# REFINED DISTRIBUTIONAL LIMIT THEOREMS
FOR COMPOUND SUMS

Vsevolod K. Malinovskii  Central Economics and Mathematics Institute (CEMI) of
Russian Academy of Science, 117418, Nakhimovskiy prosp., 47, Moscow, Russia
<EMAIL_ADDRESS>http://www.actlab.ru

###### Abstract.

The paper is a sketch of systematic presentation of distributional limit
theorems and their refinements for compound sums. When analyzing, e.g.,
ergodic semi–Markov systems with discrete or continuous time, this allows us
to separate those aspects that lie within the theory of random processes from
those that relate to the classical summation theory. All these limit theorems
are united by a common approach to their proof, based on the total probability
rule, auxiliary multidimensional limit theorems for sums of independent random
vectors, and (optionally) modular analysis.

###### Key words and phrases:

Compound sums, Clustering–type and queuing–type models, Distributional limit
theorems, Refined approximations, Modular analysis, Renewal theory, Risk
theory, ergodic Markov and semi–Markov systems.

##  1\. Introduction

In his review [46] of Petrov’s book [80], Kesten111Harry Kesten (1931–2019),
the Goldwin Smith Professor Emeritus of Mathematics, “whose insights advanced
the modern understanding of probability theory and its applications” (quoted
from obituary by Matt Hayes). wrote that “in the period 1920–1940 research in
probability theory was virtually synonymous with the study of sums of
independent random variables”, but “already in the late thirties attention
started shifting and probabilists became more interested in Markov chains,
continuous time processes, and other situations in which the independence
between summands no longer applies, and today the center of gravity of
probability theory has moved away from sums of independent random variables.”

Nevertheless, Kesten continues, “much work on sums of independent random
variables continues to be done, partly for their aesthetic appeal and partly
for the technical reason that many limit theorems, even for dependent
summands, can be reduced to the case of independent summands by means of
various tricks. The best known of these tricks is the use of regeneration
points, or ‘Dœblin’s trick’. In large part the fascination of the subject is
due to the fact that applications and the ingenuity of mathematicians continue
to give rise to new questions.” Regarding applications, Kesten mentioned
renewal theory, the theory of optimal stopping, invariance principles and
functional limit theorems, fluctuation theory, and so on.

To summarize, Kesten argues that “in all the years that these newer phenomena
were being discovered, people also continued working on more direct
generalizations and refinements of the classical limit theorems $\dots$ Many
of the proofs require tremendous skill in classical analysis $\dots$ and many
of the results, such as the Berry–Esseen estimate, the Edgeworth expansion and
asymptotic results for large deviations, are important for statistics and
theoretical purposes.”

The paper is an overview that fits into the scheme outlined by Kesten. It is
focussed on refined “distributional” (or “intrinsically analytic”222The terms
“intrinsically analytic” and “measure–theoretic” applied to limit theorems are
introduced in [19]. The former is equivalent (see, e.g., [12]) to
“distributional”, or “weak”., or “weak”) limit theorems for compound sums
which are a natural extension of ordinary sums. They are known as the key
object of renewal theory and the core component of Dœblin’s dissection of an
ordinary sum defined on a Markov chain. By “refined” limit theorems we
mean333The results for large deviations, although interesting and available,
are not presented here due to the space limitation. Berry–Esseen’s estimates
and Edgeworth’s expansions mentioned by Kesten, in contrast to approximations
without assessing their accuracy.

In contrast to “measure–theoretic” limit theorems, for which Kolmogorov’s
axiomatization is indispensable and which border on (or lie within) the theory
of random processes, “distributional” limit theorems, including those for
dependent random variables, are known long before the axiomatization. For
example, seeking in the 1900s, long before Kolmogorov’s advance, to
demonstrate that independence is not a necessary condition in “distributional”
weak law of large numbers (WLLN) and central limit theorem (CLT), Markov
introduced the Markov chains.

Without questioning the value of “measure–theoretic” limit theorems, such as
the strong law of large numbers (SLLN) and the law of iterated logarithm
(LIL), or, going further, the value of the theory of random processes, we note
that “distributional” rather than “measure–theoretic” limit theorems are
really needed for statistics, cluster analysis, etc., where an infinite number
of observations, clusters, renewals, people in queue, and so on, never occur.
Even in Bernoulli’s classical scheme, statistical inference is based (see
detailed discussion in [18]) on WLLN and CLT with its refinements, and not on
SLLN and LIL.

Dealing with complex compound sums444See classification of compound sums
below. Modular analysis is not needed for simple compound sums., we focus on
modular analysis. Speaking of “Dœblin’s trick” which reduces “many limit
theorems, even for dependent summands” to the case of independent ones, Kesten
meant this approach. Bearing in mind the ladder technique, say “Blackwell’s
trick” (see [13]), the modular analysis is even more widely used. We will
expand it to compound sums with general modular structure, but note that (see,
e.g., [39], [42], [65], [83], [87]) this is merely one, rather than the only
and the best, method for studying such sums.

Technically, “Dœblin’s trick” (or its counterpart “Blackwell’s trick”) is not
only the use of regeneration (or ladder) points. It implies (see, e.g., [20])
the use of Kolmogorov’s inequality for maximum of partial sums in order to
move from Dœblin’s (or Blackwell’s) dissection to ordinary sum of independent
modular summands. We call this approach “basic technique”.

Delving deeper, the “basic technique” in its original form fails in local
theorems and yields no “refined” approximations, such as Berry–Esseen’s
estimates and Edgeworth’s expansions. To be specific, dealing with a local
limit theorem for classical Markov chains, Kolmogorov (see [49]) pointed that
“Dœblin’s method in its original form” is suitable “only for proving integral
theorems”. We quote555In our translation from Russian. Apparently, this
article has not been translated in due course from Russian into English. more
from [49], § 2: “$\dots$ the local theorems that form the main content of the
present paper will be obtained $\dots$ with the help of some strengthening of
the method that was developed by Dœblin for proving the integral limit theorem
in the case of an infinite number of states. In order to make the development
of the method clear, we single out $\dots$ a brief exposition of Dœblin’s
method in its original form, in which it is suitable only for proving integral
theorems.”

In brief, Kolmogorov’s advance in [49] was due to a straightforward use of the
total probability formula, or, according to his terminology, “basic identity”.

Further progress was made thirty years later in [14] and in a series of papers
(see, e.g., [15], [43], [55]–[60]) that followed: Berry–Esseen’s estimates and
Edgeworth’s expansions in CLT for recurrent Markov chain were obtained by
using the total probability rule and the auxiliary multidimensional limit
theorems for sums of independent random vectors. To distinguish this approach
from “basic technique”, or “Dœblin’s method in its original form”, we call it
“advanced (modular, if switching to modular summands is needed) technique”.

Being one of the main pillars of the advanced technique, the limit theorems
for sums of independent random vectors (see, e.g., [11], [31]–[33]) are close
to perfection. In the advanced technique, analytical complexity, which never
disappears but flows from one form to another, is split into the use of,
firstly, the auxiliary theorems where (we quote from Preface of [11])
“precision and generality go hand in hand” and, secondly, tedious but fairly
standard classical analysis, e.g., approximation of integral sums by
appropriate integrals and evaluation of remainder terms.

Further presentation is arranged as follows. Section 2 is devoted to genesis
and classification of compound sums. In Section 3, we address elementary and
refined renewal theorems and refined limit theorems for cumulated rewards.
Having formulated them as limit theorems for simple compound sums, we outline
their proof using advanced technique instead of renewal equation, Laplace
transforms, and Tauberian theorems commonly used in renewal theory.

In Section 4, we address complex compound sums “with irregular summation and
independence” related to the ruin problem of collective risk theory. To get
(refined) normal approximation when such sums are proper, we use the modular
analysis based on Blackwell’s ladder idea. To get (refined) quasi–normal
approximation when such sums are defective, we resort to associated random
variables. In this context, we touch upon the inverse Gaussian approximation
which deviates somewhat from the main topic of this paper, but demonstrates
that there are more sophisticated methods than those related to asymptotic
normality.

In Section 5, seeking to use the full force of modular analysis combined with
advanced technique, we focus on compound sums with general modular structure.
Particular cases of this framework are complex compound sums “with irregular
summation and dependence”, in particular Markov dependence. The only case,
although of significant practical interest, where great impediments for
asymptotic analysis arise is defective compound sums “with irregular summation
and dependence”. This case, which is tackled by means of computer–intensive
analysis, is briefly discussed in Section 6. A concluding remark is made in
Section 7.

The article, being a review, does not contain complete proofs. Most results
are formulated in a non–strict manner, although the necessary references are
given everywhere. Striving for clarity, great emphasis is placed on discussing
the main ideas of the proofs. Overall, the aims of the advance described in
this review are the same as Loève stated in [53] writing on limit theorems of
probability theory: (i) to simplify proofs and forge general tools out of the
special ones, (ii) to sharpen and strengthen results, (iii) to find general
notions behind the results obtained and to extend their domains of validity.

##  2\. Genesis and classification of compound sums

It was observed (see [69]) that “on the Continent, if people are waiting at a
bus–stop, they loiter around in a seemingly vague fashion. When the bus
arrives, they make a dash for it $\dots$ An Englishman, even if he is alone,
forms an orderly queue of one.” In terms of mathematical modeling, this is a
“clustering–type” model opposed to a “queuing–type” model. In the former,
there is essentially no flow of time, and the objects or events of interest
are scattered in space. In the latter, there is a flow of time, and the
objects or events of interest form an ordered queue.

In models of both types, the core is the compound sum

${S}_{N_{t}}=\sum_{i=1}^{N_{t}}X_{i}$

with basis $(T_{i},X_{i})$, $i=1,2,\dots$, where $t>0$ and

$N_{t}=\inf\Big{\\{}n\geqslant 1:\sum_{i=1}^{n}T_{i}>t\Big{\\}}$ (2.1)

or $+\infty$, if the set is empty666Recall that $\inf\emptyset=+\infty$,
$\sup\emptyset=0$.. When $N_{t}=+\infty$, the sum ${S}_{N_{t}}$ is set equal
to $+\infty$. For brevity, the real–valued random variables $X_{i}$ are called
primary components ($p$–components) and $T_{i}$ secondary components
($s$–components) of the basis.

If the basis $(T_{i},X_{i})$, $i=1,2,\dots$, consists of independent random
vectors, then ${S}_{N_{t}}$ is “with independence”. Otherwise it is “with
dependence”. Examples of the latter are often found (see, e.g., [78]) in
various models with Markov modulation. If $s$–components are positive, then
${S}_{N_{t}}$ is “with regular summation”. Otherwise, it is “with irregular
summation”.

Apparently, compound sums “with regular summation and independence” fully and
distinctly came forth in “queuing–type” renewal theory and its ramifications.
Compound sums “with irregular summation and independence” appeared (see, e.g.,
[66]) in ruin problem and (see, e.g., [41]) in many settings which involve
random walks. On the other hand, Dœblin’s dissection, i.e., switching from
ordinary sum defined on a Markov chain to modular simple compound sum “with
regular summation and independence”, whose basis consists of intervals between
regeneration points and increments at these intervals, is another source of
compound sums.

The complexity of “irregular summation” is rather transparent: the sums
${V}_{n}=\sum_{i=1}^{n}T_{i}$, $n=1,2,\dots$, are not progressively
increasing, whence $\\{N_{t}=n\\}$ is equal to $\\{{V}_{1}\leqslant
t,\dots,{V}_{n-2}\leqslant t,{V}_{n-1}\leqslant t<{V}_{n}\\}$, rather than
$\\{{V}_{n-1}\leqslant t<{V}_{n}\\}$, as would be for “regular summation”.

In “dependence complexity” case, the model has to be fleshed out. In the
search of non–trivial examples which illustrate the general results, we focus
on Markov dependence and its extensions, but in fact any type of dependence
which allows an embedded modular structure with independent modules suits us
well. In terms of random processes, much of this topic is based on successive
“starting over”, as time goes on, commonly called “regeneration” or (see [47],
[78]) “regenerative phenomena”. With this approach, many cases that go beyond
the scope of Markov theory fall within the scope of this paper.

######  Remark 2.1.

Beyond the scope of this paper are the sums of the following two types. First,
the random sums ${S}_{N}$ with $N$ and $X_{1},X_{2},\dots$ not necessarily
independent of each other, but with $N$ not of the form (2.1) or the similar.
Second, the random sums ${S}_{N}=\sum_{i=1}^{N}X_{i}$ with independence
between $N$ and $X_{1},X_{2},\dots$. The former were introduced in [6], [81],
[82], and investigated, e.g., in [41]. The latter were introduced in [84] and
thoroughly studied (see, e.g., [38]) in many works.∎

###  2.1. Proper and defective compound sums

Compound sum $\mathcal{R\hskip-0.5pt/I}$ sum $\mathcal{I\hskip-0.5pt/I}$ sum
proper $\mathcal{I\hskip-0.5pt/I}$ sum defective $\mathcal{I\hskip-0.5pt/I}$
sum $\mathcal{R\hskip-0.5pt/D}$ sum $\mathcal{I\hskip-0.5pt/D}$ sum proper
$\mathcal{I\hskip-0.5pt/D}$ sum defective $\mathcal{I\hskip-0.5pt/D}$ sum

Figure 1. Classification of compound sums.

