Convergence in Skorokhod J-topology for compositions of stochastic processes
A survey on functional limit theorems for compositions of stochastic processes is presented. Applications to stochastic processes with random scaling of time, random sums, extremes with random sample size, generalised exceeding processes, sum- and max-processes with renewal stopping, and shock proce...
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| Cite this: | Convergence in Skorokhod J-topology for compositions of stochastic processes / D.S. Silvestrov // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 126–143. — Бібліогр.: 75 назв.— англ. |
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| citation_txt | Convergence in Skorokhod J-topology for compositions of stochastic processes / D.S. Silvestrov // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 126–143. — Бібліогр.: 75 назв.— англ. |
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| description | A survey on functional limit theorems for compositions of stochastic processes is presented. Applications to stochastic processes with random scaling of time, random sums, extremes with random sample size, generalised exceeding processes, sum- and max-processes with renewal stopping, and shock processes are discussed.
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Theory of Stochastic Processes
Vol. 14 (30), no. 1, 2008, pp. 126–143
D. S. SILVESTROV
CONVERGENCE IN SKOROKHOD J-TOPOLOGY FOR
COMPOSITIONS OF STOCHASTIC PROCESSES
A survey on functional limit theorems for compositions of stochastic processes is
presented. Applications to stochastic processes with random scaling of time, random
sums, extremes with random sample size, generalised exceeding processes, sum- and
max-processes with renewal stopping, and shock processes are discussed.
1. Introduction
Let us begin from short historical remarks concerned research studies on functional
limit theorems for stochastic processes.
The classical works by Khintchine (1933), Lévy (1937, 1948), Gnedenko and Kol-
mogorov (1949), and Doob (1953) created a general framework for the development of
the theory.
As far as functional limit theorems are concerned, the papers by Kolmogorov (1931,
1933), Erdös and Kac (1946a, 1946b), Doob (1949), Donsker (1951, 1952), Gikhman
(1953), Prokhorov (1953), and Kolmogorov and Prokhorov (1954) are considered to be
the precursors of the theory. In particular, Donsker (1951) gave the first universal func-
tional limit theorem, called by him an invariance principle. This theorem establishes weak
convergence of functionals (continuous in uniform topology) defined on sum-processes
constructed from i.i.d(̇independent identically distributed) random variables to the same
functionals defined on the limiting Wiener process. Kolmogorov and Prokhorov (1954)
connected the invariance principle with theorems about weak convergence of measures
in the functional metric space of continuous functions.
Prokhorov (1956) completed the general theory of weak convergence of measures
in metric spaces and gave general conditions for convergence of continuous stochastic
processes in the uniform U-topology.
Skorokhod (1955a, 1955b, 1956) invented the main topology in the space D of càdlàg
functions, the J-topology, and gave general conditions for J-convergence of càdlàg pro-
cesses. In the papers Skorokhod (1957, 1958), general conditions for J-convergence of
processes with independent increments and Markov processes were also given.
Skorokhod’s original approach was based on his representation theorem reducing the
weak convergence in space D to J-convergence of càdlàg functions. Kolmogorov (1956)
has shown that the space D can be equipped with an appropriate metric that makes the
J-convergence equivalent to the convergence in this metric. The metric, which makes
D a Polish space, was constructed by Billingsley (1968). These results permitted the
consideration of limit theorems for càdlàg processes in the framework of the general
theory of weak convergence of measures in metric spaces. This approach was used in the
book by Billingsley (1968). One can find historical remarks concerning the early period
of the development of the theory in the paper by Billingsley and Wishura (2000).
In the paper by Skorokhod (1956), some other topologies, including M-topology, were
also introduced. These topologies are not so widely used since, in most cases, càdlàg
2000 Mathematics Subject Classification. Primary 60F17.
Key words and phrases. J-topology, space D, composition of stochastic processes, functional limit
theorem, random sum, random scaling of time, generalised exceeding process, renewal-type stopping.
126
CONVERGENCE IN SKOROKHOD J-TOPOLOGY 127
processes converge in the J-topology. Nevertheless, they are useful in some special cases.
For example, the M-topology is often applied to extremal processes. The book by Whitt
(2002) gives a detailed account of the corresponding results.
The original theory of functional theorems was developed in the case where stochastic
processes are defined on a finite interval. An extension of functional limit theorems to
stochastic processes that are defined on the interval [0,∞) is needed in limit theorems
for randomly stopped stochastic processes. This is due to the possibility for the random
stopping moments to be stochastically unbounded random variables. This extension of
the theory was given by Stone (1963) and Lindvall (1973).
To complete the picture, we would also like to mention some other directions of de-
velopment of the general theory of functional limit theorems. For example, Le Cam
(1957), Varadarajan (1961), and Dudley (1966) generalised the main results on weak
convergence from metric spaces to spaces of a more general type. Borovkov (1972, 1976)
developed a version of the theory based on his methods of individual functionals. Stroock
and Varadhan (1969, 1979) developed martingale methods that cover large classes of
martingale-type stochastic processes. The general theory originated from this method is
given in Liptser and Shiryaev (1986) and Jacod and Shiryaev (1987).
A complete theory can be found in the books by Skorokhod (1961, 1964), Gikhman
and Skorokhod (1965, 1971), Parthasarathy (1967), Billingsley (1968), Pollard (1984),
Ethier and Kurtz (1986), Liptser and Shiryaev (1986), Jacod and Shiryaev (1987), David-
son (1994), Borovkov, Mogul’skij, and Sakhanenko (1995), Whitt (2002), and Silvestrov
(2004). These books also contain bibliographies of works in the area as well as the survey
paper by Mishura (2000).
One model that has attracted the attention of many researchers in this area is that
of limit theorems for randomly stopped stochastic processes and for compositions of
stochastic processes.
This model can appear in a natural way, for example: when studying limit theorems
for additive or extremal functionals of stochastic processes; in models connected with a
random change of time, change point problems and problems related to optimal stopping
of stochastic processes; and in different renewal models, particularly those which appear
in applications for risk processes, queuing systems, etc.
The model also appears in statistical applications connected with studies of samples
with a random sample size. Such sample models play an important role in sequential
analysis. They also appear in sample survey models, or in statistical models where
sample variables are associated with stochastic flows. The latter models are typical for
insurance, queuing and reliability applications, as well as many others.
The present paper is a survey of results on general conditions of J-convergence for
compositions of stochastic processes.
