is
called a _convex stochastic process_. This is the same as saying that $Z(t)$
is a real random variable for all $t\in\mathbb{R}$. If $Z(\omega)\in U$ for
all $\omega\in\Omega$, where $U$ is a subset of $C$, we say that $Z$ is a
_convex stochastic process in_ $U$ or for short _a process in_ $U$. In other
words, all trajectories of $Z$ are $U-$valued.

Let $\mathcal{B}(C):=\sigma(\mathcal{T})$ be the Borel-$\sigma$ algebra
pertaining to the topology of pointwise convergence. By Corollary 4 it
coincides with $\mathcal{B}_{uc}(C):=\sigma(\mathcal{T}_{uc})$. The following
result yields a convenient characterization of the Borel-$\sigma$ algebra.
Recall that by definition $\mathcal{C}=\sigma(\pi_{t}:t\in\mathbb{R})$.

######  Proposition 4.

$\mathcal{B}(C)=\mathcal{B}_{uc}(C)=\mathcal{C}.$

###### Proof.

Let $C^{*}:=\\{f:\mathbb{R}\rightarrow\mathbb{R};f\text{ continuous}\\}$ be
endowed with the topology $\mathcal{T}_{uc}^{*}$ of uniform convergence on
compacta. One verifies easily that
$\mathcal{T}_{uc}=C\cap\mathcal{T}_{uc}^{*}$. If $i:C\rightarrow C^{*}$ is the
natural injection into $C^{*}$, i.e., $i(f)=f$, then
$\mathcal{T}_{uc}=i^{-1}(\mathcal{T}_{uc}^{*})$, whence

$\sigma(\mathcal{T}_{uc})=\sigma(i^{-1}(\mathcal{T}_{uc}^{*}))=i^{-1}(\sigma(\mathcal{T}_{uc}^{*}))=C\cap\sigma(\mathcal{T}_{uc}^{*}),$

where the second equality is ensured by Lemma 1.2.5 in Gänssler and Stute
[15]. By Lemma A5.1 in Kallenberg [18] we have that
$\sigma(\mathcal{T}_{uc}^{*})=\sigma(\pi_{t}^{*}:t\in\mathbb{R})$, where
$\pi_{t}^{*}:C^{*}\rightarrow\mathbb{R}$ is the projection on $C^{*}$. But
$C\cap\sigma(\pi_{t}^{*}:t\in\mathbb{R})=\sigma(\pi_{t}:t\in\mathbb{R})$,
which gives the desired result. ∎

If $Z$ is a process in $U\subseteq C$ it can be regarded as a map
$Z:\Omega\rightarrow U$ with Borel$-\sigma$ algebra
$\mathcal{B}(U)=\sigma(\mathcal{T}_{U})=U\cap\mathcal{B}(C)=U\cap\mathcal{C}=\mathcal{C}_{U}$
by Proposition 4.

######  Lemma 4.

Assume that $Z$ is a convex stochastic process. Then $Z$ is
$\mathcal{A}-\mathcal{B}(C)$ measurable. If actually $Z$ is a process in
$U\subseteq C$, then $Z$ is $\mathcal{A}-\mathcal{B}(U)$ measurable.

###### Proof.

The first assertion follows from Proposition 4, which enables us to apply
Proposition 1.2.11 in Gänssler and Stute [15]. The second assertion follows
from $\mathcal{B}(U)=U\cap\mathcal{B}(C)$ in combination with the first
assertion. ∎

Recall that $S^{*}=S\cap S^{\prime}$ and
$\xi:(S^{*},\mathcal{C}_{S^{*}})\rightarrow(\mathbb{R},\mathcal{B})$ denotes
any measurable selection of $A$.

######  Corollary 6.

If $Z$ is a convex stochastic process in $S$, in $S^{\prime}$ or in $S^{*}$,
then $\sigma(Z),\;\tau(Z)$ or $\xi(Z)$, respectively, are real random
variables.

###### Proof.

Lemma 4 says that $Z:(\Omega,\mathcal{A})\rightarrow(S,\mathcal{B}(S))$ is
measurable. But
$\mathcal{B}(S)=S\cap\mathcal{B}(C)=S\cap\mathcal{C}=\mathcal{C}_{S}$, where
the second equality holds by Proposition 4. From Proposition 2 we know that
$\sigma:(S,\mathcal{C}_{S})\rightarrow(\mathbb{R},\mathcal{B})$ is measurable,
whence $\sigma(Z)=\sigma\circ Z$ is measurable as composition of measurable
maps. Replacing $S$ through $S^{\prime}$ or $S^{*}$ gives measurability of
$\tau(Z)$ or $\xi(Z)$, respectively. ∎