The classification of compound sums, where shorthand for “compound sum with
regular summation and independence” is “$\mathcal{R\hskip-0.5pt/I}$ sum”, for
“compound sum with irregular summation and independence” is
“$\mathcal{I\hskip-0.5pt/I}$ sum”, for “compound sum with regular summation
and dependence” is “$\mathcal{R\hskip-0.5pt/D}$ sum”, and for “compound sum
with irregular summation and dependence” is “$\mathcal{I\hskip-0.5pt/D}$ sum”,
is shown in Fig. 1. Under natural regularity conditions, compound sums “with
regular summation” are always proper, while compound sums “with irregular
summation” are either proper, or defective777Recall (see [37]) that a random
variable is defective, with defect $1-p$, if it takes the value $\infty$ with
probability $p>0$. Otherwise, it is proper..

The following example shows that finding the defect of
$\mathcal{I\hskip-0.5pt/I}$ sum is not easy.

######  Example 2.1.

Let $(X_{i},Y_{i})$, $i=1,2,\dots$, be a subsidiary sequence of i.i.d. random
vectors, whose components are positive and independent of each other. For
$c>0$ and $T_{i}=Y_{i}-cX_{i}$, ${S}_{N_{t}}=\sum_{i=1}^{N_{t}}X_{i}$ is
positive $\mathcal{I\hskip-0.5pt/I}$ sum with basis $(T_{i},X_{i})$,
$i=1,2,\dots$. While the components of the subsidiary sequence are independent
of each other, the components of the basis are dependent on each other. If
$X_{i}$ and $Y_{i}$, $i=1,2,\dots$, are exponentially distributed with
positive parameters $\varrho$ and $\rho$, then (see, e.g., [65], Theorem 5.8)

$\mathsf{P}\big{\\{}{S}_{N_{t}}\leqslant
x\big{\\}}=\mathsf{P}\big{\\{}{S}_{N_{t}}<\infty\big{\\}}-\frac{1}{\pi}\int_{0}^{\pi}f(z,t,x)\,dz,$
(2.2)

where

$\mathsf{P}\big{\\{}{S}_{N_{t}}<\infty\big{\\}}=\begin{cases}1,&\varrho/(c\rho)\geqslant
1,\\\\[8.0pt]
\dfrac{\varrho}{c\rho}\,\exp\big{\\{}-t(c\rho-\varrho)/c\big{\\}},&\varrho/(c\rho)<1\end{cases}$
(2.3)

and

$\displaystyle f(z,t,x)=$
$\displaystyle\,(\varrho/(c\rho))(1+\varrho/(c\rho)-2\sqrt{\varrho/(c\rho)}\cos
z)^{-1}$ (2.4)
$\displaystyle\times\exp\big{\\{}t\rho\,(\sqrt{\varrho/(c\rho)}\cos z-1)$
$\displaystyle-x\varrho(c\rho/\varrho)(1+\varrho/(c\rho)-2\sqrt{\varrho/(c\rho)}\cos
z)\big{\\}}$ $\displaystyle\times(\cos(t\rho\sqrt{\varrho/(c\rho)}\sin
z)-\cos(t\rho\sqrt{\varrho/(c\rho)}\sin z+2z)).$

Consequently, ${S}_{N_{t}}$ is proper if $\varrho/(c\rho)\geqslant 1$, which
is equivalent to $c\leqslant c^{\ast}=\mathsf{E}{Y}/\mathsf{E}{X}$, and
defective otherwise. Defect of ${S}_{N_{t}}$ is yielded in (2.3). ∎

Although the expression (2.3) for defect of ${S}_{N_{t}}$ is
complicated888This is a particular case of Cramér’s famous result on the
probability of ultimate ruin., the conditions under which ${S}_{N_{t}}$ is
proper or defective, are simple. On the one hand, if $\mathsf{E}T_{1}>0$, then
(see Theorem 2 in [37], Chapter XII) the ordinary sums ${V}_{n}$ increase a.s.
to $+\infty$, as $n\to\infty$. Consequently, for any $t$ the summation limit
$N_{t}$ is a.s. finite and ${S}_{N_{t}}$ is a.s. less than $+\infty$, whence
proper. On the other hand, if $\mathsf{E}T_{1}<0$, then the ordinary sums
${V}_{n}$ decrease a.s. to $-\infty$, as $n\to\infty$. With a positive
probability, the maximum $M=\max_{n\in\mathsf{N}}{V}_{n}$ is finite and less
than $t>0$ large enough, whence $N_{t}$ and ${S}_{N_{t}}$ are defective.

###  2.2. Renewal theory and species of compound sums

In renewal theory, $p$–components $X_{i}$, $i=1,2,\dots$, (typically positive)
are called “rewards in the moments of renewals” and $s$–components $T_{i}$,
$i=1,2,\dots$, (always positive) are called “intervals between successive
renewals”. This interpretation yields a variety of species of
$\mathcal{R\hskip-0.5pt/I}$ sums. In particular, the total number of renewals
before time $t$, defined as $\widetilde{N}_{t}=\sup\\{n\geqslant
1:{V}_{n}\leqslant t\\}$, or $0$, if $T_{1}>t$, comes to the fore. Focus
shifts from $N_{t}$ to $\widetilde{N}_{t}$, which coincides with
$\widehat{N}_{t}=N_{t}-1$ trivially related to $N_{t}$.

When moving from relatively simple $\mathcal{R\hskip-0.5pt/I}$ sums to more
complicated $\mathcal{I\hskip-0.5pt/I}$ and $\mathcal{I\hskip-0.5pt/D}$ sums,
both of them with “irregular summation”, the difference between altered (with
summation limit $\widetilde{N}_{t}$) and non–altered (with summation limit
$N_{t}$) compound sums becomes non–trivial: $\widetilde{N}_{t}$ is no more
trivially related to $N_{t}$ and the altered compound sums
${S}_{\widetilde{N}_{t}}=\sum_{i=1}^{\widetilde{N}_{t}}X_{i}$ and
${S}_{\widehat{N}_{t}}=\sum_{i=1}^{\widehat{N}_{t}}X_{i}$ are not bounded to
coincide.

As for $\mathcal{R\hskip-0.5pt/D}$ sums with Markov dependence, which
generalize $\mathcal{R\hskip-0.5pt/I}$ sums in the same way as Markov renewal
theory generalizes renewal theory, dependence is introduced by embedded Markov
chain $\\{\xi_{i}\\}_{i\geqslant 0}$. In such $\mathcal{R\hskip-0.5pt/D}$
sums, $s$–components $T_{i}$, $i=1,2,\dots$, are the time intervals between
jumps of $\\{\xi_{i}\\}_{i\geqslant 0}$ and $p$–components $X_{i}$,
$i=1,2,\dots$, which depend on $\\{\xi_{i}\\}_{i\geqslant 0}$, are rewards in
the moments of renewals.

In this way, the variety of species of compound sums echoes the diversity of
renewal processes. There are999We refer to [21], Section 2.2, or [22], Section
9.2. There are terminological collisions, e.g., in [45], Chapter 5, Section 7,
this process is called delayed renewal process. modified, equilibrium, and
ordinary renewal processes. If $T_{1}$ is distributed differently than
$T_{i}\overset{d}{=}T$, $i=2,3,\dots$, then the renewal process is called
modified. When all intervals between renewals, including the first one, are
identically distributed, the renewal process is called ordinary. Furthermore,
if

$\mathsf{P}\big{\\{}T_{1}\leqslant
x\big{\\}}=\frac{1}{\mathsf{E}T}\int_{0}^{x}\big{(}1-F_{T}(z)\big{)}\,dz,\quad
x>0,$

where $F_{T}$ denotes c.d.f. of $T$, then it is called equilibrium, or
stationary.

Regarding compound sums, modified are those in which a finite number of
elements of the basis $(T_{i},X_{i})$, $i=1,2,\dots$, e.g., the first element
$(T_{1},X_{1})$, differ from all the others. Unlike in renewal theory which
models technical repairable systems where the first repair does not
necessarily occur at the starting time zero, this modification looks awkward
for $\mathcal{R\hskip-0.5pt/I}$ sums: it boils down to a very special case of
non–identically distributed summands. The similar modification of
$\mathcal{R\hskip-0.5pt/D}$ sums with Markov dependence comes down to the
choice of the initial distribution of the embedded Markov chain
$\\{\xi_{i}\\}_{i\geqslant 0}$ and is much more sensible.

It is noteworthy that, under natural regularity conditions, switching within
the species of compound sums, e.g., from altered to unaltered compound sums
and vice versa, does not affect the main–term approximations, but affects
refinements such as Edgeworth’s expansions.

##  3\. Renewal theory and limit theorems for
$\boldsymbol{\mathcal{R\hskip-0.5pt/I}}$ sums

Founded (see [34], [35]) in the 1940s as a theoretical insight into technical
repairable systems, the classical renewal theory is focussed on “queuing–type”
models. The term “renewal”, which displaced the formerly used term “industrial
replacement” coined (see [54]) in the 1930s by Lotka, is widely used nowadays,
but was not the main or default term even in Feller’s seminal paper [36] dated
1949.

In 1948, Doob [30] noted that “renewal theory is ordinarily reduced to the
theory of certain types of integral equations $\dots$ However, it is to be
expected that a treatment in terms of the theory of probability, which uses
the modern developments of this theory, will shed new light on the subject.”
Feller agreed (see [37], Chapter VI) that in renewal theory “analytically, we
are concerned merely with sums of independent positive variables.”

###  3.1. First appearance of the advanced technique

In words, the renewal function101010The asterisk denotes the convolution
operator. $\mathsf{U}(t)=\sum_{n=0}^{\infty}F_{T}^{*n}(t)$, $t>0$, where
$F_{T}$ denotes c.d.f. of a positive random variable $T$, is the expected
number of renewals in the time interval $(0,t)$, with the origin counted as
renewal epoch111111In [22], [45], the renewal function is defined as
$\mathsf{H}(t)=\mathsf{U}(t)-1$, $t>0$ (or $\mathsf{E}\widehat{N}_{t}$), which
differs from $\mathsf{U}(t)$ by one. In [37], the function $\mathsf{U}(t)$,
$t>0$, is called the (ordinary) renewal process. In [22], [45], the renewal
process is referred to as the continuous–time random process
$\\{\widehat{N}_{t}\\}_{t\geqslant 0}$, i.e., the random number of renewals in
the time interval $(0,t)$, with the origin not being counted as a renewal
epoch.. As a formula, this is $\mathsf{U}(t)=1+\mathsf{E}\widehat{N}_{t}$.
Together with altered $\mathcal{R\hskip-0.5pt/I}$ sum ${S}_{\widehat{N}_{t}}$,
called in renewal theory “cumulated rewards”, it is built on
$\mathcal{R\hskip-0.5pt/I}$ basis $(T_{i},X_{i})\overset{d}{=}(T,X)$,
$i=1,2,\dots$, with i.i.d. components and $T$ positive.

The elementary renewal theorem (see, e.g., [41]), states that for
$0<\mathsf{E}\,T\leqslant\infty$ the approximation121212When
$\mathsf{E}\,T=\infty$, the ratio $t/\mathsf{E}\,T$ is replaced by $0$.

$\mathsf{E}\widehat{N}_{t}=\frac{t}{\mathsf{E}\,T}+o(t),\quad t\to\infty,$
(3.1)

holds. The corresponding expansion up to vanishing term is

$\mathsf{E}\widehat{N}_{t}=\frac{t}{\mathsf{E}\,T}+\frac{\mathsf{D}T-(\mathsf{E}\,T\,)^{2}}{2\,(\mathsf{E}\,T\,)^{2}}+o(1),\quad
t\to\infty,$ (3.2)

called refined elementary renewal theorem.

The elementary renewal theorem is widely known: see, e.g., Equality (3) in
[21], Section 4.2, or Equality (17) in [22], Section 9.2, or main formula in
point (b) in [45], Chapter 5, Section 6. In [22], p. 345, it is noted that “a
rigorous proof $\dots$ is possible under very weak assumptions about the
distribution of $T$, but requires difficult Tauberian arguments and will not
be attempted here”. It can also be obtained (with some caution with regard to
the definition of renewal function) from Theorem 1 in [37], Chapter XI, the
proof of which is based on the renewal equation.

The refined elementary renewal theorem and the similar results for
higher–order power moments of $\widehat{N}_{t}$ and ${S}_{\widehat{N}_{t}}$,
including expansions up to vanishing terms, were studied (see, e.g., [21],
Section 4.5, and [3]) by various methods.

We outline a proof (see details in [65], Chapter 4) that does not compete with
the proofs mentioned above in the sense of elegance or minimality of
conditions. The advantage of this proof, based on the use of refined CLT and
consisting of steps A–F, is that it will be routinely carried over to much
more complex settings.