The first book on this subject was published by me in 1974. Since that time many new
results and applications have been developed in this area. These realities have stimulated
me to begin work on a new book on this subject. This book was published in 2004. The
present survey is mainly based on the results presented in this book.
2. J-convergence of càdlàg processes
Let D be the space of real-valued càdlàg functions, i.e., functions that are continuous
from the right and have finite left limits at every point of interval [0,∞). For every
ε ≥ 0, let ζε(t), t ≥ 0 be a stochastic càdlàg process, trajectories of which belong with
probability 1 to the space D.
We refer to Gikhman and Skorokhod (1965, 1971), Billingsley (1968), Whitt (2002),
and Silvestrov (2004) for the corresponding definitions and general facts about J-conver-
gence of càdlàg stochastic processes. The symbol ζε(t), t ≥ 0 J−→ ζ0(t), t ≥ 0 as ε→ 0 is
used to show that processes ζε(t) converge in topology J to a process ζ0(t) on any interval
128 D. S. SILVESTROV
[0, T ], such that point 0 < T <∞ is a point of stochastic continuity of the process ζ0(t).
This definition is a direct translation of the original definition by Skorokhod (1956) from
finite intervals to the interval [0,∞), and a slight modification of the definitions given by
Stone (1963) and Lindvall (1973).
Let us formulate the general conditions for J-convergence of càdlàg processes on inter-
val [0,∞), which directly follow from the conditions of J-convergence on finite intervals
given by Skorohod (1956):
A: ζε(t), t ∈ S ⇒ ζ0(t), t ∈ S as ε → 0, where S is a subset of [0,∞) that is
everywhere dense in this interval and contains the point 0;
B: limc→0 limε→0 P{ΔJ(ζε(·), c, T ) > δ} = 0, δ, T > 0,
where
ΔJ(x(·), c, T ) = sup
0∨(t−c)≤t′≤t≤t′′≤(t+c)∧T
min(|x(t′)− x(t)|, |x(t′′)− x(t)|).
Note that under conditions A and B, the set S in A can be extended to the set S∪S0,
where S0 is the set of points of stochastic continuity of the process ζ0(t), t ≥ 0. Note
that S ∪ S0 is [0,∞) except for at most a countable set.
Theorem 1. Let conditions A and B hold. Then,
ζε(t), t ≥ 0 J−→ ζ0(t), t ≥ 0 as ε→ 0.
It should be noted that conditions A and B are not only sufficient but also necessary
conditions for J-convergence of càdlàg processes ζε(t), t ≥ 0, which in its turn implies
that condition A holds with the set S of points of stochastic continuity of the limiting
process, and also that the condition of J-compactness B holds.
When speaking about conditions for J-convergence of compositions of càdlàg stochas-
tic processes, one means sufficient conditions (expressed in terms of external processes
and internal stopping processes constructing a composition) which provide Skorokhod’s
general conditions for J-convergence. Components of a composition have usually sim-
pler structure than their composition. This hopefully will lead to conditions for J-
convergence, expressed in terms of components, which can be simpler than the general
conditions of J-convergence. Let us formulate the problem more precisely.
Let D+ be the subspace of non-negative and non-decreasing functions from D. For
every ε ≥ 0, let ξε(t), t ≥ 0 and νε(t), t ≥ 0 be stochastic càdlàg processes, trajectories of
which belong with probability 1, respectively, to the spaces D and D+ . We are interested
in their composition ζε(t) = ξε(νε(t)), t ≥ 0. This process is also a càdlàg process.
In the case, where the process ζε(t) = ξε(νε(t)), t ≥ 0 is a composition, it is natural
to try to find conditions sufficient for A and B expressed in terms of components of
composition.
Natural candidates that are expected to provide J-convergence of the compositions
ζε(t), t ≥ 0 are the following three conditions:
C: (νε(s), ξε(t)), (s, t) ∈ V × U ⇒ (ν0(s), ξ0(t)), (s, t) ∈ V × U as ε→ 0, where V, U
are some everywhere dense subsets in [0,∞) containing 0;
D: limc→0 limε→0P{ΔJ(ξε(·), c, T ) > δ} = 0, δ, T > 0;
E: limc→0 limε→0P{ΔJ(νε(·), c, T ) > δ} = 0, δ, T > 0.
As above, it can be noted that, under conditions C and D the sets V and U can
be extended to the sets V ∪ V0 and U ∪ U0, where V0 and U0 are the sets of points of
stochastic continuity of the processes ν0(t), t ≥ 0 and ξ0(t), t ≥ 0, respectively. Since
processes νε(t) are monotonic, it is not necessary to involve the condition E. Note that
V ∪ V0 and U ∪ U0 are [0,∞) except for at most some countable sets.
CONVERGENCE IN SKOROKHOD J-TOPOLOGY 129
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Figure 1. First-type continuity condition.
Conditions C and D provide J-convergence of processes ξε(t), t ≥ 0. Conditions C and
E provide the same for processes νε(t), t ≥ 0. But together, all three conditions C, D, and
E do not provide J-convergence, neither for the vector processes (νε(t), ξε(t)), t ≥ 0 nor
for the compositions ζε(t) = ξε(νε(t)), t ≥ 0. Some conditions additional to C, D, and
E must be assumed for the processes ξε(t) and νε(t), to provide desirable J-convergence
of compositions.
As follows from Theorem 1, the problem can be split in two subproblems: the first one
to give sufficient conditions for the weak convergence of compositions (condition A), and
the second to give sufficient conditions for the J-compactness of compositions (condition
B).
3. Weak convergence for compositions of càdlàg processes
In all examples presented below, we use a natural number n as a parameter, instead
of ε, to index the corresponding càdlàg stochastic processes. Actually, we can always
assume that ε = n−1 for n ≥ 1 and ε = 0 for n = 0.
Condition C can be expected to provide weak convergence of compositions on some
set W dense in [0,∞).
The following example shows that this hypothesis is not true. Let the stopping process
νn(t) = νn, t ≥ 0 be in fact a random variable (stopping moment) that takes values
1 − n−1 and 1 + n−1 with probability 1
2 and ξn(t) = χ[1,∞)(t), t ≥ 0, for n ≥ 1. In
this case, condition C obviously holds. The limiting stopping moment ν0 = 1 with
probability 1 and the limiting process ξ0(t) = χ[1,∞)(t), t ≥ 0. However, ξn(νn) is a
random variable that takes values 0 and 1 with probability 1
2 , while ξ0(ν0) = 1 with
probability 1. Therefore, the random variables ξn(νn) do not weakly converge. Figure 1
illustrates this example.