The concept of _convergence in distribution_ is well-known for random
variables with values in a metric space. A classical reference here is the
book of Billingsley [1]. In contrast, the extension of the concept from metric
spaces to topological spaces seems less known. It goes back to Gänssler and
Stute [15], who in turn modify the ideas of Topsøe [23]. Let $Z$ and
$Z_{\alpha},\alpha\in I$, be random variables defined on a probability space
$(\Omega,\mathcal{A},\mathbb{P})$ with values in some topological space
$(X,\mathcal{O})$, that is $Z:\Omega\rightarrow X$ and
$Z_{\alpha}:\Omega\rightarrow X$ are $\mathcal{A}-\mathcal{B}(X)$ measurable,
where $\mathcal{B}(X):=\sigma(\mathcal{O})$ denotes the Borel-$\sigma$
algebra. Then the net $(Z_{\alpha})_{\alpha\in I}$ _converges in distribution
to_ $Z$ _in_ $(X,\mathcal{O})$, if

$\liminf_{\alpha}\mathbb{P}(Z_{\alpha}\in O)\geq\mathbb{P}(Z\in
O)\quad\forall\;O\in\mathcal{O}.$ (15)

This is denoted by
$Z_{\alpha}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}Z$ in
$(X,\mathcal{O})$ and by complementation is equivalent to

$\limsup_{\alpha}\mathbb{P}(Z_{\alpha}\in F)\leq\mathbb{P}(Z\in
F)\quad\forall\;F\in\mathcal{F},$ (16)

where $\mathcal{F}$ is the family of all closed sets in $(X,\mathcal{O})$.

The following result plays an important role in what follows. For this reason
we like to state it here. The proof is comparatively simple and can be found
in Gänssler and Stute [15], p.345.

######  Theorem 5.

(Continuous Mapping) Let $(X,\mathcal{O})$ and $(E,\mathcal{G})$ be
topological spaces, $h:X\rightarrow E$ be $\mathcal{B}(X)-\mathcal{B}(E)$
measurable and $D_{h}:=\\{x\in X:h\text{ is discontinuous at }x\\}$. Suppose
$Z$ and $Z_{\alpha},\alpha\in I$, are random variables over
$(\Omega,\mathcal{A},\mathbb{P})$ with values in $(X,\mathcal{O})$, where
$\mathbb{P}^{*}(Z\in D_{h})=0$ with $\mathbb{P}^{*}$ the outer measure of
$\mathbb{P}$. Then
$Z_{\alpha}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}Z$ in
$(X,\mathcal{O})$ entails
$h(Z_{\alpha})\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}h(Z)$ in
$(E,\mathcal{G})$.

Let $(Y,\mathcal{O}_{Y})$ with $Y\subseteq X$ and
$\mathcal{O}_{Y}:=Y\cap\mathcal{O}$ be a subspace of $(X,\mathcal{O})$. Notice
that $\mathcal{B}(Y)=\sigma(\mathcal{O}_{Y})=Y\cap\mathcal{B}(X)$. So, a map
$Z:\Omega\rightarrow X$ with range contained in $Y$ is
$\mathcal{A}-\mathcal{B}(Y)$ measurable (considered as a map into $Y$) if and
only if it is $\mathcal{A}-\mathcal{B}(X)$ measurable. Suppose $Z$ and
$Z_{\alpha}$ for all $\alpha\in I$ are random variables with values in the
subspace. Then
$Z_{\alpha}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}Z$ in
$(Y,\mathcal{O}_{Y})$ is equivalent to
$Z_{\alpha}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}Z$ in
$(X,\mathcal{O})$. Indeed, the natural injection
$i:(Y,\mathcal{O}_{Y})\rightarrow(X,\mathcal{O})$ given by $i(x)=x$ is
continuous. Thus the _Contiuous Mapping Theorem (CMT)_ shows sufficiency. The
necessity follows from (15) upon noticing that $\\{Z_{\alpha}\in Y\cap
O\\}=\\{Z_{\alpha}\in O\\}$ and $\\{Z\in Y\cap O\\}=\\{Z\in O\\}$ for all
$O\in\mathcal{O}$. We call this equivalence the _Subspace-lemma_.

For further properties including the _Portmanteau-Theorem_ we refer to chapter
8.4 in Gänssler and Stute [15].

Recall the left- and right-order topologies $\mathcal{O}_{<}$ and
$\mathcal{O}_{>}$, which are not metrizible. If a net $(x_{\alpha})_{\alpha\in
I}$ converges in $(\mathbb{R},\mathcal{O}_{<})$ and in
$(\mathbb{R},\mathcal{O}_{>})$, then it converges in the natural topology
$\mathcal{O}_{n}$, and vice versa. The following example shows that there is a
counterpart for distributional convergence.