Step A: use of fundamental identity. The identity

$\displaystyle\mathsf{E}\widehat{N}_{t}=$
$\displaystyle\,\sum_{n=1}^{\infty}n\;\mathsf{P}\big{\\{}\widehat{N}_{t}=n\big{\\}}=\sum_{n=1}^{\infty}n\;\mathsf{P}\bigg{\\{}\sum_{i=1}^{n}T_{i}\leqslant
t<\sum_{i=1}^{n+1}T_{i}\bigg{\\}}$ (3.3) $\displaystyle=$
$\displaystyle\,\sum_{n=1}^{\infty}n\big{(}F_{T}^{*n}(t)-F_{T}^{*(n+1)}(t)\big{)}$

is obvious. If the probability density function (p.d.f.) $f_{T}$ exists (we
assume this for simplicity of presentation), then (3.3) can be rewritten as

$\displaystyle\mathsf{E}\widehat{N}_{t}=\sum_{n=1}^{\infty}n\int_{0}^{t}f_{T}^{*n}(t-z)\,\mathsf{P}\big{\\{}T_{n+1}>z\big{\\}}\,dz.$
(3.4)

Step B: reduction of range of summation. At this step, we cut out from the sum
on the right side of (3.3) those terms that correspond to small $n$ (i.e.,
$n<n_{t}$ with properly selected $n_{t}\to\infty$, as $t\to\infty$). This is
intuitively clear: for large $t$, the random variable $\widehat{N}_{t}$ is
likely to be large. Therefore, the probabilities
$\mathsf{P}\big{\\{}\widehat{N}_{t}=n\big{\\}}$ for small $n$ are small.
Technically, it is done using the well–known probability inequalities, such as
Markov’s inequality.

Step C: reduction of range of integration. At this step, relying on the moment
conditions, we cut out from the integral in (3.4) that part that corresponds
to large $z$ (i.e., $z>z_{t}$ with properly selected $z_{t}\to\infty$, as
$t\to\infty$). This is also intuitively clear: any single random variable,
including $T_{n+1}$, is small compared to the sum $\sum_{i=1}^{n}T_{i}$, as
$n$ is sufficiently large.

Step D: application of refined CLT. Writing $\mu_{T}=\mathsf{E}\,T$,
$\sigma_{T}^{2}=\mathsf{D}\,T$ and switching to standardized random variables
$\tilde{T}_{i}=(T_{i}-\mu_{T})/\sigma_{T}$, $i=1,2,\dots$, we have

$f_{T}^{*n}(t-z)=\frac{1}{\sigma_{T}\sqrt{n}}\,f_{\,n^{-1/2}\sum_{i=1}^{n}\tilde{T}_{i}}\big{(}\lambda_{T,n}{(t-z)}\big{)},$

where $\lambda_{T,n}{(t-z)}=(t-z-\mu_{T}n)/(\sigma_{T}\sqrt{n})$. Assuming
that natural moment conditions are satisfied, we apply Berry–Esseen’s estimate
(see Theorem 11 in [80], Chapter VII, § 2) in the proof of (3.1) and
Edgeworth’s expansion (see Theorem 17 in [80], Chapter VII, § 3) in the proof
of (3.2). In both cases, these results are taken with non–uniform remainder
terms.

Step E: analysis of approximating term. In the proof of (3.1), where
Berry–Esseen’s estimate was used, the approximating term is

$\mathcal{I}_{t}=\frac{1}{\sigma_{T}}\sum_{n>n_{t}}n^{1/2}\int_{0}^{z_{t}}\varphi_{\left({0},{1}\right)}\big{(}\lambda_{T,n}{(t-z)}\big{)}\,\mathsf{P}\big{\\{}T>z\big{\\}}\,dz,$

where $\varphi_{\left({0},{1}\right)}$ denotes p.d.f. of standard normal
distribution. In the proof of (3.2), where Edgeworth’s expansion was used, the
approximating term is the sum of

$\displaystyle\mathcal{I}_{t}^{\,{\dagger}}=$
$\displaystyle\,\frac{1}{\sigma_{T}}\sum_{n>n_{t}}\sqrt{n}\int_{0}^{z_{t}}\varphi_{\left({0},{1}\right)}(\lambda_{T,n}{(t-z)})\,\mathsf{P}\big{\\{}T>z\big{\\}}\,dz$
$\displaystyle\mathcal{I}_{t}^{\,{\ddagger}}=$
$\displaystyle\,\frac{1}{\sigma_{T}}\frac{\mathsf{E}(T^{3})}{6\,\sigma_{T}^{3}}\sum_{n>n_{t}}\int_{0}^{z_{t}}\,\big{(}\lambda_{T,n}^{3}{(t-z)}-3\,\lambda_{T,n}{(t-z)}\big{)}$
$\displaystyle\times\varphi_{\left({0},{1}\right)}\big{(}\lambda_{T,n}{(t-z)}\big{)}\,\mathsf{P}\big{\\{}T>z\big{\\}}\,dz.$

Using Riemann summation formula with nodal points
$\lambda_{T,n}{(t)}{}<\lambda_{T,n-1}{(t)}{}<\dots<\lambda_{T,2}{(t)}{}<\lambda_{T,1}{(t)}{}$,
we seek to reduce $\mathcal{I}_{t}$ to ${t}/{\mathsf{E}\,T}$ with the required
accuracy, and $\mathcal{I}_{t}^{\,{\dagger}}+\mathcal{I}_{t}^{\,{\ddagger}}$
to
$t/\mathsf{E}\,T+(\mathsf{D}T-(\mathsf{E}\,T)^{2})/(2\,(\mathsf{E}\,T)^{2})$
with the required accuracy. This is a tedious but fairly standard classical
analysis left to the reader.

Step F: analysis of remainder term. In the proof of (3.1), the remainder term
is

$\mathcal{R}_{t}={C}_{1}\sum_{n>n_{t}}\delta_{n}\int_{0}^{z_{t}}\big{(}1+|\,\lambda_{T,n}{(t-z)}\,|\,\big{)}^{-3}\,\mathsf{P}\big{\\{}T>z\big{\\}}\,dz.$

In the proof of (3.2), the remainder term is

$\displaystyle\mathcal{R}_{t}=$
$\displaystyle\,{C}_{2}\sum_{n>n_{t}}\frac{\delta_{n}}{\sqrt{n}}\int_{0}^{z_{t}}\big{(}1+\big{|}\,\lambda_{T,n}{(t-z)}\,\big{|}\,\big{)}^{-4}\,\mathsf{P}\big{\\{}T>z\big{\\}}\,dz,$

with $\delta_{n}\to 0$, as $n\to\infty$. The former is $o(t)$, $t\to\infty$.
The latter is $o(1)$, $t\to\infty$, which is shown by direct analytical
methods. This is also a tedious but fairly standard classical analysis left to
the reader.

###  3.2. Refined limit theorems for cumulated rewards

Moving from renewal function (or power moments of $\widehat{N}_{t}$) to the
distribution of altered $\mathcal{R\hskip-0.5pt/I}$ sum
${S}_{\widehat{N}_{t}}=\sum_{i=1}^{\widehat{N}_{t}}X_{i}$, called in renewal
theory “cumulated rewards”, we use the shorthand notation
$\mu_{X}=\mathsf{E}X$, $\sigma_{X}^{2}=\mathsf{D}X$,
$\varkappa_{XT}^{\,(i,j)}=\mathsf{E}((X-\mu_{X})^{i}(T-\mu_{T})^{j})$ for $i$
and $j$ integers, and $\varkappa_{XT}=\varkappa_{XT}^{\,(1,1)}$. We put

$M_{{S}}{\,\\!}=\mu_{X}\mu_{T}^{-1},\quad
D_{{S}}^{2}{\,}=\mathsf{E}\,(\mu_{X}T-\mu_{T}X)^{2}\mu_{T}^{-3}.$

By $\varPhi_{\left({0},{1}\right)}(x)$, we denote c.d.f. of standard normal
distribution.

######  Theorem 3.1 (Berry–Esseen’s estimate).

If p.d.f. of $T$ is bounded above by a finite constant131313In Theorem 3.1,
this condition is excessive. We repeat that we do not strive for maximum
generality (or rigor) in this presentation., $D_{{S}}^{2}{\,}>0$, and
$\mathsf{E}(T^{3})<\infty$, $\mathsf{E}(X^{3})<\infty$, then

$\sup_{x\in\mathsf{R}}\,\Big{|}\,\mathsf{P}\big{\\{}{S}_{\widehat{N}_{t}}-M_{{S}}{\,t}\leqslant
x\,D_{{S}}{\,t^{1/2}}\big{\\}}-\varPhi_{\left({0},{1}\right)}(x)\,\Big{|}=O\big{(}t^{-1/2}\big{)},\quad
t\to\infty.$

######  Theorem 3.2 (Edgeworth’s expansion).

If the conditions of Theorem 3.1 are satisfied and $\mathsf{E}(X^{k})<\infty$,
$\mathsf{E}(T^{k})<\infty$, then there exist polynomials
$\mathsf{Q}_{\,r}(x)$, $r=1,2,\dots,k-3$, of degree $3r-1$, such that

$\displaystyle\sup_{x\in\mathsf{R}}\bigg{|}\,\mathsf{P}\big{\\{}{S}_{\widehat{N}_{t}}-M_{{S}}{\,t}\leqslant
xD_{{S}}{\,t^{1/2}}\big{\\}}-\varPhi_{\left({0},{1}\right)}(x)$
$\displaystyle\hskip
90.0pt-\sum_{r=1}^{k-3}\frac{\mathsf{Q}_{\,r}(x)}{t^{r/2}}\,\,\varphi_{\left({0},{1}\right)}(x)\bigg{|}=O\big{(}t^{-(k-2)/2}\big{)},\quad
t\to\infty.$

In particular,

$\mathsf{Q}_{\,1}(x)=-\frac{1}{6}\,\big{(}K_{(3,0)}\mathsf{H}_{2}(x)+3\hskip
0.2pt{I}_{1}\big{)},$

where $\mathsf{H}_{2}(x)=x^{2}-1$ is Chebyshev–Hermite’s polynomial,

$\displaystyle K_{(3,0)}=$
$\displaystyle\,\big{(}\big{(}\varkappa_{XT}^{\,(3,0)}-3\sigma_{X}^{2}\varkappa_{XT}\mu_{T}^{-1}+6\sigma_{X}^{2}\sigma_{T}^{2}\mu_{T}\mu_{X}^{-2}-6\sigma_{T}^{2}\varkappa_{XT}\mu_{X}^{2}\mu_{T}^{-3}\big{)}\,\mu_{T}^{-1}$
$\displaystyle-3\,\big{(}\varkappa_{XT}^{\,(2,1)}-2\varkappa_{XT}^{2}\mu_{T}^{-1}+\sigma_{X}^{2}\sigma_{T}^{2}\mu_{T}^{-1}\big{)}\,\mu_{T}\mu_{X}^{-2}$
$\displaystyle+3\,\big{(}\varkappa_{XT}^{\,(1,2)}-\sigma_{T}^{2}\varkappa_{XT}\mu_{T}^{-1}\big{)}\,\mu_{X}^{2}\mu_{T}^{-3}$
$\displaystyle-\big{(}\varkappa_{XT}^{\,(0,3)}-3\sigma_{T}^{4}\mu_{T}^{-1}\big{)}\,\mu_{X}^{3}\mu_{T}^{-4}\big{)}\,D_{{S}}^{-3}{\,},$

and
${I}_{1}=\mu_{X}\big{(}\sigma_{T}^{2}\,\mu_{T}^{-2}+1\big{)}\,D_{{S}}^{-1}{\,}$.

Both Theorems 3.1 and 3.2 are proved (see details in [65], Chapter 4) by means
of practically the same advanced technique which was used in the proof of
(3.1) and (3.2), respectively. It consists of steps A–F, which we will draw.

Step A: use of fundamental identity. Equalities (3.3) and (3.4) are replaced
by

$\displaystyle\mathsf{P}\big{\\{}{S}_{\widehat{N}_{t}}\leqslant x\big{\\}}=$
$\displaystyle\,\sum_{n=1}^{\infty}\int_{0}^{t}\mathsf{P}\bigg{\\{}\sum_{i=1}^{n}X_{i}\leqslant
x\,\bigg{|}\,\sum_{i=1}^{n}T_{i}=t-z\,\bigg{\\}}$ (3.5) $\displaystyle\times
f_{T}^{*n}(t-z)\,\mathsf{P}\big{\\{}T_{n+1}>z\big{\\}}\,dz,$

which is the total probability formula.

Steps B and C remain essentially the same.

Step D: application of refined CLT. The integrand in (3.5) is ready for
application of (see [31]–[33]) two–dimensional hybrid integro–local (for
density) CLT. In more detail, in the proof of Theorem 3.1, we use Berry–Esseen
estimate, in the proof of Theorem 3.2, we use Edgeworth’s expansion in this
CLT, both with non–uniform remainder terms.

Steps E and F remain nearly the same. Technically, they are more complicated
than steps E and F in the proof of (3.1) and (3.2), but use essentially the
same analytical techniques.

###  3.3. Garbage and deficiency of basic technique

When studying altered $\mathcal{R\hskip-0.5pt/I}$ sum ${S}_{\widehat{N}_{t}}$,
the main point of basic technique consists in approximating
${S}_{\widehat{N}_{t}}$ by the ordinary sum
${S}_{\,[\mathsf{E}\\!\widehat{N}_{t}]}$, whose asymptotical normality is
evident: the real number $[\mathsf{E}\\!\widehat{N}_{t}]$ is (see (3.1)) of
the order $t/\mathsf{E}\,T\to\infty$, as $t\to\infty$, and the summands are
i.i.d. In other words, although technically the basic technique applies
Kolmogorov’s inequality for maximum of partial sums, its fundamental idea is
to throw the difference
$\mathcal{G}_{\,t}={S}_{\widehat{N}_{t}}-{S}_{\,[\mathsf{E}\\!\widehat{N}_{t}]}$,
called garbage, in the trash.