In this example, the limiting stopping moment is a discontinuity point for the corre-
sponding limiting external process. The oscillation of stopping moments in a neighbour-
hood of this discontinuity point causes that compositions ξn(νn) do not weakly converge.
Let us denote R[x(·)] the set of discontinuity points for a càdlàg function x(t), t ≥ 0.
The example considered above leads to the following hypothesis. In order to provide
desirable weak convergence of compositions on some set W it is enough to add to C the
condition that the limiting process ξ0(t), t ≥ 0 is continuous at a random point ν0(w)
with probability 1 for every w from set W , i.e., to assume that the following condition
holds:
FW : P{ν0(w) ∈ R[ξ0(·)]} = 0 for w ∈ W .
130 D. S. SILVESTROV
The following more sophisticated example shows that this hypothesis is also not true.
Conditions C and FW together are not sufficient to provide weak convergence of com-
positions on set W without some additional assumptions.
Let ξk, k = 0, 1, . . . be a sequence of non-negative i.i.d. random variables with a
continuous distribution function F (x) and ζn = max 0≤k≤n ξk be the maximum of the
first n + 1 random variables of this sequence. Let us introduce the random variables
μn = min (r : ξr = ζn). By the definition, ζn = ξμn . The last representation can
be rewritten in the following form: ζn/n = ξn(νn), where ξn(t) = ξ[tn]/n, t ≥ 0 and
νn(t) = νn = μn/n, t ≥ 0.
It is easy to see that the random variable μn takes values 0, . . . , n with probability 1
n+1
and, hence, νn ⇒ ν0 as n → ∞, where ν0 is a random variable uniformly distributed in
[0, 1]. Since the random variables ξk, k = 0, 1, . . . are i.i.d. random variables, ξn(t) P−→ 0
as n → ∞, for t ≥ 0. From Slutsky theorem and the above convergence relations, it
follows that (νn, ξn(t)), t ≥ 0⇒ (ν0, 0), t ≥ 0 as n→∞.
So, condition C holds. Condition FW also holds, with the set W = [0,∞), since the
limiting process ξ0(t) = 0, t ≥ 0 is continuous.
In this case, ξ0(ν0) = 0. However, the random variables ζn/n = ξn(νn) may not
converge weakly to 0 as n → ∞. For example, let F (x) = χ[1,∞)(x)(1 − 1/x). Then
P{ζn/n < x} = F (xn)n → exp {−x−1} as n → ∞, for x > 0. This means that the
random variables ζn/n⇒ ζ as n→∞, where ζ is a non-negative random variable which
has the distribution function P{ζ ≤ x} = χ[0,∞)(x) exp {−x−1}.
An explanation of the example above is that the processes ξn(t), t ≥ 0 weakly converge
to the zero-process ξ0(t) = 0, t ≥ 0, but these processes do not converge in the topology
J or even in the weaker topology M. They can possess too large oscillations in small
intervals. This effect has the result that the max-processes ξ+n (t) = sups≤t ξn(s), t ≥ 0
do not converge weakly to the corresponding zero max-process ξ+0 (t) = sups≤t ξ0(s) =
0, t ≥ 0. At the same time, the stopping moments νn and the external processes ξn(t),
t ≥ 0 are connected so that ξn(νn) = ξ+n (νn).
The above example shows that, in order to provide weak convergence of compositions
on set W , one should add, to the condition of join weak convergence C and the continuity
condition FW , an additional compactness condition, for example, this can be the J-
compactness condition D.
Let us use notation Fw for the condition FW in the case where set W = {w} contains
only the point w. Let also denote by W0 the set of all points w such that condition Fw
holds.
Let also V0 be the set of all points of stochastic continuity of the process ν0(t), t ≥ 0.
Note that, due to monotonicity of the processes νε(t), t ≥ 0, and the assumption that
the set V is dense in [0,∞), the set V can be extended to the set V ∪ V0 in condition C.
The set V ∪ V0 coincides with [0,∞) except for at most a countable set.
The following result was obtained in Silvestrov (1971a, 1972a).
Theorem 2. Let conditions C and D hold. Then, for the set S = (V ∪ V0) ∩W0,
ζε(t) = ξε(νε(t)), t ∈ S ⇒ ζ0(t) = ξ0(ν0(t)), t ∈ S as ε→ 0.
The set W0 can be empty, but, under some natural assumptions, this set is dense in
[0,∞), more over, it coincides with [0,∞) except for at most a countable set.
The problem has an additional new aspect if one does not prescribe a set of weak
convergence but would like to guarantee only the weak convergence of compositions
ξε(νε(t)) on some subset S dense in the interval [0,∞). This is an important problem
since this is required in the basic condition A.
Let us introduce the following condition:
G: P{ν0(t′) = ν0(t′′) ∈ R[ξ0(·)]} = 0 for 0 ≤ t′ < t′′ <∞.
CONVERGENCE IN SKOROKHOD J-TOPOLOGY 131
The following useful lemma from Silvestrov and Teugels (2004) gives the necessary
and sufficient condition for the set W0 to be dense in [0,∞).
Lemma 1. The condition G is necessary for FW to hold for some set W , which is dense
in [0,∞), and sufficient for FW to hold for some set W , which is [0,∞) except for at
most a countable set.
Condition G holds, for example, if the process ν0(t), t ≥ 0 is strictly monotonic with
probability 1.
This condition also holds if the process ξ0(t) = ξ′0(t) + ξ′′0 (t), t ≥ 0 can be decomposed
in a sum of two càdlàg processes such that the process ξ′0(t), t ≥ 0 is continuous with
probability 1 while the process ξ′′0 (t), t ≥ 0 is stochastically continuous and independent
of the process ν0(t), t ≥ 0.
Condition G guarantees that the set of weak convergence S = (V ∪ V0) ∩W0, which
appears in Theorem 2, coincides with [0,∞) except for at most a countable set.
It should be noted that condition G does not guarantee that set S contains a pre-
assigned point w ∈ [0,∞). In order to include a point w in the set of convergence, one
should assume that condition Fw holds and also to require that w ∈ V ∪ V ′
0 . Since the
point 0 ∈ V , in this case, one should require condition F0 to hold, in order to include
this point in the set of weak convergence.