######  Example 2.

$Z_{\alpha}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}Z\;\text{ in
}\;(\mathbb{R},\mathcal{O}_{>})\quad\Longleftrightarrow\quad\liminf_{\alpha}\mathbb{P}(Z_{\alpha}>x)\geq\mathbb{P}(Z>x)\quad\forall\;x\in\mathcal{R}.$
(17) $Z_{\alpha}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}Z\;\text{
in
}\;(\mathbb{R},\mathcal{O}_{<})\quad\Longleftrightarrow\quad\liminf_{\alpha}\mathbb{P}(Z_{\alpha}<x)\geq\mathbb{P}(Z<x)\quad\forall\;x\in\mathcal{R}.$
(18) $Z_{\alpha}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}Z\;\text{
in }\;(\mathbb{R},\mathcal{O}_{>})\text{ and in
}\;(\mathbb{R},\mathcal{O}_{<})\quad\Longleftrightarrow\quad\;Z_{\alpha}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}Z\;\text{
in }\;(\mathbb{R},\mathcal{O}_{n})$ (19)

Here, (17) and (18) are immediate consequences of the definitions. In (19) the
sufficiency of the right side follows from
$\mathcal{O}_{n}\supseteq\mathcal{O}_{<}$ and
$\mathcal{O}_{n}\supseteq\mathcal{O}_{>}$. To see necessity let
$x\in\mathbb{R}$. Then we obtain:

$\displaystyle\mathbb{P}(Z<x)$ $\displaystyle\leq$
$\displaystyle\liminf_{\alpha}\mathbb{P}(Z_{\alpha}<x)\quad\hskip
56.9055pt\text{ by }(\ref{right})$ $\displaystyle\leq$
$\displaystyle\limsup_{\alpha}\mathbb{P}(Z_{\alpha}<x)$ $\displaystyle\leq$
$\displaystyle\limsup_{\alpha}\mathbb{P}(Z_{\alpha}\leq x)\leq\mathbb{P}(Z\leq
x)\quad\text{ by }(\ref{left})\text{ and complementation}.$

Thus, if the distribution function of $Z$ is continuous at $x$, i.e.,
$\mathbb{P}(Z<x)=\mathbb{P}(Z\leq x)$ it follows that
$\lim_{\mathbb{\alpha}}\mathbb{P}(Z_{\alpha}\leq x)=\mathbb{P}(Z\leq x)$ as
required.

Deduce from (17): If $Z_{\alpha}\leq Z_{\alpha}^{*}\;\mathbb{P}-$almost surely
(a.s.) for all $\alpha\geq\alpha_{0}\in I$ or if $Z\geq Z^{*}$ a.s., then
$Z_{\alpha}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}Z\;\text{ in
}\;(\mathbb{R},\mathcal{O}_{>})$ entails
$Z_{\alpha}^{*}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}Z^{*}\;\text{
in }\;(\mathbb{R},\mathcal{O}_{>})$. (property 1)

Deduce from (18): If $Z_{\alpha}\geq Z_{\alpha}^{*}$ (a.s.) for all
$\alpha\geq\alpha_{0}\in I$ or if $Z\leq Z^{*}$ a.s., then
$Z_{\alpha}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}Z\;\text{ in
}\;(\mathbb{R},\mathcal{O}_{<})$ entails
$Z_{\alpha}^{*}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}Z^{*}\;\text{
in }\;(\mathbb{R},\mathcal{O}_{<})$. (property 2)

In particular, this shows that the limit variables are not unique.

Our next result characterizes distributional convergence for random variables
with values in the function space $(C,\mathcal{T})$. Below the euclidian space
$\mathbb{R}^{k}$ is endowed with the product-topology $\mathcal{O}_{n}^{k}$.

######  Proposition 6.

(Functional limits) Let $Z$ and $Z_{\alpha},\alpha\in I$, be convex stochastic
processes. Then they are random variables in $(C,\mathcal{T})$ and the
following statements (20) and (21) are equivalent:

$Z_{\alpha}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}Z\;\text{ in
}\;(C,\mathcal{T})$ (20)

$(Z_{\alpha}(t_{1}),\ldots,Z_{\alpha}(t_{k}))\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}(Z(t_{1}),\ldots,Z(t_{k}))\;\text{
in }(\mathbb{R}^{k},\mathcal{O}_{n}^{k})$ (21)

for every $k\in\mathbb{N}$ and for each collection of points
$t_{1},\ldots,t_{k}\in D$, where $D$ is a countable and dense subset of
$\mathbb{R}$.

###### Proof.

The first assertion holds by Lemma 4. Assume (20) holds. By definition of
$\mathcal{T}$ every projection
$\pi_{t}:(C,\mathcal{T})\rightarrow(\mathbb{R},\mathcal{O}_{n})$ is
continuous, whence the product map
$\pi:=(\pi_{t_{1}},\ldots,\pi_{t_{k}}):(C,\mathcal{T})\rightarrow(\mathbb{R}^{k},\mathcal{O}_{n}^{k})$
is continuous as well. Since
$(Z_{\alpha}(t_{1}),\ldots,Z_{\alpha}(t_{k}))=\pi(Z_{\alpha})$ and
$(Z(t_{1}),\ldots,Z(t_{k}))=\pi(Z)$, an application of the CMT yields (21).