Constrained by this idea, the basic technique does not allow further
refinements, such as in Theorems 3.1 and 3.2, since $\mathcal{G}_{\,t}$ is a
random variable of order $t^{1/4}$: under natural regularity conditions,
including $\mathsf{E}(T^{3})<\infty$, $\mathsf{E}(X^{3})<\infty$,

$\displaystyle\sup_{x\in\mathsf{R}}\bigg{|}\,\mathsf{P}\bigg{\\{}\frac{\mu_{T}^{3/4}}{\sigma_{X}\sigma_{T}^{1/2}t^{1/4}}\,\mathcal{G}_{\,t}\leqslant
x\bigg{\\}}$ (3.6) $\displaystyle\hskip
40.0pt-\int_{-\infty}^{\infty}\varPhi_{\left({0},{1}\right)}\bigg{(}\frac{x}{\sqrt{|z|}}\bigg{)}\varphi_{\left({0},{1}\right)}(z)\,dz\bigg{|}=O\big{(}t^{-1/4}\big{)},\quad
t\to\infty.$

The proof of (3.6) (see details in [57] and [58]) by means of advanced
technique follows the scheme sketched in the proof of (3.1) and (3.2) and
Theorems 3.1 and 3.2. It consists of steps A–F, applies refined CLT with
non–uniform remainder terms, and requires mere standard classical analysis at
steps B, C, E, and F.

Figure 2. Graph of
$f(y)={\displaystyle\int_{-\infty}^{\infty}}\varphi_{\left({0},{1}\right)}\Big{(}\frac{y}{\sqrt{|z|}}\Big{)}\frac{1}{\sqrt{|z|}}\,\varphi_{\left({0},{1}\right)}(z)\,dz$.

For greater clarity, note that p.d.f. of the limit distribution in (3.6) is

$f(y)=\int_{-\infty}^{\infty}\varphi_{\left({0},{1}\right)}\bigg{(}\frac{y}{\sqrt{|z|}}\bigg{)}\frac{1}{\sqrt{|z|}}\;\varphi_{\left({0},{1}\right)}(z)\,dz,$

its maximal value is

$f(0)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{1}{\sqrt{|z|}}\;\varphi_{\left({0},{1}\right)}(z)\,dz=\frac{\Gamma(1/4)}{2^{3/4}\pi}\approx
0.686,$

and its graph is drawn in Fig. 2.

##  4\. Limit theorems for $\boldsymbol{\mathcal{I\hskip-0.5pt/I}}$ sums and
collective risk theory

The reader familiar with collective risk theory readily recognizes in
$\mathsf{P}\\{{S}_{N_{t}}\leqslant x\\}$ of Example 2.1 the probability of
ruin within time $x$, where141414Under the assumption of independence from
each other. $X_{i}$, $i=1,2,\dots$, are intervals between claims, $Y_{i}$,
$i=1,2,\dots$, are claim amounts, $t$ is the initial capital, and $c$ is the
premium intensity. For $0<c\leqslant c^{\ast}$, the
$\mathcal{I\hskip-0.5pt/I}$ sum ${S}_{N_{t}}$ is proper, while (in terms of
the risk reserve model) insurance is ruinous. For
$c>c^{\ast}=\mathsf{E}{Y}/\mathsf{E}{T}$, the $\mathcal{I\hskip-0.5pt/I}$ sum
${S}_{N_{t}}$ is defective, while insurance is profitable.

###  4.1. Modular approach: a way to manage complexity

A system is modular if its components can be separated and recombined, with
the advantage of flexibility and variety in use. Breaking such a system into
independent or nearly independent modules is done in order (see [10]) “to hide
the complexity of each part behind an abstraction and interface”. Modularity
is implemented in two ways: by splitting an orderly queue (for “queuing–type”
models) by events that occur one after another in time, or by sorting spatial
items (for “clustering–type” models) into loosely related and similar sets or
groups.

An example of the latter is the development of the OS/360 operating system for
the original IBM 360 line of computers. At first (see [16]), it was organized
in a relatively indecomposable way. It was deemed that each programmer should
be familiar with all the material, i.e., should have a copy of the workbook in
his office. But over time, this led to a rapid growth of troubles. For
example, even maintaining the workbook, whose volume constantly and
significantly increased, began to take up significant time from each working
day. Therefore, it became necessary to figure out whether programmers
developing one particular module really need to know about the structure of
the entire system.

It was concluded that, especially in large projects, the answer is negative
and managers (quoting from [79]) “should pay attention to minimizing
interdependencies. If knowledge is hidden or encapsulated within a module,
that knowledge cannot affect, and therefore need not be communicated to, other
parts of a system.” This idea has been framed as “information hiding”, a key
concept in the modern object–oriented approach to computer programming.

While the modular approach is not needed for $\mathcal{R\hskip-0.5pt/D}$ sums,
it is useful when studying $\mathcal{I\hskip-0.5pt/I}$,
$\mathcal{R\hskip-0.5pt/D}$, and proper $\mathcal{I\hskip-0.5pt/D}$ sums. In
“queuing–type” models, it is applied by splitting an orderly queue by events
(see, e.g., [78]: “regenerative phenomena”, or the like) that occur one after
another in time. In “clustering–type” models, it is applied by sorting spatial
items into loosely related (and similar) sets or groups.

###  4.2. Normal approximation for proper
$\boldsymbol{\mathcal{I\hskip-0.5pt/I}}$ sums

In this case, modular structure is defined by the random indices
$0\equiv\mathbb{\ell}_{0}<\mathbb{\ell}_{1}<\mathbb{\ell}_{2}\dots$, where

$\mathbb{\ell}_{k}=\inf\big{\\{}i>\mathbb{\ell}_{k-1}:{V}_{i}>{V}_{\mathbb{\ell}_{k-1}}\big{\\}}\overset{d}{=}\mathbb{\ell},\quad
k=1,2,\dots.$

The key point is the following “exact”151515This means that, unlike, e.g.,
Dœblin’s dissection, it does not contain incomplete initial and final modules.
partition of the proper $\mathcal{I\hskip-0.5pt/I}$ sum ${S}_{N_{t}}$, called
Blackwell’s dissection161616This dissection was introduced by Blackwell in
[13].:

${S}_{N_{t}}=\sum_{i=1}^{\mathcal{N}_{t}}{\mathcal{X}}_{\hskip 1.0pti},$ (4.1)

where $\mathcal{N}_{t}=\inf\big{\\{}n>0:\sum_{i=1}^{n}{\mathcal{T}}_{\hskip
1.0pti}>t\big{\\}}$, or $\infty$, if the set is empty, ${\mathcal{X}}_{\hskip
1.0pti}=\sum_{j=\mathbb{\ell}_{i-1}+1}^{\mathbb{\ell}_{i}}X_{j}$, and
${\mathcal{T}}_{\hskip
1.0pti}=\sum_{j=\mathbb{\ell}_{i-1}+1}^{\mathbb{\ell}_{i}}T_{j}>0$ by
definition. Bearing in mind that $t$ can be exceeded only at a ladder index,
equality (4.1) is obvious.

Since $({\mathcal{T}}_{\hskip 1.0pti},{\mathcal{X}}_{\hskip
1.0pti})\overset{d}{=}({\mathcal{T}}_{\hskip 1.0pt},{\mathcal{X}}_{\hskip
1.0pt})$, $i=1,2,\dots$, are i.i.d. and ${\mathcal{T}}_{\hskip 1.0pt}$ is
positive by definition, the sum on the right–hand side of (4.1) is a modular
$\mathcal{R\hskip-0.5pt/I}$ sum which is examined as above. Using (4.1),
nearly all results on asymptotic normality and its refinements available for
$\mathcal{R\hskip-0.5pt/I}$ sums can be transferred to proper
$\mathcal{I\hskip-0.5pt/I}$ sums.

In particular (see details in [65], Chapter 9), in Example 2.1 with
$0<c<c^{\ast}$ the normal approximation under natural regularity conditions is

$\lim_{t\to\infty}\sup_{x\in\mathsf{R}}\,\big{|}\,\mathsf{P}\big{\\{}{S}_{N_{t}}\leqslant
x\big{\\}}-\varPhi_{\left({m_{{}_{\scriptscriptstyle\triangledown}}t},{D_{{\hskip-1.0pt\scriptscriptstyle\triangledown}}^{2}t}\right)}(x)\,\big{|}=0,$
(4.2)

where $m_{{}_{\scriptscriptstyle\triangledown}}=\mu_{X}\mu_{T}^{-1}$ and
$D_{{\hskip-1.0pt\scriptscriptstyle\triangledown}}^{2}=\mathsf{E}(\mu_{T}X-\mu_{X}T\,)^{2}\mu_{T}^{-3}$.
Its refinements, i.e., Berry–Esseen’s estimate and Edgeworth’s expansion, are
similar to those in Theorems 3.1 and 3.2 and are omitted in this presentation.

Regarding $m_{{}_{\scriptscriptstyle\triangledown}}$ and
$D_{{\hskip-1.0pt\scriptscriptstyle\triangledown}}^{2}$ expressed above in the
original (rather than modular) terms, the following should be noted. The
refined normal approximation (4.2) obtained by modular approach is first
written in terms of the modular random variables $\mathbb{\ell}$,
${\mathcal{T}}_{\hskip 1.0pt}$, and ${\mathcal{X}}_{\hskip 1.0pt}$. Converting
them to the original terms requires additional effort. There are several ways
to do this. First, Wald’s identities can bring a great release since certain
combinations of moments of block random variables ${\mathcal{T}}_{\hskip
1.0pt}$ and ${\mathcal{X}}_{\hskip 1.0pt}$ can be represented through the
moments of the original random variables $T$ and $X$. Second, moments of
ladder index $\mathbb{\ell}$ and ladder modules ${\mathcal{T}}_{\hskip
1.0pt}$, ${\mathcal{X}}_{\hskip 1.0pt}$ can be calculated analytically, using
(see, e.g., [37]) Spitzer’s sums.

###  4.3. Quasi–normal approximation for defective
$\boldsymbol{\mathcal{I\hskip-0.5pt/I}}$ sums

For $c>c^{\ast}$, under natural regularity conditions (see details in [65],
Chapter 9), quasi–normal approximation171717This name emphasizes the presence
of a normal distribution function. In risk theory, it is known as
Cramér–Lundberg’s approximation. is

$\lim_{t\to\infty}\sup_{x>0}\,\big{|}\,e^{\varkappa
t}\,\mathsf{P}\big{\\{}{S}_{N_{t}}\leqslant
x\big{\\}}-\mathbb{C}\;\varPhi_{\left({m_{{}_{\scriptscriptstyle\vartriangle}}t},{D_{{\hskip-1.0pt\scriptscriptstyle\vartriangle}}^{2}t}\right)}(x)\,\big{|}=0.$
(4.3)

It follows from (4.3) that
$\mathsf{P}\\{{S}_{N_{t}}<\infty\\}\approx\,\mathbb{C}\,e^{-\varkappa t}$,
which is an asymptotic formula for defect (cf. (2.3)) of defective
$\mathcal{I\hskip-0.5pt/I}$ sum ${S}_{N_{t}}$.

Here $\varkappa$ is a positive solution (w.r.t. $x$) to the nonlinear
equation181818In risk theory, equation (4.4) is called Lundberg’s equation.
Its positive solution $\varkappa$ is called Lundberg’s exponent, or adjustment
coefficient.

$\mathsf{E}{\,e^{xT}\,}=1$ (4.4)

and in terms of the associated random variables $\mskip
1.5mu\overline{\mskip-2.0mu{T}\mskip-1.20001mu}\mskip
0.90001mu_{i}\overset{d}{=}\mskip
1.5mu\overline{\mskip-2.0mu{T}\mskip-1.20001mu}\mskip 0.90001mu$ and $\mskip
3.59999mu\overline{\mskip-4.79999mu{X}\mskip-2.0mu}\mskip
1.5mu_{i}\overset{d}{=}\mskip
3.59999mu\overline{\mskip-4.79999mu{X}\mskip-2.0mu}\mskip 1.5mu$,
$i=1,2,\dots$,