The following variant of Theorem 2 gives conditions that guarantee that the basic
condition of weak convergence A holds for compositions of càdlàg processes.
Theorem 3. Let conditions C, D, G, and F0 hold. Then the set S = (V ∪ V0) ∩W0 is
[0,∞) except for at most a countable set, contains the point 0, and,
ζε(t) = ξε(νε(t)), t ∈ S ⇒ ζ0(t) = ξ0(ν0(t)), t ∈ S as ε→ 0.
Theorem 3 gives conditions that, together, imply the desirable weak convergence of
randomly stopped càdlàg processes. These conditions are: the condition C of joint
weak convergence of random stopping moments and external stochastic processes; the
condition D of J-compactness of external stochastic processes; and the conditions of
continuity G and F0, which guarantee that the limiting process ν0(s) stops the limiting
external process ξ0(t) at its points of continuity for every s from some set dense in [0,∞).
This combination makes a good balance between the conditions imposed on the pre-
limiting external and stopping processes, on the one hand, and the limiting external and
stopping processes, on the other. Pre-limiting joint distributions of stopping and ex-
ternal processes usually have a complicated structure. However, these distributions are
involved only in the simplest and most natural way via the condition of their joint weak
convergence. The second J-compactness condition involves only the external processes
themselves and not the stopping processes. This condition is a standard one. It was
thoroughly studied for various classes of càdlàg stochastic processes. The continuity
conditions involve joint distributions of the limiting stopping process and the limiting
external process. These limiting joint distributions are usually simpler than the cor-
responding pre-limiting joint distributions. This permits one to check the continuity
conditions in various practically important cases. Due to a balance between the condi-
tions imposed on the pre-limiting and limiting external processes and internal stopping
processes Theorems 2 and 3 are effective tools for use in limit theorems for randomly
stopped stochastic processes.
The continuity condition G is effective in the cases where the process ν0(s) mostly
stops the process ξ0(t) at points of continuity of this process. However, there are cases,
where the process ν0(s) stops the ξ0(t) at its discontinuity point for all s from some
interval [s′, s′′]. This may happen, for example, for compositions with renewal type
stopping. In such cases condition G does not hold and should be replaced by some
weaker condition.
132 D. S. SILVESTROV
Let us denote by α
(δ)
εk the successive moments of jumps of the process ξε(t), t ≥ 0,
with absolute values of jumps greater than or equal to δ.
Let also Z0 be the set of all δ > 0 such that the process ξ0(t), t ≥ 0 has no jumps with
the absolute value equal to δ with probability 1.
Let us introduce the following condition:
HW : There exist a sequence δl ∈ Z0, δl → 0 as l→∞, and a sequence 0 < Tr →∞ as
r → ∞, such that, for every l, k, r ≥ 1, lim0<c→0 limε→0 P{α(δl)
εk − c ≤ νε(w) <
α
(δl)
εk , α
(δl)
εk < Tr} = 0 for w ∈ W .
Lemma 2. Condition FW implies condition HW .
Let us use notation Hw for the condition HW in the case where set W = {w} contain
only the point w. Let also denote by W ′
0 the set of all points w such that condition Hw
holds.
Note that, according Lemma 2, the set W0 ⊆W ′
0.
The following theorem from Silvestrov (2004) improves the result given in Theorem
2.
Theorem 4. Let conditions C and D hold. Then for the set S = (V ∪ V0) ∩W ′
0,
ζε(t) = ξε(νε(t)), t ∈ S ⇒ ζ0(t) = ξ0(ν0(t)), t ∈ S as ε→ 0.
Let us now introduce the following condition:
I: There exist a sequence δl ∈ Z0, δl → 0 as l → ∞, and a sequence 0 < Tr →
∞ as r → ∞, such that, for every l, k, r ≥ 1, lim0<c→0 limε→0 P{α(δl)
εk − c ≤
νε(t′), νε(t′′) < α
(δl)
εk , α
(δl)
εk < Tr} = 0 for all 0 ≤ t′ < t′′ <∞.
The following lemma is an analogue of Lemma 1.
Lemma 3. The condition I is necessary for HW to hold for some set W dense in
[0,∞), and sufficient for HW to hold for some set W which is [0,∞) except for at most
a countable set.
Condition I guarantees that the set of weak convergence S = (V ∪ V0) ∩W ′
0, which
appears in Theorem 4, coincides with [0,∞) except for at most a countable set.
It should be noted that condition I does not guarantee that set S contains a pre-
assigned point w ∈ [0,∞). In order to include a point w in the set of convergence, one
should assume that condition Hw holds and also to require that w ∈ V ∪ V ′
0 . Since the
point 0 ∈ V , in this case, one should require condition H0 to hold, in order to include
this point in the set of weak convergence.
The following variant of Theorem 4 gives conditions (weaker than in Theorem 3) that
guarantee that the basic condition of weak convergence A holds for compositions of
càdlàg processes.
Theorem 5. Let conditions C, D, I, and H0 hold. Then the set S = (V ∪ V0) ∩W ′
0 is
[0,∞) except for at most a countable set, contains the point 0, and,
ζε(t) = ξε(νε(t)), t ∈ S ⇒ ζ0(t) = ξ0(ν0(t)), t ∈ S as ε→ 0.
As was mentioned above, Theorems 2 and 3, based on the continuity condition G, do
not cover the cases where the limiting internal process stops the limiting external process
at its discontinuity points. This case is covered by Theorems 4 and 5. In Theorem 5,
condition G is replaced by the weaker condition I that ensures the right positioning of
the pre-limiting stopping moments on the right-hand side of the moments where the pre-
limiting external processes experience large jumps. This condition does involve the pre-
limiting joint distributions of stopping processes and external processes not only via their
joint distributions but also via the joint distributions of stopping processes and moments
CONVERGENCE IN SKOROKHOD J-TOPOLOGY 133
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�
�
�
�
�
�
��
1− 1
n
1
2
1 + 1
n
�
�
�
(b) νn(t), t ≥ 0.
Figure 2. Second-type continuity condition.
of large jumps for external processes. The latter distributions are not so complicated
and the corresponding conditions can effectively be verified in some important cases.
In conclusion of discussion concerned the conditions for weak convergence of compo-
sitions of càdlàg processes, let us consider the following example shown in Figures 2, 3,
and 4.