For the converse first note that by countability we have that
$D=\\{t_{1},t_{2},\ldots\\}$. Recall the projection map $H$ given in Lemma 3.
It follows from Example 2.6 in Billingsley [2] that (21) entails that
$H(Z_{\alpha})\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}H(Z)$ in
$(\mathbb{R}^{\mathbb{N}},\Pi)$. Now, the Subspace-lemma says that
$H(Z_{\alpha})\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}H(Z)$ in
$(R,\mathcal{R})$. By Lemma 3 the inverse
$H^{-1}:(R,\mathcal{R})\rightarrow(C,\mathcal{T})$ is continuous, so that
another application of the CMT yields (20). ∎

The second statement (21) is known as _convergence of the finite dimensional
distributions (on D)_ and denoted by $Z_{\alpha}\stackrel{{\scriptstyle
fd}}{{\longrightarrow}}_{D}Z$ (in short: _convergence of the fidis_).

We are now in the position to formulate several so-called _Argmin-Theorems_
for convex stochastic processes.

######  Theorem 7.

Let $D$ be countable and dense in $\mathbb{R}$. Consider convex stochastic
processes $Z$ and $Z_{\alpha},\alpha\in I$, in $U$. Suppose that
$Z_{\alpha}\stackrel{{\scriptstyle fd}}{{\longrightarrow}}_{D}Z$. Then the
following statements hold:

  * (1)

If $U=S$, then
$\sigma(Z_{\alpha})\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}\sigma(Z)$
in $(\mathbb{R},\mathcal{O}_{>})$.
If in addition $Z$ is a process in $S^{*}$ with $Z\in S_{u}$ a.s., then
$\sigma(Z_{\alpha})\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}\sigma(Z)$
in $(\mathbb{R},\mathcal{O}_{n})$.

  * (2)

If $U=S^{\prime}$, then
$\tau(Z_{\alpha})\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}\tau(Z)$
in $(\mathbb{R},\mathcal{O}_{<})$.
If in addition $Z$ is a process in $S^{*}$ with $Z\in S_{u}$ a.s., then
$\tau(Z_{\alpha})\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}\tau(Z)$
in $(\mathbb{R},\mathcal{O}_{n})$.

  * (3)

If $U=S^{*}$, then
$\xi(Z_{\alpha})\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}\sigma(Z)$
in $(\mathbb{R},\mathcal{O}_{>})$ and
$\xi(Z_{\alpha})\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}\tau(Z)$
in $(\mathbb{R},\mathcal{O}_{<})$.
If in addition $\sigma(Z)\stackrel{{\scriptstyle\mathcal{D}}}{{=}}\tau(Z)$,
then
$\xi(Z_{\alpha})\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}\sigma(Z)$
in $(\mathbb{R},\mathcal{O}_{n})$.

###### Proof.

First notice that by Corollary 6 all involved maps are real random variables.
By Proposition 6
$Z_{\alpha}\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}Z$ in
$(C,\mathcal{T})$, whence by the Subspace-lemma

$Z_{\alpha}\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}Z\text{ in
}(S,\mathcal{T}_{S}).$ (22)

From Corollary 1 we know that
$\sigma:(S,\mathcal{T}_{S})\rightarrow(\mathbb{R},\mathcal{O}_{>})$ is
continuous and consequently the CMT yields the first convergence in (1).

By Lemma 1 $S_{u}\in\mathcal{C}_{S^{*}}$. According to Lemma 4 $Z$ is
$\mathcal{A}-\mathcal{C}_{S^{*}}$ measurable upon noticing that
$\mathcal{B}(S^{*})=S^{*}\cap\mathcal{B}(C)=S^{*}\cap\mathcal{C}=\mathcal{C}_{S^{*}}$
by Proposition 4. Thus $\\{Z\in S_{u}\\}\in\mathcal{A}$, whence by Corollary 3
it follows that $0\leq\mathbb{P}^{*}(Z\in
D_{\sigma})\leq\mathbb{P}^{*}(Z\notin S_{u})=\mathbb{P}(Z\notin S_{u})=0$.
Finally, by Propositions 2 and 4
$\sigma:(S,\mathcal{T}_{S})\rightarrow(\mathbb{R},\mathcal{O}_{n})$ is
$\mathcal{B}(S)-\mathcal{B}(\mathbb{R})$ measurable. Consequently by (22)
another application of the CMT gives the second convergence in (1).