$\displaystyle m_{{}_{\scriptscriptstyle\vartriangle}}=\mu_{\mskip
3.59999mu\overline{\mskip-4.79999mu{X}\mskip-2.0mu}\mskip 1.5mu}\mu_{\mskip
1.5mu\overline{\mskip-2.0mu{T}\mskip-1.20001mu}\mskip 0.90001mu}^{-1},\quad
D_{{\hskip-1.0pt\scriptscriptstyle\vartriangle}}^{2}=\mathsf{E}\big{(}\mu_{\mskip
1.5mu\overline{\mskip-2.0mu{T}\mskip-1.20001mu}\mskip 0.90001mu}\,\mskip
3.59999mu\overline{\mskip-4.79999mu{X}\mskip-2.0mu}\mskip 1.5mu-\mu_{\mskip
3.59999mu\overline{\mskip-4.79999mu{X}\mskip-2.0mu}\mskip 1.5mu}\,\mskip
1.5mu\overline{\mskip-2.0mu{T}\mskip-1.20001mu}\mskip
0.90001mu\big{)}^{2}\mu_{\mskip
1.5mu\overline{\mskip-2.0mu{T}\mskip-1.20001mu}\mskip 0.90001mu}^{-3},$
$\displaystyle\mathbb{C}=\frac{1}{\varkappa\,\mu_{\mskip
1.5mu\overline{\mskip-2.0mu{T}\mskip-1.20001mu}\mskip
0.90001mu}}\exp\bigg{\\{}-\sum_{n=1}^{\infty}\frac{1}{n}\,\mathsf{P}\big{\\{}{V}_{n}>0\big{\\}}-\sum_{n=1}^{\infty}\frac{1}{n}\,\mathsf{P}\big{\\{}{\mskip
1.5mu\overline{\mskip-2.0mu{V}\mskip-2.40002mu}\mskip 1.5mu}_{n}\leqslant
0\big{\\}}\bigg{\\}},$

where ${\mskip 1.5mu\overline{\mskip-2.0mu{V}\mskip-2.40002mu}\mskip
1.5mu}_{n}=\sum_{i=1}^{n}\mskip
1.5mu\overline{\mskip-2.0mu{T}\mskip-1.20001mu}\mskip 0.90001mu_{i}$. Recall
(see, e.g., Example (b) in [37], Chapter XII, Section 4) that the associated
random variables we define as follows. Starting with
$\mathcal{I\hskip-0.5pt/I}$ basis $(T_{i},X_{i})\overset{d}{=}(T,X)$,
$i=1,2,\dots$, we switch from $F_{TX}(t,x)=\mathsf{P}\big{\\{}T\leqslant
t,X\leqslant x\big{\\}}$ to $F_{\mskip
1.5mu\overline{\mskip-2.0mu{T}\mskip-1.20001mu}\mskip 0.90001mu\mskip
3.59999mu\overline{\mskip-4.79999mu{X}\mskip-2.0mu}\mskip
1.5mu}(t,x)=\mathsf{P}\big{\\{}\mskip
1.5mu\overline{\mskip-2.0mu{T}\mskip-1.20001mu}\mskip 0.90001mu\leqslant
t,\mskip 3.59999mu\overline{\mskip-4.79999mu{X}\mskip-2.0mu}\mskip
1.5mu\leqslant x\big{\\}}$ defined by the integral
$\int_{-cx}^{t}\int_{0}^{x}e^{\varkappa u}\,F_{TX}(du,dv)$. The latter
probability distribution is proper and $\mathsf{E}\,\mskip
1.5mu\overline{\mskip-2.0mu{T}\mskip-1.20001mu}\mskip 0.90001mu>0$. Commonly
used shorthand for it is $F_{\mskip
1.5mu\overline{\mskip-2.0mu{T}\mskip-1.20001mu}\mskip 0.90001mu\mskip
3.59999mu\overline{\mskip-4.79999mu{X}\mskip-2.0mu}\mskip
1.5mu}(dt,dx)=e^{\varkappa z}F_{TX}(dt,dx)$.

######  Remark 4.1.

The proof of (4.3) carried out using the modular approach is based (see [9]
and [63]) on the following counterpart of equality (4.1):

$\mathsf{P}\big{\\{}{S}_{N_{t}}\leqslant
x\\}=\mathsf{E}\bigg{(}\exp\bigg{\\{}-\varkappa\;\sum_{i=1}^{{\bar{\mathcal{N}}}_{t}}\,{\bar{\mathcal{T}}}_{i}\bigg{\\}}\,\boldsymbol{\mathsf{1}}_{(-\infty,x]}\bigg{(}\sum_{i=1}^{{\bar{\mathcal{N}}}_{t}}\,{\bar{\mathcal{X}}}_{i}\bigg{)}\bigg{)},$
(4.5)

where ${\bar{\mathcal{N}}}_{t}=\inf\big{\\{}n\geqslant
1:\sum_{i=1}^{n}{{\bar{\mathcal{T}}}_{i}}>t\big{\\}}$, or $\infty$, if the set
is empty, and the simple modular basis
$({\bar{\mathcal{T}}}_{k},{\bar{\mathcal{X}}}_{k})$, $k=1,2,\dots$, is given
by

${\bar{\mathcal{T}}}_{k}={\mskip
1.5mu\overline{\mskip-2.0mu{V}\mskip-2.40002mu}\mskip 1.5mu}_{\,\mskip
0.59999mu\overline{\mskip-1.20001mu{\mathbb{\ell}}\mskip 1.20001mu}\mskip
1.5mu_{k}}-{\mskip 1.5mu\overline{\mskip-2.0mu{V}\mskip-2.40002mu}\mskip
1.5mu}_{\,\mskip 0.59999mu\overline{\mskip-1.20001mu{\mathbb{\ell}}\mskip
1.20001mu}\mskip
1.5mu_{k-1}}>0\quad\text{and}\quad{\bar{\mathcal{X}}}_{k}={\mskip
1.5mu\overline{\mskip-2.0mu{S}\mskip-2.40002mu}\mskip 1.5mu}_{\,\mskip
0.59999mu\overline{\mskip-1.20001mu{\mathbb{\ell}}\mskip 1.20001mu}\mskip
1.5mu_{k}}-{\mskip 1.5mu\overline{\mskip-2.0mu{S}\mskip-2.40002mu}\mskip
1.5mu}_{\,\mskip 0.59999mu\overline{\mskip-1.20001mu{\mathbb{\ell}}\mskip
1.20001mu}\mskip 1.5mu_{k-1}},\quad k=1,2,\dots,$

where ${\mskip 1.5mu\overline{\mskip-2.0mu{S}\mskip-2.40002mu}\mskip
1.5mu}_{n}=\sum_{i=1}^{n}\mskip
3.59999mu\overline{\mskip-4.79999mu{X}\mskip-2.0mu}\mskip 1.5mu_{i}$ and the
ladder indices generated by the associated random variables are $\,\mskip
0.59999mu\overline{\mskip-1.20001mu{\mathbb{\ell}}\mskip 1.20001mu}\mskip
1.5mu_{0}\equiv 0$ and $\,\mskip
0.59999mu\overline{\mskip-1.20001mu{\mathbb{\ell}}\mskip 1.20001mu}\mskip
1.5mu_{k}=\inf\big{\\{}i>\,\mskip
0.59999mu\overline{\mskip-1.20001mu{\mathbb{\ell}}\mskip 1.20001mu}\mskip
1.5mu_{k-1}:{\mskip 1.5mu\overline{\mskip-2.0mu{V}\mskip-2.40002mu}\mskip
1.5mu}_{i}>{\mskip 1.5mu\overline{\mskip-2.0mu{V}\mskip-2.40002mu}\mskip
1.5mu}_{\,\mskip 0.59999mu\overline{\mskip-1.20001mu{\mathbb{\ell}}\mskip
1.20001mu}\mskip 1.5mu_{k-1}}\big{\\}}$, $k=1,2,\dots$.∎

######  Remark 4.2.

Edgeworth’s expansions are known for both the normal approximation (4.2) and
(see [63]) the quasi–normal approximation (4.3). Further details see, e.g., in
[65], Chapter 9.∎

######  Example 4.1 (Example 2.1 continued).

Let the random variables $X_{i}\overset{d}{=}X$ and $Y_{i}\overset{d}{=}Y$,
$i=1,2,\dots$, be exponentially distributed with parameters $\varrho>0$ and
$\rho>0$. Clearly, $\mathsf{E}{\,e^{-xcX}}=\varrho/(\varrho+cx)$ and
$\mathsf{E}{\,e^{xY}}=\rho/(\rho-x)$. It follows that
$\mathsf{E}{\,e^{xT}\,}=\mathsf{E}{\,e^{xY}}\mathsf{E}{\,e^{-xcX}}$, whence
$\mathsf{E}{\,e^{xT}\,}=\rho\varrho/((\rho-x)(\varrho+cx))$, and equation
(4.4) is the quadratic equation $\rho\varrho/((\rho-x)(\varrho+cx))=1$ with
respect to $x$. For $c>c^{\ast}=\varrho/\rho$, its positive solution is
$\varkappa=\rho\,(1-\varrho/(c\rho))$. Further calculations yield
$\mathbb{C}=\varrho/(c\rho)$ and

$\begin{gathered}m_{{}_{\scriptscriptstyle\triangledown}}=-\frac{1}{c\,(1-\varrho/(c\rho))},\quad
D_{{\hskip-1.0pt\scriptscriptstyle\triangledown}}^{2}=-\frac{2\,(\varrho/(c\rho))}{c^{2}\rho\,(1-\varrho/(c\rho))^{3}},\\\\[0.0pt]
m_{{}_{\scriptscriptstyle\vartriangle}}=\frac{\varrho/(c\rho)}{c\,(1-\varrho/(c\rho))},\quad
D_{{\hskip-1.0pt\scriptscriptstyle\vartriangle}}^{2}=\frac{2\,(\varrho/(c\rho))}{c^{2}\rho\,(1-\varrho/(c\rho))^{3}}.\end{gathered}$

Figure 3. Graph ($X$-axis is $c$) of $\mathsf{P}\big{\\{}{S}_{N_{t}}\leqslant
x\big{\\}}$ (solid line), which is a monotone decreasing function calculated
numerically, using equalities (2.2)–(2.4), and the approximation (4.2), (4.3)
(dash–dotted line), when $T$ and $Y$ are exponentially distributed with
parameters $\delta=2$ and $\rho=1$, respectively, and $t=200$, $u=10$.
Horizontal line: $\mathsf{P}\big{\\{}{S}_{N_{t}}\leqslant x\big{\\}}$ with $c$
equal to $c^{\ast}$, which is $0.699$. Figure 4. Graphs ($X$-axis is $c$) of
$\mathsf{P}\big{\\{}{S}_{N_{t}}\leqslant x\big{\\}}$ (solid line) calculated
numerically, using equalities (2.2)–(2.4), and approximation (4.6)
(dash–dotted line), when $T$ and $Y$ are exponentially distributed with
parameters $\delta=2$ and $\rho=1$, respectively, and $t=200$, $u=10$.
Horizontal line: $\mathsf{P}\big{\\{}{S}_{N_{t}}\leqslant x\big{\\}}$ with $c$
equal to $c^{\ast}$, which is $0.699$.

Approximations (4.2) and (4.3) are compared (see Fig. 3) with the exact values
of $\mathsf{P}\big{\\{}{S}_{N_{t}}\leqslant x\big{\\}}$ calculated
numerically. They are unsatisfactory in the vicinity of the point $c^{\ast}$
and satisfactory elsewhere. ∎

######  Remark 4.3.

The assumption that a positive solution $\varkappa$ to the equation (4.4)
exists is a serious constraint on $T=Y-cX$. It follows that
$\mathsf{E}{\,e^{xT}\,}$ must exist in the right neighborhood of $0$ and that
$\mathsf{P}\big{\\{}T>x\big{\\}}$ is exponentially bounded from above for
$x>0$. The latter follows from Markov’s inequality, as follows:
$\mathsf{P}\big{\\{}T>x\big{\\}}\leqslant e^{-\varkappa
x}\,\mathsf{E}{\,e^{\varkappa T}}=e^{-\varkappa x}$.∎

###  4.4. Inverse Gaussian approximation for
$\boldsymbol{\mathcal{I\hskip-0.5pt/I}}$ sums

The asymptotic (as $t\to\infty$) behavior of $\mathcal{I\hskip-0.5pt/I}$ sum
does not necessarily have to be related to a normal distribution. The
following inverse Gaussian approximation (see details in [65], [66]) for
$\mathsf{P}\\{{S}_{N_{t}}\leqslant x\\}$ has great advantages. In the
framework of Example 2.1, introduce $M=\mathsf{E}{X}/\mathsf{E}{Y}$ and
$D^{\,2}=((\mathsf{E}{X})^{2}\mathsf{D}{Y}+(\mathsf{E}{Y})^{2}\mathsf{D}{X})/(\mathsf{E}{Y})^{3}$.
Under certain regularity conditions, we have

$\lim_{t\to\infty}\sup_{x>0}\big{|}\,\mathsf{P}\big{\\{}{S}_{N_{t}}\leqslant
x\big{\\}}-{\mathcal{M}}_{t,c}(x)\,\big{|}=0,$ (4.6)

where

$\displaystyle{\mathcal{M}}_{t,c}(x)$
$\displaystyle=\int_{0}^{\frac{cx}{t}}\frac{1}{z+1}\,\varphi_{\left({cM(z+1)},{\frac{c^{2}D^{2}}{u}(z+1)}\right)}(z)\,dz,$

or, in equivalent form,

$\displaystyle{\mathcal{M}}_{t,c}(x)$
$\displaystyle=\begin{cases}\big{(}F(z+1;\mu,\lambda)\\\\[2.0pt] \hskip
30.0pt-F(1;\mu,\lambda)\big{)}\Big{|}_{z=\frac{cx}{t},\mu=\frac{1}{1-cM},\lambda=\frac{t}{c^{2}D^{2}}},&0<c\leqslant
c^{\ast},\\\\[2.0pt]
\exp\bigg{\\{}-\dfrac{2\lambda}{\hat{\mu}}\bigg{\\}}\,\big{(}F(z+1;\hat{\mu},\lambda)\\\\[6.0pt]
\hskip
30.0pt-F(1;\hat{\mu},\lambda)\big{)}\Big{|}_{z=\frac{cx}{t},\hat{\mu}=\frac{1}{cM-1},\lambda=\frac{t}{c^{2}D^{2}}},&c>c^{\ast}\end{cases}$

and

$\displaystyle F(z;\mu,\lambda)=$
$\displaystyle\,\varPhi_{\left({0},{1}\right)}\bigg{(}\sqrt{\frac{\lambda}{z}}\,\bigg{(}\frac{z}{\mu}-1\bigg{)}\bigg{)}$
(4.7)
$\displaystyle+\exp\bigg{\\{}\frac{2\lambda}{\mu}\bigg{\\}}\,\varPhi_{\left({0},{1}\right)}\bigg{(}-\sqrt{\frac{\lambda}{z}}\,\bigg{(}\frac{z}{\mu}+1\bigg{)}\bigg{)},\quad
z>0.$

The approximation (4.6) is called inverse Gaussian because (4.7) is c.d.f. of
the inverse Gaussian distribution with parameters $\mu>0$ and $\lambda>0$.