Let ξn(t) = χ[1,∞)(t), t ≥ 0, for n ≥ 1. Let also, for n ≥ 1, the process νn(t), t ≥
0 have three possible realisations that occur with probabilities pn, qn and rn, where
pn + qn + rn = 1. These realisations are 1
2 for t ≥ 0, 1 − n−1 for t ≥ 0, and 1 + n−1
for t ≥ 0.
We assume that probabilities pn, qn and rn converge as n→∞ to the limiting values
p0, q0 and r0, respectively.
In this case, condition C obviously holds. The limiting process ξ0(t) = χ[1,∞)(t), t ≥ 0.
The limiting stopping process ν0(t), t ≥ 0 has only two possible realisations that occur
with the probabilities p0 and q0 + r0, respectively. These realisations are 1
2 for t ≥ 0, and
1 for t ≥ 0. The condition of J-compactness D also holds. For n ≥ 1, the composition
ξn(νn(t)), t ≥ 0 has two possible realisations that occur with the probabilities pn + qn
and rn, respectively. These realisations are 0 for t ≥ 0, and 1 for t ≥ 0. The limiting
composition ξ0(ν0(t)), t ≥ 0 has the same two possible realisations but they occur with
the probabilities p0 and q0 + r0, respectively.
Conditions G and F0 hold if and only if (a) p0 = 1. In this case, the limiting
composition ξ0(ν0(t)), t ≥ 0 has with probability 1 only one realisation, 0 for t ≥ 0. The
compositions ξn(νn(t)) weakly converge to ξ0(ν0(t)) on the set S = [0,∞).
134 D. S. SILVESTROV
�
�
t1
�
�
ξ0(t)
�
�
1
�
�
(a) ξ0(t), t ≥ 0.
�
�
t
ν0(t)
�
�
�
q0 + r0
p0
�
�
�
�
�
�
�
��
�
�
�
�
�
�
�
��
1
1
2
�
�
(b) ν0(t), t ≥ 0.
Figure 3. Second-type continuity condition.
Conditions I and H0 hold if and only if (b) q0 = 0. If also p0 < 1, then conditions I
and H0 hold but G and F0 do not. If q0 = 0, the limiting composition ξ0(ν0(t)), t ≥ 0 has
two possible realisations 0 for t ≥ 0, and 1 for t ≥ 0. They occur with the probabilities
p0 and r0, respectively. Again, the ξn(νn(t)) weakly converge to ξ0(ν0(t)) on the set
S = [0,∞).
Neither conditions G and F0 nor conditions I and H0 hold, if (c) q0 > 0. In this case
pn + qn � →p0 and rn � →q0 + r0 as n → ∞. The compositions ξn(νn(t)) do not weakly
converge to ξ0(ν0(t)) for every t ∈ [0,∞).
These statements are consistent with the remarks above.
There exists a huge literature concerned weak limit theorems for randomly stopped
stochastic processes mainly related to the models, where stochastic processes stopped
at random moments that are independent or asymptotically independent of the external
processes. The originating works here are Wald (1945), Robbins (1948), Kolmogorov
and Prokhorov (1949), Anscombe (1952), Dobrushin (1955), and Rényi (1957). We refer
to books by Kruglov and Korolev (1990), Gnedenko and Korolev (1996), and Bening
and Korolev (2002) which give an extended presentation of results and bibliographies
of works related to these models. An extended bibliography is also given in Silvestrov
(2004).
As far as the model, where the limiting process is the composition of the limiting
external process and the limiting internal stopping process that can be dependent in
an arbitrary way, is concerned, the first general result was given in Billingsley (1968).
There, the author deal with the case when the external limiting process is continuous.
This result was improved to the form of Theorem 2, where the external limiting process
is a càdlàg process, in Silvestrov (1971a, 1972a). Theorems 4 and 5 present new results
from Silvestrov (2004).
4. J-convergence for compositions of càdlàg processes
As was mentioned in above, conditions C, D, and E, which are natural candidates
that are expected to provide J-convergence of compositions of càdlàg processes are not
sufficient for this. As follows from the examples given in Section 3, the continuity condi-
tion G or the weaker continuity condition I should be additionally assumed to provide
the required basic condition A of weak convergence for compositions on some set dense
in [0,∞).
The following example let us clarify situation with another basic condition B of J-
compactness for compositions of càdlàg processes.
CONVERGENCE IN SKOROKHOD J-TOPOLOGY 135
�
�
t
ξn(νn(t))
�
�
�
rn
pn + qn
�
�
�
�
�
�
�
��
�
�
�
�
�
�
�
�
�
���
1
�
�
(a) ξn(νn(t)), t ≥ 0.
�
�
t
ξ0(ν0(t))
�
�
�
q0 + r0
p0
�
�
�
�
�
�
�
��
�
�
�
�
�
�
�
�
�
���
1
�
�
(b) ξ0(ν0(t)), t ≥ 0.
Figure 4. Second-type continuity condition.
Let us consider the following example illustrated in Figures 5 and 6. We define ξn(t) =
χ[1, 7
4 )(t), t ≥ 0, for n ≥ 1, and νn(t) = t+ n−1 if t < 1 and t+ 1 if t ≥ 1, for n ≥ 1.
In this case, condition C obviously holds. The corresponding limiting process ξ0(t) =
χ[1, 7
4 )(t), t ≥ 0, and the limiting stopping process ν0(t) = t if t < 1 and t + 1 if t ≥ 1.
Conditions D and E also hold.
In this case, the composition ξn(νn(t)) = χ[1−n−1,1)(t), t ≥ 0, while ξ0(ν0(t)) = 0,
t ≥ 0.
The process ξn(νn(t)), t ≥ 0 has two jumps with the values 1 and −1 in the close
points 1− n−1 and 1, respectively. So, ΔJ (ξn(νn(·)), c, T ) = 1 if n−1 < c and, therefore,
limc→0 limn→∞ ΔJ (ξn(νn(·)), c, T ) = 1. This shows that the condition of J-compactness
B does not hold for the processes ξn(νn(t)), t ≥ 0 since the left limiting value of the
limiting stopping process ν0(t), t ≥ 0, at point 1, which is a point of discontinuity for the
limiting stopping process, is ν0(1− 0) = 1. This value is a point of discontinuity for the
external limiting processes ξ0(t), t ≥ 0. The example can be easily modified such that
the right limiting value of the limiting stopping process at a discontinuity point would
cause the same effect.