The second part (2) follows exactly the same way. Regarding part (3) it should
be noted that $S^{*}\subseteq S$ and $S^{*}\subseteq S^{\prime}$. Therefore
$\sigma(Z_{\alpha})\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}\sigma(Z)$
in $(\mathbb{R},\mathcal{O}_{>})$ by (1) and
$\tau(Z_{\alpha})\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}\tau(Z)$
in $(\mathbb{R},\mathcal{O}_{<})$ by (2). This shows the first assertions in
part (3), because $\sigma(Z_{\alpha})\leq\xi(Z_{\alpha})\leq\tau(Z_{\alpha})$
for all $\alpha\in I$ and so we can use property 1 and property 2 from Example
2. Finally, the second assertion in (3) follows from (19) in Example 2. ∎

Our Argmin-Theorems (1)-(3) involve two types of uniqueness assumptions. In
(1) and (2) the requirement $Z\in S_{u}$ a.s. is the same as
$\sigma(Z)=\tau(Z)$ a.s., which in turn implies
$\sigma(Z)\stackrel{{\scriptstyle\mathcal{D}}}{{=}}\tau(Z)$ as in part (3).
However, if $\sigma(Z)$ and $\tau(Z)$ are $\mathbb{P}-$integrable, then the
reverse implication holds. Indeed, in this case by linearity
$\mathbb{E}[\tau(Z)-\sigma(Z)]=0$. But since the integrand is non-negative, it
must be equal to zero a.s.

A comparison shows that on the one hand in (3) the uniqueness condition is
weaker than in (1) or in (2), on the other hand in (3) the demands on the
stochastic processes are more stringent.

In the following we give some interesting equivalent characterizations for
almost sure uniqueness of the minimizing point.

######  Proposition 8.

Suppose $Z$ is a convex stochastic process in $S^{*}$. Then the following
three statements are equivalent:

  * (1)

$\sigma(Z)=\tau(Z)$ a.s.

  * (2)

$x\notin A(Z)$ a.s. for Lebesgue-almost every $x\in\mathbb{R}$.

  * (3)

$0\notin[D^{-}Z(x),D^{+}Z(x)]$ a.s. for Lebesgue-almost every
$x\in\mathbb{R}$.

###### Proof.

We write briefly $\sigma=\sigma(Z)$ and $\tau=\tau(Z)$. Moreover, let
$\lambda$ denote the Lebegue-measure on $\mathbb{R}$. Note that
$\tau-\sigma\geq 0$, whence by Corollary 6 the expectation exists and is equal
to:

$\displaystyle\mathbb{E}[\tau-\sigma]$ $\displaystyle=$
$\displaystyle\mathbb{E}[\lambda([\sigma,\tau])]=\int_{\Omega}\int_{\mathbb{R}}1_{\\{\sigma\leq
x,\tau\geq x\\}}\lambda(dx)\mathbb{P}(d\omega)$ $\displaystyle=$
$\displaystyle\int_{\Omega}\int_{\mathbb{R}}1_{\\{D^{-}Z(x)\leq 0\leq
D^{+}Z(x)\\}}\lambda(dx)\mathbb{P}(d\omega)\quad\text{by Theorem
\ref{location}}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\mathbb{P}(D^{-}Z(x)\leq 0\leq
D^{+}Z(x))\lambda(dx)\hskip 21.33955pt\text{by Fubini}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\mathbb{P}(x\in A(Z))\lambda(dx)\hskip
85.35826pt\text{by }(\ref{minset})$

Observe that all occurring integrands are non-negative. Thus the equivalence
can be deduced from a well-known result from integration theory, confer, e.g.,
Lemma 1.15, p. 304 in Dshalalow [4]. ∎

If $Z$ in the above proposition is in addition differentiable, then
$\sigma(Z)=\tau(Z)$ a.s. if and only if $Z^{\prime}(x)\neq 0$ a.s. for
$\lambda-$ every $x\in\mathbb{R}$.

Next, we establish an Argmin-Theorem for almost sure convergence. Just as in
Theorem 7, there are also convergence statements here for the non-unique case.
This extends results known so far, confer Theorem 7.77 in Liese and Mieschke
[19], which in turn rely on the unpublished preprint of Hjort and Pollard
[12].

######  Theorem 9.

Let $Z$ and $Z_{\alpha},\alpha\in I$, be convex stochastic processes in $U$
defined on a complete probability space $(\Omega,\mathcal{A},\mathbb{P})$.
Assume that $Z_{\alpha}(t)\rightarrow Z(t)$ a.s. for every $t\in D$ with $D$ a
countable and dense subset of $\mathbb{R}$.

  * (1)

If $U=S$, then $\liminf_{\alpha}\sigma(Z_{\alpha})\geq\sigma(Z)$ a.s.

  * (2)

If $U=S^{\prime}$, then $\limsup_{\alpha}\tau(Z_{\alpha})\leq\tau(Z)$ a.s.