######  Remark 4.4.

In [65], Chapter 7, refinements (i.e., analogues of the Berry–Esseen estimate
and the Edgeworth expansion) of the inverse Gaussian approximation (4.6) are
obtained.∎

Comparing approximations (4.2) and (4.3) with (4.6), one can argue191919Using
heuristic arguments and referring to exact formulas (2.2)–(2.4). that the
results for proper and defective $\mathcal{I\hskip-0.5pt/I}$ sums in Example
2.1 must border seamlessly. But the normal (with
$0<c<c^{\ast}=\mathsf{E}{Y}/\mathsf{E}{T}$) and quasi–normal (with
$c>c^{\ast}$) approximations (4.2) and (4.3) do not border seamlessly.
Moreover (see Fig. 3), both of them are either poor, or
invalid202020Quasi–normal approximation (4.3) formally requires $c>c^{\ast}$.
The existence of $\varkappa>0$ is possible only if the tail of $Y$ is
exponentially decreasing. for $c$ at or near the critical value $c^{\ast}$.

This distressing fact only indicates that the normal and quasi–normal
approximations are deficient at $c^{\ast}$. In contrast, the inverse Gaussian
approximation (see Fig. 4) does not have such a drawback. The case when $c$ is
close to $c^{\ast}$ or even equals $c^{\ast}$ is routinely covered by the
inverse Gaussian approximation.

######  Remark 4.5.

The intuitive explanation of deficiency of normal and quasi–normal
approximations in the vicinity of the critical value $c^{\ast}$ is given in
[64], Section 9.2.4. It is as follows. For exponentially distributed $X_{i}$
and $Y_{i}$, $i=1,2,\dots$, and $c$ equal to $c^{\ast}$ the compound sum is
made up of the random variables with power moments of order no greater than
$1/2$. Therefore, there is no reason to expect validity of normal and
quasi–normal approximations.∎

The flaw of (4.2) and (4.3) in the vicinity od $c^{\ast}$ is merely a drawback
of a particular mathematical technique. However, it has had a major negative
impact on insurance modeling: while the insurance process in a close
neighborhood of the critical point $c^{\ast}$ is of great importance (see
details in [64]), it was overlooked for a long time partly due to this flaw.
Moreover, the inverse Gaussian approximation allows (see [66]) to develop a
new approach to risk measures different from value at risk (VaR).

##  5\. Limit theorems for $\boldsymbol{\mathcal{R\hskip-0.5pt/D}}$ and proper
$\boldsymbol{\mathcal{I\hskip-0.5pt/D}}$ sums

Kesten mentioned the part of modular analysis that deals with the lack of
independence. It is best known because the area of “Markov chains, continuous
time processes, and other situations in which the independence between
summands no longer applies” is of particular interest. Among the tricks which
reduce limit theorems for dependent summands to the case of independent
summands, Kesten mentioned “Dœblin’s trick”.

Studying limit theorems for $\mathcal{R\hskip-0.5pt/D}$ and proper
$\mathcal{I\hskip-0.5pt/D}$ sums, we will combine “Dœblin’s trick” and
“Blackwell’s trick” and extend it to compound sums with general modular
structure.

###  5.1. Compound sums with general modular structure

For $t>0$ and a complex basis $(T_{i},X_{i})$, $i=1,2,\dots$, we focus on
unaltered complex compound sum ${S}_{N_{t}}$. We denote by
$0<\mathcal{J}_{0}<\mathcal{J}_{1}<\dots$ a sequence of random indices, put

$\begin{gathered}{\mathcal{\tau}}_{\hskip
1.0pt0}=\mathcal{J}_{0},\quad{\mathcal{T}}_{\hskip
1.0pt0}=\sum_{i=1}^{\mathcal{J}_{0}}\;T_{i},\quad{\mathcal{X}}_{\hskip
1.0pt0}=\sum_{i=1}^{\mathcal{J}_{0}}\;X_{i},\\\\[0.0pt]
{\mathcal{\tau}}_{\hskip
1.0ptk}=\mathcal{J}_{k}-\mathcal{J}_{k-1},\quad{\mathcal{T}}_{\hskip
1.0ptk}=\sum_{i=\mathcal{J}_{k-1}+1}^{\mathcal{J}_{k}}T_{i},\quad{\mathcal{X}}_{\hskip
1.0ptk}=\sum_{i=\mathcal{J}_{k-1}+1}^{\mathcal{J}_{k}}X_{i},\end{gathered}$

where $k=1,2,\dots$, and introduce the following definition.

######  Definition 5.1.

We say that the basis $(T_{i},X_{i})$, $i=1,2,\dots$, allows a modular
structure if there exist a.s. finite random indices
$0<\mathcal{J}_{0}<\mathcal{J}_{1}<\dots$, such that

  * (i)

the random vectors $({\mathcal{\tau}}_{\hskip 1.0ptk},{\mathcal{T}}_{\hskip
1.0ptk},{\mathcal{X}}_{\hskip 1.0ptk})$ are independent for $k=0,1,\dots$, and
identically distributed for $k=1,2,\dots$,

  * (ii)

${V}_{\mathcal{J}_{k}}>\max_{0\leqslant i\leqslant\mathcal{J}_{k}-1}{V}_{i}$,
$k=0,1,\dots$, i.e., $\mathcal{J}_{i}$, $i=0,1,\dots$, are upper record
moments of the sequence ${V}_{n}=\sum_{i=1}^{n}T_{i}$, $n=1,2,\dots$.

For modular i.i.d. random vectors $({\mathcal{X}}_{\hskip
1.0ptk},{\mathcal{T}}_{\hskip 1.0ptk},{\mathcal{\tau}}_{\hskip
1.0ptk})\overset{d}{=}({\mathcal{X}}_{\hskip 1.0pt},{\mathcal{T}}_{\hskip
1.0pt},{\mathcal{\tau}}_{\hskip 1.0pt})$, $k=1,2,\dots$, we use the shorthand
notation $\mu_{{\mathcal{\tau}}_{\hskip
1.0pt}}=\mathsf{E}{\mathcal{\tau}}_{\hskip 1.0pt}$,
$\mu_{{\mathcal{T}}_{\hskip 1.0pt}}=\mathsf{E}{\mathcal{T}}_{\hskip 1.0pt}$,
$\mu_{{\mathcal{X}}_{\hskip 1.0pt}}=\mathsf{E}{\mathcal{X}}_{\hskip 1.0pt}$,
$\sigma_{{\mathcal{\tau}}_{\hskip
1.0pt}}^{2}=\mathsf{D}{\mathcal{\tau}}_{\hskip 1.0pt}$,
$\sigma_{{\mathcal{T}}_{\hskip 1.0pt}}^{2}=\mathsf{D}{\mathcal{T}}_{\hskip
1.0pt}$, $\sigma_{{\mathcal{X}}_{\hskip
1.0pt}}^{2}=\mathsf{D}{\mathcal{X}}_{\hskip 1.0pt}$. For
$\hat{\mathcal{\tau}}^{\hskip 1.0pt}={\mathcal{\tau}}_{\hskip
1.0pt}-\mu_{{\mathcal{\tau}}_{\hskip 1.0pt}}$, $\hat{\mathcal{T}}^{\hskip
1.0pt}={\mathcal{T}}_{\hskip 1.0pt}-\mu_{{\mathcal{T}}_{\hskip 1.0pt}}$,
$\hat{\mathcal{X}}^{\hskip 1.0pt}={\mathcal{X}}_{\hskip
1.0pt}-\mu_{{\mathcal{X}}_{\hskip 1.0pt}}$, and write
$\varkappa_{{\mathcal{T}}_{\hskip 1.0pt}{\mathcal{\tau}}_{\hskip
1.0pt}}=\mathsf{E}\hat{\mathcal{T}}^{\hskip
1.0pt}\\!\hat{\mathcal{\tau}}^{\hskip 1.0pt}$,
$\varkappa_{{\mathcal{X}}_{\hskip 1.0pt}{\mathcal{T}}_{\hskip
1.0pt}}=\mathsf{E}\hat{\mathcal{X}}^{\hskip 1.0pt}\hat{\mathcal{T}}^{\hskip
1.0pt}$, $\varkappa_{{\mathcal{X}}_{\hskip 1.0pt}{\mathcal{\tau}}_{\hskip
1.0pt}}=\mathsf{E}\hat{\mathcal{X}}^{\hskip 1.0pt}\hat{\mathcal{\tau}}^{\hskip
1.0pt}$, $\rho_{{\mathcal{T}}_{\hskip 1.0pt}{\mathcal{\tau}}_{\hskip
1.0pt}}=\varkappa_{{\mathcal{T}}_{\hskip 1.0pt}{\mathcal{\tau}}_{\hskip
1.0pt}}/(\sigma_{{\mathcal{T}}_{\hskip 1.0pt}}\sigma_{{\mathcal{\tau}}_{\hskip
1.0pt}})$ $\rho_{{\mathcal{X}}_{\hskip 1.0pt}{\mathcal{T}}_{\hskip
1.0pt}}=\varkappa_{{\mathcal{X}}_{\hskip 1.0pt}{\mathcal{T}}_{\hskip
1.0pt}}/(\sigma_{{\mathcal{X}}_{\hskip 1.0pt}}\sigma_{{\mathcal{T}}_{\hskip
1.0pt}})$ $\rho_{{\mathcal{X}}_{\hskip 1.0pt}{\mathcal{\tau}}_{\hskip
1.0pt}}=\varkappa_{{\mathcal{X}}_{\hskip 1.0pt}{\mathcal{\tau}}_{\hskip
1.0pt}}/(\sigma_{{\mathcal{X}}_{\hskip 1.0pt}}\sigma_{{\mathcal{\tau}}_{\hskip
1.0pt}})$. We further introduce zero–mean block random variables212121Note
that $\mathsf{E}\big{(}{\mathcal{R_{missing}}{}}_{\hskip
1.0pt\\!{\mathcal{X}}_{\hskip 1.0pt}\\!{\mathcal{T}}_{\hskip
1.0pt}}^{2}\big{)}$ is the same as
$\mathsf{D}\big{(}{\mathcal{R_{missing}}{}}_{\hskip
1.0pt\\!{\mathcal{X}}_{\hskip 1.0pt}\\!{\mathcal{T}}_{\hskip 1.0pt}}\big{)}$
and $\mathsf{E}\big{(}{\mathcal{R_{missing}}{}}_{\hskip
1.0pt\\!{\mathcal{\tau}}_{\hskip 1.0pt}\\!{\mathcal{T}}_{\hskip
1.0pt}}^{2}\big{)}$ is the same as
$\mathsf{D}\big{(}{\mathcal{R_{missing}}{}}_{\hskip
1.0pt\\!{\mathcal{\tau}}_{\hskip 1.0pt}\\!{\mathcal{T}}_{\hskip
1.0pt}}\big{)}$.

$\displaystyle{\mathcal{R_{missing}}{}}_{\hskip 1.0pt\\!{\mathcal{T}}_{\hskip
1.0pt}{\mathcal{\tau}}_{\hskip 1.0pt}}$
$\displaystyle=\mu_{{\mathcal{T}}_{\hskip 1.0pt}}\,{\mathcal{\tau}}_{\hskip
1.0pt}-\mu_{{\mathcal{\tau}}_{\hskip 1.0pt}}{\mathcal{T}}_{\hskip 1.0pt}$
$\displaystyle{\mathcal{R_{missing}}{}}_{\hskip 1.0pt\\!{\mathcal{X}}_{\hskip
1.0pt}\\!{\mathcal{T}}_{\hskip 1.0pt}}=\mu_{{\mathcal{T}}_{\hskip
1.0pt}}{\mathcal{X}}_{\hskip 1.0pt}-\mu_{{\mathcal{X}}_{\hskip
1.0pt}}{\mathcal{T}}_{\hskip 1.0pt}$ and
$\displaystyle=\mu_{{\mathcal{T}}_{\hskip 1.0pt}}\hat{\mathcal{\tau}}^{\hskip
1.0pt}-\mu_{{\mathcal{\tau}}_{\hskip 1.0pt}}\hat{\mathcal{T}}^{\hskip 1.0pt},$
$\displaystyle=\mu_{{\mathcal{T}}_{\hskip 1.0pt}}\hat{\mathcal{X}}^{\hskip
1.0pt}-\mu_{{\mathcal{X}}_{\hskip 1.0pt}}\hat{\mathcal{T}}^{\hskip 1.0pt}$

and put

$\begin{gathered}\mathcal{M}_{N}{\,\\!}=\mu_{{\mathcal{\tau}}_{\hskip
1.0pt}}\mu_{{\mathcal{T}}_{\hskip
1.0pt}}^{-1},\quad\mathcal{D}_{N}^{\,2}{\,}=\mathsf{E}\big{(}{\mathcal{R_{missing}}{}}_{\hskip
1.0pt\\!{\mathcal{T}}_{\hskip 1.0pt}{\mathcal{\tau}}_{\hskip
1.0pt}}^{2}\big{)}\,\mu_{{\mathcal{T}}_{\hskip 1.0pt}}^{-3},\\\\[2.0pt]
\mathcal{M}_{{S}}{\hskip 0.6pt\\!}=\mu_{{\mathcal{X}}_{\hskip
1.0pt}}\mu_{{\mathcal{T}}_{\hskip
1.0pt}}^{-1},\quad\mathcal{D}_{{S}}^{\,2}{\,}=\mathsf{E}\big{(}{\mathcal{R_{missing}}{}}_{\hskip
1.0pt\\!{\mathcal{X}}_{\hskip 1.0pt}\\!{\mathcal{T}}_{\hskip
1.0pt}}^{2}\big{)}\mu_{{\mathcal{T}}_{\hskip 1.0pt}}^{-3},\\\\[0.0pt]
\mathcal{B}_{{S}N}{\,\\!}=\mathsf{E}\big{(}{\mathcal{R_{missing}}{}}_{\hskip
1.0pt\\!{\mathcal{T}}_{\hskip 1.0pt}{\mathcal{\tau}}_{\hskip
1.0pt}}{\mathcal{R_{missing}}{}}_{\hskip 1.0pt\\!{\mathcal{X}}_{\hskip
1.0pt}\\!{\mathcal{T}}_{\hskip 1.0pt}}\big{)}\,\mu_{{\mathcal{T}}_{\hskip
1.0pt}}^{-3},\quad\mathcal{\rho}_{{S}\\!N}{\,\\!}=\mathcal{B}_{{S}N}{\,}\big{(}\mathcal{D}_{{S}}^{\,}{\,}\mathcal{D}_{N}^{\,}{\,\\!}\big{)}^{-1}.\end{gathered}$