The example considered above leads to the following hypothesis. In order to provide
J-compactness of compositions it is enough to add, to the conditions C, D and E, the
following condition:
J: P{ν0(t± 0) /∈ R[ξ0(·)] for t ∈ R[ν0(·)]} = 1.
This hypothesis is true as shows the following theorem from Silvestrov (1974).
Theorem 6. Let conditions C, D, E, and J hold. Then,
lim
c→0
lim
ε→0
P{ΔJ(ξε(νε(·)), c, T ) > δ} = 0, δ, T > 0.
Now we can give general conditions for J-convergence of compositions of càdlàg
processes. This can be done by combining conditions for weak convergence of com-
positions given in Theorem 3 or in Theorem 5, with conditions for J-compactness given
in Theorem 6.
The following theorem is from Silvestrov (1974).
Theorem 7. Let conditions C, D, E, G, F0, and J hold. Then,
ζε(t) = ξε(νε(t)), t ≥ 0 J−→ ζ0(t) = ξ0(ν0(t)), t ≥ 0 as ε→ 0.
136 D. S. SILVESTROV
�
�
t
ξn(t)
1 7
4
1 �
� �
�
�
�
�
��
�
�
�
���
�
�
�
���
�
�
�
���� �
�
(a) ξn(t), t ≥ 0.
�
�
1
1 + 1
n
2
νn(t)
1 t
�
�
��
�
�
�
�
�
��
�
�
�
�
���
�
���
(b) νn(t), t ≥ 0.
Figure 5. Third-type continuity condition.
�
�
t
ξn(νn(t))
1− 1
n 1
1
�
�
�
�
�
�� �
�
��
� ��
�
�
(a) ξn(νn(t)), t ≥ 0.
�
�
t
ξ0(ν0(t))
�
� ��
(b) ξ0(ν0(t)), t ≥ 0.
Figure 6. Third-type continuity condition.
In the example given above, the vector processes (νn(t), ξn(t)), t ≥ 0 J-converge.
However, the compositions ξn(νn(t)), t ≥ 0 can J-converge even if the vector processes
(νn(t), ξn(t)), t ≥ 0 do not J-converge.
Let us modify the example considered above and shown in Figures 5 and 6. Figure 7
illustrates this modified example. We use the same external processes ξn(t) = χ[1, 74 )(t),
t ≥ 0, for n ≥ 1, but define new internal stopping processes νn(t) = 1
2 (1 − n−1)−1t if
t < 1− n−1 and t+ 1 + n−1 if t ≥ 1− n−1, for n ≥ 1.
In this case, the corresponding limiting process ξ0(t) = χ[1, 7
4 )(t), t ≥ 0, and the limiting
stopping process ν0(t) = 1
2 t if t < 1 and t + 1 if t ≥ 1. Hence, ξn(νn(t)) = 0, t ≥ 0, for
n ≥ 1 as well as for n = 0. Therefore, the compositions ξn(νn(t)), t ≥ 0 J-converge. This
is consistent with Theorem 7, since all conditions of this theorem hold.
However, the vector processes (νn(t), ξn(t)), t ≥ 0 do not J-converge, since the process
(νn(t), ξn(t)) has two large jumps with the absolute values 3
2 and 1 in the close points
1− n−1 and 1, respectively.
Theorem 7 serves properly for the models, where limiting external processes are ran-
domly stopped at continuity points. The models, where external processes are randomly
stopped at discontinuity points, do require to improve this result. In this case, the fol-
lowing theorem from Silvestrov (2004) provides effective conditions for J-convergence of
compositions of càdlàg processes.
CONVERGENCE IN SKOROKHOD J-TOPOLOGY 137
�
�
t
ξn(t)
1 7
4
1 �
� �
�
�
�
�
��
�
�
�
���
�
�
�
���
�
�
�
���� �
�
(a) ξn(t), t ≥ 0.
�
�
1
1
2
2
νn(t)
11− 1
n t
�
�
��
�
�
�
�
�
�
�
��
�
�
�
���
�
���
(b) νn(t), t ≥ 0.
Figure 7. Third-type continuity condition.
Theorem 8. Let conditions C, D, E, I, H0, and J hold. Then,
ζε(t) = ξε(νε(t)), t ≥ 0 J−→ ζ0(t) = ξ0(ν0(t)), t ≥ 0 as ε→ 0.
Let us go back to the example considered in Subsection 3 and shown in Figures 2, 3,
and 4. In this example, conditions C, D, E, and the third-type continuity conditions J
hold.
In the case (a) p0 = 1, conditions G and F0 hold. Therefore, according to Theorem
7, the compositions ξn(νn(t)), t ≥ 0 J-converge to ξ0(ν0(t)), t ≥ 0 as n→∞.
In the case (b) q0 = 0, p0 < 1, conditions G and F0 do not hold. However, in this
case, the weaker conditions I and H0 hold. Therefore, according to Theorem 8, the
compositions ξn(νn(t)), t ≥ 0 J-converge to ξ0(ν0(t)), t ≥ 0 as n→∞.
This example is, of course, an artificial one. In the next two sections, we illustrate the
results given in Theorems 7 and 8, in particular, clarify the difference in the conditions of
these theorems, by applying these theorems for two classical models of randomly stopped
processes with random scaling of time and renewal type stopping.
The preceding results mainly relate to Theorem 7. The simplest case, where both
limiting external and stopping processes are continuous was considered in Billingsley
(1968). This result was extended in various directions in Silvestrov (1971b, 1972c, 1972d),
Serfozo (1973), and Whitt (1980). The case, where the limiting external process in
continuous, was considered in Silvestrov (1974) and Whitt (1980). The case, where the
limiting stopping process is continuous, is important for many applications. This case was
considered in Silvestrov (1972b, 1974). The general case, where both the limiting external
and stopping processes can be discontinuous, was considered in Silvestrov (1974). These
results are presented in Theorem 7. Theorem 8 presents a new result from Silvestrov
(2004).
5. J-convergence of compositions with random scaling of time
In this section, we consider the model, where stopping process has the following simple
form νε(t) = tνε, t ≥ 0, where νε is a non-negative random variable. In this case, the
composition has the form ζε(t) = ξε(tνε), t ≥ 0.