  * (3)

If $U=S^{*}$ and $Z\in S_{u}$ a.s., then
$\lim_{\alpha}\sigma(Z_{\alpha})=\sigma(Z)=\tau(Z)$ a.s. and
$\lim_{\alpha}\tau(Z_{\alpha})=\tau(Z)=\sigma(Z)$ a.s.

  * (4)

If $U=S^{*}$, then $\liminf_{\alpha}\xi(Z_{\alpha})\geq\sigma(Z)$ a.s. and
$\limsup_{\alpha}\xi(Z_{\alpha})\leq\tau(Z)$ a.s. If in addition $Z\in S_{u}$
a.s., then $\lim_{\alpha}\xi(Z_{\alpha})=\sigma(Z)=\tau(Z)$ a.s.

###### Proof.

Put $\Omega_{t}:=\\{Z_{\alpha}(t)\rightarrow Z(t)\\},t\in D$. Then
$\Omega_{t}\in\mathcal{A}$ and $\mathbb{P}(\Omega_{t})=1$ by assumption and by
completeness. Thus $\Omega_{0}:=\bigcap_{t\in D}\Omega_{t}\in\mathcal{A}$ and
$\mathbb{P}(\Omega_{0})=1$, because $D$ is countable. Corollary 5 yields that
$\Omega_{0}\subseteq\\{\liminf_{\alpha}\sigma(Z_{\alpha})\geq\sigma(Z)\\}=:\Omega_{1}$.
It follows from completeness that $\Omega_{1}\in\mathcal{A}$, and by
monotonicity of $\mathbb{P}$ we arrive at $\mathbb{P}(\Omega_{1})=1$, which
shows (1). In the same way one obtains (2). Furthermore, Corollary 5 ensures
that $\Omega_{0}\cap\\{Z\in S_{u}\\}$ is a subset of
$\\{\lim_{\alpha}\sigma(Z_{\alpha})=\sigma(Z)\\}$ and of
$\\{\lim_{\alpha}\tau(Z_{\alpha})=\tau(Z)\\}$. Again by completeness this
gives (3). Since $S^{*}\subseteq S$ and $S^{*}\subseteq S^{\prime}$ as well as
$\sigma(Z_{\alpha})\leq\xi(Z_{\alpha})\leq\tau(Z_{\alpha})$ the first part of
(4) follows from (1) and (2). Finally, the second part of (4) follows from (3)
in combination with the sandwich theorem. ∎

######  Remark 4.

If $(Z_{\alpha})_{\alpha\in\mathbb{N}}$ in Theorem 9 in fact is a sequence,
then we can drop the completeness assumption about the underlying probability
space $(\Omega,\mathcal{A},\mathbb{P})$. The reason for this is Corollary 6,
which guarantees that the sets $\Omega_{t},\Omega_{0}$ and $\Omega_{1}$ are
elements of $\mathcal{A}$.

The Argmin-Theorem for convergence in probability takes the form as described
below. The reader recognises that it is only formulated for sequences and not
more generally for nets. This is because we carry out the proof via the
subsequence criterion. To the best of our knowledge, there is no counterpart
for nets.

######  Theorem 10.

Let $Z_{n},n\in\mathbb{N}$, be convex stochastic processes in $U$ and let
$Z\in S^{*}$ have an almost surely unique minimizer. Suppose
$Z_{n}(t)\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}Z(t)$ for each
$t\in D$, where $D$ is a countable and dense subset of $\mathbb{R}$.

  * (1)

If $U=S$, then
$\sigma(Z_{n})\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}\sigma(Z)\stackrel{{\scriptstyle
a.s.}}{{=}}\tau(Z)$.

  * (2)

If $U=S^{\prime}$, then
$\tau(Z_{n})\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}\tau(Z)\stackrel{{\scriptstyle
a.s.}}{{=}}\sigma(Z)$.

  * (3)

If $U=S^{*}$, then
$\xi(Z_{n})\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}\sigma(Z)\stackrel{{\scriptstyle
a.s.}}{{=}}\tau(Z)$