Assumption (i) tackles “dependence complexity”, while (ii) takes hold of
“irregularity complexity”, in both cases by reducing the original complex
compound sum to a simple modular compound sum with simple modular basis
$({\mathcal{T}}_{\hskip 1.0ptk},{\mathcal{X}}_{\hskip 1.0ptk})$,
$k=1,2,\dots$. It consists of i.i.d. modular random vectors with
$s$–components positive. Moreover, if ${V}_{\mathcal{J}_{k}}<t$, then crossing
level $t$ cannot occur inside $k$th block or inside blocks with a smaller
index.

With
$\mathsf{R}_{\,}(x,y)=\dfrac{x-\mathcal{\rho}_{{S}\\!N}{\,\\!}y}{1-\mathcal{\rho}_{{S}\\!N}^{2}{\,}}$,
we will use the following notation:

$\mathcal{K}_{(3,0)}=\,\mathsf{E}\big{(}{\mathcal{R_{missing}}{}}_{\hskip
1.0pt\\!{\mathcal{X}}_{\hskip 1.0pt}\\!{\mathcal{T}}_{\hskip
1.0pt}}^{3}\big{)}\mu_{{\mathcal{T}}_{\hskip
1.0pt}}^{-4}\mathcal{D}_{{S}}^{\,-3}{\,}-3\mathsf{R}_{\,}({L}_{1},{L}_{2}),$

where

$\displaystyle{L}_{1}=$
$\displaystyle\,\big{(}\varkappa_{{\mathcal{X}}_{\hskip
1.0pt}{\mathcal{\tau}}_{\hskip 1.0pt}}(\mu_{{\mathcal{\tau}}_{\hskip
1.0pt}}\sigma_{{\mathcal{T}}_{\hskip 1.0pt}}^{2}-\mu_{{\mathcal{T}}_{\hskip
1.0pt}}\varkappa_{{\mathcal{T}}_{\hskip 1.0pt}{\mathcal{\tau}}_{\hskip
1.0pt}})-\sigma_{{\mathcal{\tau}}_{\hskip
1.0pt}}^{2}(\mu_{{\mathcal{X}}_{\hskip 1.0pt}}\sigma_{{\mathcal{T}}_{\hskip
1.0pt}}^{2}-\mu_{{\mathcal{T}}_{\hskip 1.0pt}}\varkappa_{{\mathcal{X}}_{\hskip
1.0pt}{\mathcal{T}}_{\hskip 1.0pt}})$
$\displaystyle+\varkappa_{{\mathcal{T}}_{\hskip 1.0pt}{\mathcal{\tau}}_{\hskip
1.0pt}}(\mu_{{\mathcal{X}}_{\hskip 1.0pt}}\varkappa_{{\mathcal{T}}_{\hskip
1.0pt}{\mathcal{\tau}}_{\hskip 1.0pt}}-\mu_{{\mathcal{\tau}}_{\hskip
1.0pt}}\varkappa_{{\mathcal{X}}_{\hskip 1.0pt}{\mathcal{T}}_{\hskip
1.0pt}})\big{)}\mu_{{\mathcal{T}}_{\hskip
1.0pt}}^{-3}\mathcal{D}_{{S}}^{\,-1}{\,}\mathcal{D}_{N}^{\,-2}{\,},$
$\displaystyle{L}_{2}=$
$\displaystyle\,\big{(}-\varkappa_{{\mathcal{X}}_{\hskip
1.0pt}{\mathcal{\tau}}_{\hskip 1.0pt}}(\mu_{{\mathcal{X}}_{\hskip
1.0pt}}\sigma_{{\mathcal{T}}_{\hskip 1.0pt}}^{2}-\mu_{{\mathcal{T}}_{\hskip
1.0pt}}\varkappa_{{\mathcal{X}}_{\hskip 1.0pt}{\mathcal{T}}_{\hskip
1.0pt}})+\sigma_{{\mathcal{X}}_{\hskip
1.0pt}}^{2}(\mu_{{\mathcal{\tau}}_{\hskip 1.0pt}}\sigma_{{\mathcal{T}}_{\hskip
1.0pt}}^{2}-\mu_{{\mathcal{T}}_{\hskip 1.0pt}}\varkappa_{{\mathcal{T}}_{\hskip
1.0pt}{\mathcal{\tau}}_{\hskip 1.0pt}})$
$\displaystyle-\varkappa_{{\mathcal{X}}_{\hskip 1.0pt}{\mathcal{T}}_{\hskip
1.0pt}}(\mu_{{\mathcal{\tau}}_{\hskip 1.0pt}}\varkappa_{{\mathcal{X}}_{\hskip
1.0pt}{\mathcal{T}}_{\hskip 1.0pt}}-\mu_{{\mathcal{X}}_{\hskip
1.0pt}}\varkappa_{{\mathcal{T}}_{\hskip 1.0pt}{\mathcal{\tau}}_{\hskip
1.0pt}})\big{)}\mu_{{\mathcal{T}}_{\hskip
1.0pt}}^{-3}\mathcal{D}_{{S}}^{\,-2}{\,}\mathcal{D}_{N}^{\,-1}{\,}.$

It can be shown by direct algebra that
$\mathsf{R}_{\,}\big{(}{L}_{1},{L}_{2}\big{)}=-\big{(}\mu_{{\mathcal{X}}_{\hskip
1.0pt}}\sigma_{{\mathcal{T}}_{\hskip 1.0pt}}^{2}-\mu_{{\mathcal{T}}_{\hskip
1.0pt}}\varkappa_{{\mathcal{X}}_{\hskip 1.0pt}{\mathcal{T}}_{\hskip
1.0pt}}\big{)}\,\mathcal{D}_{{S}}^{\,-1}{\,}\mu_{{\mathcal{T}}_{\hskip
1.0pt}}^{-2}$. Finally, we introduce

$\displaystyle\eta_{\hskip 1.0pt1}$
$\displaystyle=\mathsf{E}{\mathcal{X}}_{\hskip
1.0pt0}+\mu_{{\mathcal{T}}_{\hskip
1.0pt}}^{-1}\sum_{s=1}^{\infty}\int_{-\infty}^{\infty}x\int_{0}^{\infty}\mathsf{P}\big{\\{}{\mathcal{\tau}}_{\hskip
1.0pt1}\geqslant s,{\mathcal{N}}_{\hskip 1.0pt}(t)=s,{\mathcal{X}}_{\hskip
1.0pt0}^{\hskip 1.0pt(s)}\in dx\big{\\}}\,dt,$ $\displaystyle\eta_{\hskip
1.0pt2}$ $\displaystyle=\mathsf{E}{\mathcal{T}}_{\hskip
1.0pt0}+\mu_{{\mathcal{T}}_{\hskip
1.0pt}}^{-1}\sum_{s=1}^{\infty}\int_{0}^{\infty}t\,\mathsf{P}\big{\\{}{\mathcal{\tau}}_{\hskip
1.0pt1}\geqslant s,{\mathcal{N}}_{\hskip 1.0pt}(t)=s\big{\\}}\,dt,$
$\displaystyle\eta_{\hskip 1.0pt3}$
$\displaystyle=\mathsf{E}{\mathcal{\tau}}_{\hskip
1.0pt0}+\mu_{{\mathcal{T}}_{\hskip
1.0pt}}^{-1}\sum_{s=1}^{\infty}s\int_{0}^{\infty}\mathsf{P}\big{\\{}{\mathcal{\tau}}_{\hskip
1.0pt1}\geqslant s,{\mathcal{N}}_{\hskip 1.0pt}(t)=s\big{\\}}\,dt,$

where ${\mathcal{X}}_{\hskip 1.0pt0}^{\hskip
1.0pt(s)}=\sum_{k=\mathcal{J}_{0}+1}^{\mathcal{J}_{0}+s}X_{k}$ and
${\mathcal{N}}_{\hskip 1.0pt}(t)=\inf\big{\\{}n\geqslant
1:\sum_{k=\mathcal{J}_{0}+1}^{\mathcal{J}_{0}+n}T_{k}>t\big{\\}}$, or
$\infty$, if the set is empty.

Theorem 5.1 below is formulated under the following conditions.

Condition $\boldsymbol{V}$ (Non–degenerate correlation matrix)
$\mathsf{det}\,{{{Q}}}>0$ for the correlation matrix

${{{Q}}}={Cor}\big{(}{\mathcal{X}}_{\hskip 1.0pt1},{\mathcal{T}}_{\hskip
1.0pt1},{\mathcal{\tau}}_{\hskip
1.0pt1}\big{)}=\left(\begin{matrix}1&\rho_{{\mathcal{X}}_{\hskip
1.0pt}{\mathcal{T}}_{\hskip 1.0pt}}&\rho_{{\mathcal{X}}_{\hskip
1.0pt}{\mathcal{\tau}}_{\hskip 1.0pt}}\\\\[0.0pt] \rho_{{\mathcal{X}}_{\hskip
1.0pt}{\mathcal{T}}_{\hskip 1.0pt}}&1&\rho_{{\mathcal{T}}_{\hskip
1.0pt}{\mathcal{\tau}}_{\hskip 1.0pt}}\\\\[0.0pt] \rho_{{\mathcal{X}}_{\hskip
1.0pt}{\mathcal{\tau}}_{\hskip 1.0pt}}&\rho_{{\mathcal{T}}_{\hskip
1.0pt}{\mathcal{\tau}}_{\hskip 1.0pt}}&1\end{matrix}\right).$

Condition $\boldsymbol{P}_{\boldsymbol{}}$ (Bounded density condition) For an
integer $n\geqslant 1$, there exists a bounded convolution power
$f_{{\mathcal{T}}_{\hskip 1.0pt},{\mathcal{\tau}}_{\hskip 1.0pt}}^{*n}(x,n)$
w.r.t. Lebesgue measure.

Condition $\boldsymbol{L}$ (Lattice condition) The integer–valued random
variable ${\mathcal{\tau}}_{\hskip 1.0pt1}$ assumes values in the set of
natural numbers with maximal span $1$.

Condition $\boldsymbol{C}$ (Uniform Cramér’s condition)

$\overline{\lim}_{|t_{1}|\to\infty}\big{|}\,\mathsf{E}\exp\big{\\{}it_{1}{\mathcal{X}}_{\hskip
1.0pt}+it_{2}{\mathcal{T}}_{\hskip 1.0pt}+it_{3}{\mathcal{\tau}}_{\hskip
1.0pt}\big{\\}}\big{|}<1$

for all $t_{2},t_{3}\in\mathsf{R}$.

Condition $\boldsymbol{B}_{\boldsymbol{(r,s)}}$ (Block moments condition) For
$r,s>0$

  * (i)

$\mathsf{E}\hskip 1.0pt\big{(}\mathcal{J}_{0}^{\hskip 1.0ptr}\big{)}<\infty$,
$\mathsf{E}\big{(}{\mathcal{\tau}}_{\hskip 1.0pt1}^{s}\big{)}<\infty$,

  * (ii)

$\mathsf{E}\bigg{(}\bigg{[}\displaystyle\sum_{k=1}^{\mathcal{J}_{0}}\big{|}\,T_{k}\big{|}\,\bigg{]}^{r}\bigg{)}<\infty$,
$\mathsf{E}\bigg{(}\bigg{[}\displaystyle\sum_{k=\mathcal{J}_{0}+1}^{\mathcal{J}_{1}}\big{|}\,T_{k}\big{|}\,\bigg{]}^{\hskip-1.0pts}\bigg{)}<\infty$,

  * (iii)

$\mathsf{E}\bigg{(}\bigg{[}\displaystyle\sum_{k=1}^{\mathcal{J}_{0}}\big{|}\,X_{k}\big{|}\,\bigg{]}^{r}\bigg{)}<\infty$,
$\mathsf{E}\bigg{(}\bigg{[}\displaystyle\sum_{k=\mathcal{J}_{0}+1}^{\mathcal{J}_{1}}\big{|}\,X_{k}\big{|}\,\bigg{]}^{\hskip-1.0pts}\,\bigg{)}<\infty$.