The most well known variant is represented by the model of random sums. Let, for
every ε > 0, ξε,n, n = 1, 2, . . . be a sequence of real-valued random variables and με
be a non-negative random variable. Further, we need a non-random function nε > 0 of
parameter ε such that nε → ∞ as ε → 0. Consider a sum-process with non-random
stopping index, ξε(t) =
∑
k≤tnε
ξε,k, t ≥ 0, and an analogue of this process the stopping
138 D. S. SILVESTROV
index of which is random, ζε(t) =
∑
k≤tμε
ξε,k, t ≥ 0. Denote by νε = με/nε the
normalised random stopping index. Then the process ζε(t) = ξε(tνε), t ≥ 0 can be
represented in the form of composition of two processes ξε(t), t ≥ 0 and νε(t) = tνε,
t ≥ 0.
Another classical variant is represented by the model of extremes with random sample
size. It is defined in the way similar with the case of random sums. The only difference
is that the external process is defined as ξε(t) = maxk≤tnε ξε,k, t ≥ 0 while the stopping
process again is νε(t) = tνε, t ≥ 0.
Consider the following weak convergence condition:
K: (νε, ξε(t)), t ∈ U ⇒ (ν0, ξ0(t)), t ∈ U as ε → 0, where (a) ν0 is a non-negative
random variable, (b) ξ0(t), t ≥ 0 is a càdlàg process, (c) U is a subset of [0,∞)
that is dense in this interval and contains the point 0.
Condition K obviously implies condition C to hold. The processes νε(t) = tνε, t ≥ 0
are monotonic and the corresponding limiting process ν0(t) = tν0, t ≥ 0 is continuous.
These imply that the processes νε(t) are compact in J topology, i.e., condition E holds.
Also conditions G and F0 hold because of the process ν0(t) is continuous. Therefore,
conditions of Theorems 3 and 7 are reduced to only two conditions K and D.
In this case, we can give an explicit description of the set of weak convergence S =
(V0 ∪ V ′
0) ∩ W0. Obviously, the sets V0 = V ′
0 = [0,∞) and, thus, the set of weak
convergence S = W0.
By the definition, set W0 is the set of t ≥ 0 such that P{τkn/ν0 = t} = 0 for all k,
n = 1, 2, . . ., where τkn, k = 1, 2, . . . are successive moments of jumps of the process ξ0(t),
t ≥ 0, with absolute values of the jumps lying in the interval [ 1
n ,
1
n−1 ). Note that the
random variables τkn take values in the interval (0,∞] and the random variable ν0 takes
values in the interval [0,∞). So, the random variable τkn/ν0 takes values in the interval
(0,∞], that is, it is positive and, possibly, improper.
The set W0 coincides with [0,∞) except for at most a countable set. Also, 0 ∈ W0.
Indeed, the set W 0 = [0,∞) \W0 coincides with the set of all atoms of the distribution
functions of the random variables τkn/ν0, k, n = 1, 2, . . .. This set is at most countable
and 0 /∈ W 0. Therefore, the set W0 equals [0,∞) except for the set W 0. Also, 0 ∈ W0.
Note also that W0 is the set of points of stochastic continuity for the process ξ0(tν0),
t ≥ 0.
The following theorem is a direct corollary of the results in Silvestrov (1971a, 1972a,
1972b).
Theorem 9. Let conditions K and D hold. Then,
ζε(t) = ξε(tνε), t ≥ 0 J−→ ζ0(t) = ξ0(tν0), t ≥ 0 as ε→ 0.
We consider a general model where the corresponding external processes ξε(t), t ≥ 0
and stopping indices νε can be dependent in an arbitrary way. As follows from Theorem
9, weak convergence as well as J-convergence of the corresponding randomly stopped
processes ξε(tνε), t ≥ 0 can be obtained under only two conditions. The first one is
the condition K of joint weak convergence of the random stopping indices and exter-
nal processes with non-random stopping indices, and the second the condition D of
J-compactness of the external processes. No extra assumptions on their independence
or even on their asymptotic independence are required.
Note also that in the classical case of sum-processes with random indices and i.i.d.
summands, the condition of J-compactness D can be omitted since it is implied, in this
case, by condition K.
In some sense, the result of Theorem 9 is surprising. It gives a unified approach to
various concrete models including those mentioned above sum- and max-processes with
CONVERGENCE IN SKOROKHOD J-TOPOLOGY 139
random indices. Being applied to these models, Theorem 9 covers the most of the pre-
ceding results, without any additional assumptions about independence or asymptotical
independence of random stopping indices and external processes.
6. J-convergence of compositions with renewal type stopping
Let αε(t) = (κε(t), ξε(t)), t ≥ 0 be, for every ε ≥ 0, a two-dimensional càdlàg process
with real-valued components. In this section we consider the model, where stopping
process has the form νε(t) = inf(s ≥ 0 : κε(s) > t), t ≥ 0 and, therefore, the composition
has the form ζε(t) = ξε(νε(t)), t ≥ 0. In Silvestrov (1974, 2000, 2004) the process νε(t) is
referred as an exceeding time process and ζε(t) as a generalised exceeding time process.
The most well known variant is represented by the model of generalised renewal
process. Let, for every ε > 0, αε,n = (κε,n, ξε,n), n = 1, 2, . . . be a sequence of ran-
dom vectors with real-valued components. Further, we need a non-random function
nε > 0 of parameter ε such that nε → ∞ as ε → 0. Consider a sum-process αε(t) =
(κε(t), ξε(t)) =
∑
k≤tnε
αε,k, t ≥ 0. Then, the process νε(t) = inf(s ≥ 0 : κε(s) > t) is a
renewal process and ζε(t) is a generalised renewal process.
Another variant of a generalised exceeding process is an extremal processes with
renewal stopping. It is defined in the way similar with the case of generalised re-
newal process. The only difference is that the external process is defined as ξε(t) =
maxk≤tnε ξε,k, t ≥ 0. In this case, the process ζε(t) can be referred as a max-process
with renewal stopping.
An interesting is also model, in which the process ξε(t) is defined as a sum-process
while the process κε(t) = maxk≤tnε κε,k, t ≥ 0 is a max-process. In this case, the process
ζε(t) is usually referred as a shock process.
Let us also introduce the process κ+
ε (t) = sup0≤s≤t κε(s), t ≥ 0.
To avoid some additional side effects at time point 0 and the need of considering the
case, where the random variables νε(t) can be improper, let us assume the following
condition:
L: (a) κε(0) ≡ 0, and (b) κ+
ε (t) P−→∞ as t→∞, for every ε ≥ 0.