###### Proof.

We will use the subsequence criterion, confer, e.g., Lemma 5.2 in Kallenberg
[17]. So, let $(n_{0k})_{k\in\mathbb{N}}$ be a subsequence of $\mathbb{N}$. As
a countable set $D$ has the form $D=\\{t_{1},t_{2},\ldots\\}$. From
$Z_{n}(t_{1})\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}Z(t_{1})$ it
follows with the subsequence criterion that there exists a subsequence
$(n_{1i})_{i\in\mathbb{N}}$ of $(n_{0k})_{k\in\mathbb{N}}$ such that on a set,
$\Omega_{1}$ say, with probability one we have that
$Z_{n_{1i}}(t_{1})\rightarrow Z(t_{1}),i\rightarrow\infty$. Another
application of the subsequence criterion ensures that there exists a
subsequence $(n_{2i})_{i\in\mathbb{N}}$ of $(n_{1i})_{i\in\mathbb{N}}$ such
$Z_{n_{2i}}(t_{2})\rightarrow Z(t_{2}),i\rightarrow\infty$, on a set,
$\Omega_{2}$ say, which has probability one. Continuing in this way we find
for every $j\geq 1$ a subsequence $(n_{ji})_{i\in\mathbb{N}}$ of
$(n_{j-1,i})_{i\in\mathbb{N}}$ and a set $\Omega_{j}$ with
$\mathbb{P}(\Omega_{j})=1$ such that $Z_{n_{ji}}(t_{j})\rightarrow
Z(t_{j}),i\rightarrow\infty$, on $\Omega_{j}$. Set $\Omega_{0}:=\bigcap_{j\geq
1}\Omega_{j}$. Observe that $\mathbb{P}(\Omega_{0})=1$. The ”diagonal”
$(n_{ii})_{i\in\mathbb{N}}$ is a subsequence of the given sequence
$(n_{0k})_{k\in\mathbb{N}}$. Consider an arbitrary index $j\geq 1$. Apart from
the first $j-1$ terms the sequence $(n_{ii})_{i\in\mathbb{N}}$ is a
subsequence of $(n_{ji})_{i\in\mathbb{N}}$. But on $\Omega_{0}$, along that
sequence we have convergence at point $t_{j}$, whence in particular
$Z_{n_{ii}}(t_{j})\rightarrow Z(t_{j}),i\rightarrow\infty$. Thus we arrive at
$Z_{n_{ii}}(t)\rightarrow Z(t),i\rightarrow\infty$, for all $t\in D$ a.s.
Especially, the sequence $(Z_{n_{ii}})_{i\in\mathbb{N}}$ fulfils the
requirements of Theorem 9, which yields that
$\sigma(Z_{n_{ii}})\rightarrow\sigma(Z)$ a.s. or that
$\tau(Z_{n_{ii}})\rightarrow\tau(Z)$ a.s., respectively, and another
application of the subsequence criterion gives (1) and (2). The assertion (3)
follows from Theorem 9 part (4). ∎

Concluding remarks

Argmin theorems for convex processes (and distributional convergence) are
known since Davis, Knight and Liu (1992) [3], Hjort and Pollard (1993) [12],
Geyer (1996) [11] or Knight (2002) [14]. Here the processes can even be
defined on $\mathbb{R}^{d}$ and not merely on $\mathbb{R}$ as in this paper.
Apart from that, however, there are two notable differences. First, the
publications mentioned above only consider sequences, whereas we more
generally allow for nets of processes. Secondly, a further main difference is
that we also provide results when the limit process does not have a unique
minimizing point. Incidentally, both points also apply to the Argmin theorems
for almost sure convergence. The idea behind this is to understand semi-
continuity of $\sigma$ or $\tau$ as continuity with respect to the order
topologies in place of the natural topology on $\mathbb{R}$. After that the
Continuous Mapping Theorem does the rest. Another way of looking at our
approach is this: As long as a unique minimizer of the limit process exists,
it is a natural candidate for the limit variable. If this is not the case, we
do not search for new candidates, but simply make the topology on $\mathbb{R}$
smaller. This also differs from Ferger’s [7] innovative approach, which
retains the natural topology but more generally allows Choquet-capacities to
play the part of limit ”distributions”. The applicability of the Argmin
theorem for convex processes lies in the fact that here, in contrast to such
processes in larger function spaces, the only prerequisite is that of
convergence of the finite dimensional distributions, confer Ferger [9] for an
application in $M-$estimation. For example let
$C^{*}=\\{f:\mathbb{R}\rightarrow\mathbb{R};f\text{ is continuous}\\}\supset
C$ be endowed with the topology $\mathcal{T}_{uc}^{*}$ of uniform convergence
on compacta. Here, even if you only consider sequences, one needs not only a
functional limit theorem
$Z_{n}\stackrel{{\scriptstyle\mathcal{D}}}{{\mathcal{\rightarrow}}}Z$ in
$(C^{*},\mathcal{T}_{uc}^{*})$, but also stochastic boundedness of the
$\xi(Z_{n})$:

$\lim_{d\rightarrow\infty}\limsup_{n\rightarrow\infty}\mathbb{P}(|\xi(Z_{n})|>d)=0,$
(23)

see Ibragimov and Hasminski (1981) [13], van der Vaart and Wellner (1996) [24]
or Ferger (2015) [8]. For the proof of the functional limit theorem alone,
besides convergence of the fidis also tightness of the sequence $(Z_{n})$ is
required, which is usually done through maximal inequalities. Not to forget
the proof of (23), usually by upper estimates for the tail probabilities. This
means that the programme that has to be worked through is much more extensive
and demanding than in the convex case.