The following theorem is Theorem 1 (III) in [62].

######  Theorem 5.1.

Assume that the basis $(T_{i},X_{i})$, $i=1,2,\dots$, of the complex compound
sum ${S}_{N_{t}}$ allows a modular structure. If conditions ${V}$, ${P}$,
${L}$, ${C}$, and ${B}_{(k-\frac{3}{2},k)}$ with $k>3$ are satisfied, then
there exist polynomials $\mathsf{Q}_{\,r}(x)$, $r=1,2,\dots,k-3$, of degree
$3r-1$ such that

$\displaystyle\sup_{x\in\mathsf{R}}\,\bigg{|}\,\mathsf{P}\big{\\{}{S}_{N_{t}}-\mathcal{M}_{{S}}{\hskip
0.6ptt}\leqslant
x\mathcal{D}_{{S}}^{\,}{\,t^{1/2}}\big{\\}}-\varPhi_{\left({0},{1}\right)}(x)$
$\displaystyle\hskip
90.0pt-\sum_{r=1}^{k-3}\frac{\mathsf{Q}_{\,r}(x)}{t^{r/2}}\,\varphi_{\left({0},{1}\right)}(x)\,\bigg{|}=O\big{(}t^{(k-2)/2}\big{)},\quad
t\to\infty.$

In particular,

$\mathsf{Q}_{\,1}(x)=-\frac{1}{6}\;\mathcal{K}_{(3,0)}\mathsf{H}_{2}(x)+\mathsf{R}_{\,}({L}_{1},{L}_{2})-\frac{\mu_{{\mathcal{T}}_{\hskip
1.0pt}}\eta_{\hskip 1.0pt1}-\mu_{{\mathcal{X}}_{\hskip 1.0pt}}\eta_{\hskip
1.0pt2}}{\mathcal{D}_{{S}}^{\,}{\,}\mu_{{\mathcal{T}}_{\hskip 1.0pt}}}.$

###### Sketch of proof of Theorem 5.1.

The proof, similarly to the proof of Theorem 3.2, consists of steps A–F.

Step A: use of fundamental identity. Similar to (3.5), we use the total
probability rule. Writing

$\begin{gathered}{\mathcal{X}}_{\hskip 1.0pt}^{[r]}=\sum_{i=1}^{r}X_{i},\
{\mathcal{T}}_{\hskip 1.0pt}^{[r]}=\sum_{i=1}^{r}T_{i},\ {\mathcal{X}}_{\hskip
1.0pt}^{(s)}=\sum_{i=\mathcal{J}_{m}+1}^{\mathcal{J}_{m}+s}X_{i},\
{\mathcal{T}}_{\hskip
1.0pt}^{(s)}=\sum_{i=\mathcal{J}_{m}+1}^{\mathcal{J}_{m}+s}T_{i}\end{gathered}$

and ${\mathcal{N}}_{\hskip 1.0pt}(t)=\inf\big{\\{}s\geqslant
1:{\mathcal{T}}_{\hskip 1.0pt}^{(s)}>t\big{\\}}$, we have

$\displaystyle\mathsf{P}\big{\\{}{S}_{N_{t}}\leqslant
x\big{\\}}=\,\sum_{n=1}^{\infty}\mathsf{P}\big{\\{}{S}_{N_{t}}\leqslant
x,N_{t}=n\big{\\}},$

where the right–hand side is the sum of, first, the expression

$\displaystyle\sum_{n=1}^{\infty}\sum_{r}\iint_{\begin{subarray}{c}0\leqslant
t_{1}\leqslant t\\\\[1.0pt] 0\leqslant x_{1}\leqslant
x\end{subarray}}\mathsf{P}\Big{\\{}{\mathcal{\tau}}_{\hskip
1.0pt0}=r,{\mathcal{T}}_{\hskip 1.0pt}^{[r]}\in dt_{1},{\mathcal{X}}_{\hskip
1.0pt}^{[r]}\in dx_{1}\Big{\\}}$ $\displaystyle\hskip
30.0pt\times\mathsf{P}\Big{\\{}{\mathcal{\tau}}_{\hskip 1.0pt1}\geqslant
n-r,{\mathcal{N}}_{\hskip 1.0pt}(t-t_{1})=n-r,{\mathcal{X}}_{\hskip
1.0pt}^{(n-r)}\leqslant x-x_{1}\Big{\\}},$

which corresponds to the absence of complete blocks, and, second, the
expression

$\displaystyle\sum_{n=1}^{\infty}\sum_{r,s}\sum_{m=1}^{n-(r+s)}\idotsint_{\begin{subarray}{c}0\leqslant
t_{1}+t_{2}\leqslant t\\\\[1.0pt] 0\leqslant x_{1}+x_{2}\leqslant
x\end{subarray}}\mathsf{P}\bigg{\\{}\sum_{i=1}^{m}{\mathcal{X}}_{\hskip
1.0pti}\leqslant x-\big{(}x_{1}+x_{2}\big{)},$ (5.1) $\displaystyle\hskip
100.0pt\sum_{i=1}^{m}{\mathcal{\tau}}_{\hskip
1.0pti}=n-(s+r)\,\bigg{|}\,\sum_{i=1}^{m}{\mathcal{T}}_{\hskip
1.0pti}=t-(t_{1}+t_{2})\big{)}\bigg{\\}}$ $\displaystyle\hskip 30.0pt\times
f_{{\mathcal{T}}_{\hskip
1.0pt}}^{*m}\big{(}t-(t_{1}+t_{2})\big{)}\,\mathsf{P}\Big{\\{}{\mathcal{\tau}}_{\hskip
1.0pt0}=r,{\mathcal{T}}_{\hskip 1.0pt}^{[r]}\in dt_{1},{\mathcal{X}}_{\hskip
1.0pt}^{[r]}\in dx_{1}\Big{\\}}$ $\displaystyle\hskip
30.0pt\times\mathsf{P}\Big{\\{}{\mathcal{\tau}}_{\hskip 1.0pt1}\geqslant
s,{\mathcal{N}}_{\hskip 1.0pt}(t_{2})=s,{\mathcal{X}}_{\hskip 1.0pt}^{(s)}\in
dx_{2}\Big{\\}}\,dt_{2}.$

It is clear that for large $t$ only the expression (5.1) matters.

In words, in (5.1) the first incomplete block is made up by $r$ summands and
the value $t$ cannot be exceeded all over this block because $t_{1}<t$; there
are $m$ complete blocks, and exceeding the value $t$ still cannot happen, but
it does occurs on the $s$th summand of the the $(m+1)$th complete block
consisting of more than $s$ summands.

Steps B and C remain essentially the same as in the proof of Theorem 3.2.

Step D: application of refined CLT. The integrand in (5.1) is ready for
application of Edgeworth’s expansion in three–dimensional hybrid
integro–local–local CLT with non–uniform remainder term. These theorems see in
[31]–[33].

Steps E and F are technically much more complicated than in the proof of
refined elementary renewal theorem (3.2) and Theorem 3.2 (they require a
number of special identities), but use essentially the same analytical
methods. ∎

######  Remark 5.1.

Theorem 3.2 is a corollary of Theorem 5.1. For $\mathcal{I\hskip-0.5pt/I}$
basis $(T_{i},X_{i})$, $i=1,2,\dots$, the modular structure is yielded by
random indices $0\equiv\mathbb{\ell}_{0}<\mathbb{\ell}_{1}<\dots$ and, using
Wald’s identities, $\mathcal{M}_{{S}}{\hskip 0.6pt}$ is transformed into
$M_{{S}}{\,}$, and $\mathcal{K}_{(3,0)}$ is transformed into $K_{(3,0)}$.∎

###  5.2. Limit theorems for $\boldsymbol{\mathcal{R\hskip-0.5pt/D}}$ sums
with Markov dependence

Markov renewal process (see, e.g., [24], [78]) is a homogeneous
two–dimensional Markov chain $(\xi_{k},T_{k})$, $k=0,1,\dots$, which takes
values in a general state space
$(\mathit{E}\times\mathsf{R}^{+},\mathcal{E}\otimes\mathcal{R})$. Its
transition functions are defined by a semi–Markov transition kernel. Markov
renewal processes and their counterpart, semi–Markov processes, are among the
most popular models of applied probability222222See, e.g., two bibliographies
on semi–Markov processes [85] and [86]. The former consists of about 600
papers by some 300 authors and the later of almost a thousand papers by more
than 800 authors..

In the same way as for renewal process, this model can be reformulated in
terms of $\mathcal{R\hskip-0.5pt/D}$ basis $(T_{i},X_{i})$, $i=1,2,\dots$, and
$\mathcal{R\hskip-0.5pt/D}$ sum with Markov dependence. Under certain
regularity conditions, the sequence $0<\mathcal{J}_{0}<\mathcal{J}_{1}<\dots$
from Definition 5.1 is built in [7], [72]. Theorem analogous to Theorem 5.1
see in [60].

######  Remark 5.2 (Refined CLT for Markov chains).

In the special case $T_{i}\equiv 1$, the $\mathcal{R\hskip-0.5pt/D}$ sum with
Markov dependence becomes the ordinary sum ${S}_{n}$ of random variables
defined on the Markov chain $\\{\xi_{i}\\}_{i\geqslant 0}$.

Originally, “Dœblin’s trick” was applied to ${S}_{n}$ with discrete
$\\{\xi_{i}\\}_{i\geqslant 0}$. Subsequently, it was repeatedly noted (see,
e.g., [25] and [68]) that not the full countable space structure is often
needed, but just the existence of one single “proper” point. The results then
carry over with only notational changes to the countable case and, moreover,
to (by means of the “splitting technique” and its counterparts, see [8], [72],
[73], and [68]) irreducible recurrent general state space Markov chain
$\\{\xi_{i}\\}_{i\geqslant 0}$ which satisfies strong minorization condition.
Using modular advanced technique, for discrete (see [14]) and general state
space Markov chains (see [15], [57], [59]) Berry–Esseen’s estimate,
Edgeworth’s expansions, and asymptotic results for large deviations in CLT
have been obtained.∎

###  5.3. Limit theorems for proper $\boldsymbol{\mathcal{I\hskip-0.5pt/D}}$
sums with Markov dependence

Markov additive process (see, e.g., [23], [78]) is a generalization of Markov
renewal process with $(\xi_{k},T_{k})$, $k=0,1,\dots$, taking values in
$(\mathit{E}\times\mathsf{R},\mathcal{E}\otimes\mathcal{R})$ rather than in
$(\mathit{E}\times\mathsf{R}^{+},\mathcal{E}\otimes\mathcal{R})$. Under
certain regularity conditions, for proper $\mathcal{I\hskip-0.5pt/D}$ sums
with Markov dependence, the sequence $0<\mathcal{J}_{0}<\mathcal{J}_{1}<\dots$
from Definition 5.1 is built in [4] and [5]. Loosely speaking, these are the
ladder points at which regeneration occurs.

##  6\. Analysis of defective $\boldsymbol{\mathcal{I\hskip-0.5pt/D}}$ sums

For defective $\mathcal{I\hskip-0.5pt/D}$ sums, a deep analytical insight,
such as in the case of defective $\mathcal{I\hskip-0.5pt/I}$ sums, is (see,
e.g., [70], [74]–[77]) hardly possible. An alternative is computer–intensive
analysis (see, e.g., [2], [50]).

Note that the papers devoted to defective $\mathcal{I\hskip-0.5pt/D}$ sums
with Markov dependence are formulated as a study of the ruin probabilities for
risk processes of Markovian type. This link with the ruin theory not only
suggests rational examples of defective $\mathcal{I\hskip-0.5pt/D}$ sums, but
also indicates areas in which they are of major interest.

##  7\. A concluding remark

Due credits must be given to Dœblin (1915–1940), who (see [51], [52], [67])
put forth (see [26]–[29]) the major innovative ideas of coupling,
decomposition and dissection232323The English translation of the title of [26]
is “On two problems of Kolmogorov concerning countable Markov chains”, with
reference to [48]. Therefore, Dœblin’s glory in this regard is partly shared
with Kolmogorov. in Markov chains. The contributions of many scientists after
him are mere a development of his seminal ideas. In this regard, compelling is
the phrase from [40], that “our goal is to isolate the key ingredients in
Dœblin’s proof and use them to derive ergodic theorems for much more general
Markov processes on a typically uncountable state space”.

Having created a tool for sequential analysis, Anscombe (see [6]) rediscovered
that part of “Dœblin’s trick” (as Kesten called it in [46]), or “Dœblin’s
method” (as Kolmogorov called it in [49]), which is the use of Kolmogorov’s
inequality for maximum of partial sums. Rényi (see [81], [82]), when he
adapted Anscombe’s advance to compound sums ${S}_{N_{t}}$ with $N_{t}$ more
general242424In Rényi’s sums (see Remark 2.1), $N_{t}$ is dependent on i.i.d.
summands and
$N_{t}/\,t\overset{\scriptscriptstyle{\mathsf{P}}}{\rightarrow}\varrho$,
$t\to\infty$, but is of a general form. than in (2.1), acknowledged (see [82])
that “K.L. Chung kindly called my attention to the fact that the main idea of
the proof (the application of the inequality of Kolmogorov $\dots$) given in
[81] is due to Dœblin (see [26]). It has been recently proved $\dots$ that the
method in question can lead to the proof of the most general form of
Anscombe’s theorem.” Just the way it is in the world of ideas.

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