There are two answers to the question why this is so. Firstly, by Proposition
6 it is already true under the sole assumption that the fidis converge, that
then even a functional limit theorem applies. Secondly, there is no
counterpart of Corollary 3, which says that $\sigma$ is continuous on $S_{u}$.
For example, let us consider $(C^{*},\mathcal{T}_{uc}^{*})$ and let
$\sigma^{*}(f)$ be the smallest minimizing point of $f$ (existence assumed).
Then one can construct a sequence $(f_{n})$ such that $f_{n}$ converges to $f$
uniformly on every compact $K\subseteq\mathbb{R}$, i.e., $f_{n}\rightarrow f$
in $(C^{*},\mathcal{T}_{uc}^{*})$, but $\sigma^{*}(f_{n})\rightarrow-\infty$.
In particular, $\sigma^{*}$ is far from being continuous on $S_{u}$.

Declarations

Compliance with Ethical Standards: I have read and I understand the provided
information.

Competing Interests: The author has no competing interests to declare that are
relevant to the content of this article.

## References

  * [1] P. Billingsley, Convergence of Probability Measures, New York: John Wiley & Sons, 1968.
  * [2] P. Billingsley, Convergence of Probability Measures, Second Edition, New York: John Wiley & Sons, 1999.
  * [3] R. A. Davis, K. Knight, and J. Liu, M-estimation for autoregressions with innite variance, Stochastic Process. Appl. 40 (1992), 145–180.
  * [4] J. H. Dshalalow, Real Analysis, Boca Raton, London, New York, Washington, D.C.: Chapman & Hall/CRC, 2001.
  * [5] P. Embrechts and M. Hofert, A note on generalized inverses, Mathematical Methods of Operations Research 77 (2013), 423–432.
  * [6] C. Feng, H. Wang, X. M. Tu and J. Kowalski, A note on generalized inverses of distribution function and quantile transformation, Applied Mathematics 3 (2012), 2098–2100.
  * [7] D. Ferger, A Continuous Mapping Theorem for the argmin-set functional with applications to convex stochastic processes, Kybernetika 57 (2021), 426–445.
  * [8] D. Ferger, Arginf-sets of multivariate cadlag processes and their distributional convergence in hyperspace topologies, Theory of Stochastic Processes 20(36), No.2 (2015), 13–41.
  * [9] D. Ferger, Distributional hyperspace-convergence of Argmin-sets in convex M-estimation, Theor. Probability and Math. Statist. (2023), in press.
  * [10] A. de La Fortelle, A study on generalized inverses and increasing functions Part I: generalized inverses, hal-01255512 (2016), 1–14.
  * [11] J. Geyer, On the asymptotics of convex stochastic optimization, Unpublished manuscript (1996).
  * [12] N. L. Hjort and D. Pollard, Asymptotic for minimizers of convex processes, Preprint, Dept. of Statistics, Yale University (1993). arxiv:1107.3806v1
  * [13] I. A. Ibragimov and R. Z. Has’minskii, Statistical Estimation: Asymptotic Theory, New York: Springer-Verlag, 1981.
  * [14] K. Knight, What are the limiting distributions of quantile estimators?, In Statistical Data Analysis Based on the $L_{1}$-Norm and Related Methods (Y. Dodge, ed.) 47–65. Series Statistics for Industry and Technology. Basel: Birkhäuser, 2002.
  * [15] P. Gänssler and W. Stute, Wahrscheinlichkeitstheorie, Berlin, Heidelberg, Germany: Springer-Verlag, 1977.
  * [16] H. Heuser, Lehrbuch der Analysis. Teil 1, 14-th Edition, Wiesbaden, Germany: B.G. Teubner Verlag, 2004.
  * [17] O. Kallenberg, Foundations of Modern Probability, Volume 1, Third Edition, Springer Nature Switzerland AG, 2021.
  * [18] O. Kallenberg, Foundations of Modern Probability, Volume 2, Third Edition, Springer Nature Switzerland AG, 2021.
  * [19] F. Liese and K.-J. Mieschke, Statistical Decision Theory, New York: Springer Science+Business Media, LLC, 2008.
  * [20] 2006 C. P. Niculescu and L.-E. Persson, Convex Functions and Their Applications: A Contemporary Approach, New York: Springer Science+Business Media, Inc., 2006.
  * [21] R. T. Rockafellar, Convex Analysis, Princeton, New Jersey: Princeton University Press, 1970.
  * [22] T. B. Singh, Introduction to Topology, Singapore: Springer Nature, 2019.
  * [23] F. Topsøe, Topology and Measure, Lecture Notes in Mathematics Vol. 133, Berlin-Heidelberg-New York: Springer-Verlag, 1970.
  * [24] A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes. With Applications to Statistics, New York: Springer-Verlag, 1996.