Let us also assume that the processes αε(t), t ≥ 0 J-converge, i.e., the following
conditions hold:
M: αε(t) ∈ Y ⇒ α0(t), t ∈ Y as ε → 0, where Y is a subset of [0,∞), dense in this
interval and containing the point 0;
N: limc→0 limε→0P{ΔJ(αε(·), c, T ) > δ} = 0, δ, T > 0.
In this paper, we also restrict consideration by the most important case, where the
following condition holds:
O: κ+
0 (t), t ≥ 0 is a strictly monotonic process with probability 1.
Using the definition of the stopping process νε(t) = inf(s ≥ 0 : κε(s) > t) = inf(s ≥
0 : κ+
ε (s) > t), t ≥ 0 it is not difficult to show that conditions L, M, and O imply
that condition C holds for the processes (νε(t), ξε(t)), t ≥ 0. Moreover, in this case, the
limiting random variable ν0(0) = 0 with probability 1. Since, the process ξ0(t), t ≥ 0 is a
càdlàg process, condition F0 and therefore condition H0 holds. Condition N obviously
implies condition D. Condition O implies that the limiting process ν0(t), t ≥ 0 is contin-
uous with probability 1. The processes νε(t), t ≥ 0 are monotonic, and, therefore, their
weak convergence to a continuous process implies that these processes are J-compact,
i.e., condition E holds. Also condition J automatically holds due to continuity of the
process ν0(t), t ≥ 0.
The following useful lemma from Silvestrov (2004) completes the analysis of conditions
which should provide J-convergence of generalised exceeding processes.
Lemma 4. Conditions L, M, N, and O imply condition I to hold.
140 D. S. SILVESTROV
In fact, condition O can be weaken in this lemma and replaced by the assumption
that ν0(0) = 0 with probability 1.
The following theorem proved in Silvestrov (2004) follows from Theorem 8 and the
remarks above.
Theorem 10. Let conditions L, M, N, and O hold. Then,
ζε(t) = ξε(νε(t)), t ≥ 0 J−→ ζ0(t) = ξ0(ν0(t)), t ≥ 0 as ε→ 0.
Theorem 10 gives very natural and general conditions of weak and J-convergence for
generalised exceeding processes.
We refer to the works Silvestrov (2000, 2004), where one can find a detailed analysis
of conditions for weak and J-convergence of generalised exceeding processes, including
models, where condition O does not hold.
The generalised exceeding processes supply examples of the models, where the first-
type continuity condition G does not work while the second-type continuity condition I
can be used.
Let us assume that there exist δ > 0 and a random moment τ such that P(Aδ) = p > 0,
where Aδ = {κ+
0 (τ) − κ+
0 (τ − 0) ≥ δ, |ξ0(τ) − ξ0(τ − 0)| ≥ δ, τ < ∞}. Then, obviously,
there exist t′ < t′′ such that P{Aδ, κ
+
0 (τ−0) ≤ t′, κ+
0 (τ) > t′′} = p′ > 0. But, in this case,
P{ν0(t′) = ν0(t′′) = τ ∈ R[ξ0(·)]} = p′′ > 0. Thus, the first-type continuity condition G
does not hold.
In applications to generalised renewal processes constructed from the sequences of
i.i.d. random vectors (κε,n, ξε,n), n = 1, 2, . . . with non-negative first components, the
limiting process α0(t) = (κ0(t), ξ0(t)), t ≥ 0 is a càdlàg homogeneous process with
independent increments, which the first component is non-negative. In this case, the
processes κ0(t) = κ′0(t) + κ′′0(t), t ≥ 0 and ξ0(t) = ξ′0(t) + ξ′′0 (t), t ≥ 0 can be decomposed
in the sums of continuous and jump components. If the jump components κ′′0(t), t ≥ 0
and ξ′′0 (t), t ≥ 0 are independent then condition G holds. However, in the case, where
the jump components κ′′0(t), t ≥ 0 and ξ′′0 (t), t ≥ 0 are dependent, there exists δ > 0 such
that moment τ = inf(s > 0 : κ0(s) − κ0(s − 0) ≥ δ, |ξ0(s) − ξ0(s − 0)| ≥ δ) < ∞ with
probability 1. Thus, condition G does not hold, while condition I, according Lemma 4,
can hold.
The works by Silvestrov and Teugels (1998, 2004) and Silvestrov (1974, 2000, 2004)
contain a detailed presentation of applications to generalised renewal processes, extremal
processes with renewal stopping and shock processes, including models constructed from
sequences of i.i.d. random variables, as well as bibliographical remarks related to publi-
cations in this area.
7. Conclusion
The theory of limit theorems for randomly stopped stochastic processes still contains
many problems that do require further studies. These are limit theorems for vector
compositions of stochastic processes and randomly stopped stochastic fields. Also the
models with Markov and martingale type stopping do require additional studies. These
limit theorems may be applied to models of optimal stopping for stochastic processes and
have important financial and statistical applications. Inverse limit theorems for randomly
stopped stochastic processes do also require additional investigation. These theorems
would allow deducing convergence of non-randomly stopped stochastic processes from
their counterpart for randomly stopped processes. Such theorems play a key role in
necessity statements of the limit theorems for processes with semi-Markov modulation.
The latter processes themselves supply examples of many interesting models of randomly
stopped stochastic processes with queuing, reliability, and other applications.
In conclusion, we would like to point out, together with books by Silvestrov (1974,
2004), used as the basis for the present survey, other books that contain results on limit
CONVERGENCE IN SKOROKHOD J-TOPOLOGY 141
theorems for randomly stopped processes. These are Csörgő and Révész (1981), Anisimov
(1988, 2007), Gut (1988), Kruglov and Korolev (1990), Csörgő and Horváth (1993, 1997),
Rahimov (1995), Gnedenko and Korolev (1996), Kalashnikov (1997), Bening and Korolev
(2002), Whitt (2002), and Korolyuk and Limnios (2005).
The book by Silvestrov (2004) also contains a comprehensive bibliography of works
related to limit theorems for randomly stopped stochastic processes, with more than 800
references.
References
1. Anisimov, V.V. (1988) Random Processes with Discrete Components. Vysshaya Shkola and
Izdatel’stvo Kievskogo Universiteta, Kiev
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E-mail address: dmitrii.silvestrov@mdh.se
|
| id | nasplib_isofts_kiev_ua-123456789-4543 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-07T15:52:55Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
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| spelling | Silvestrov, D.S. 2009-11-25T11:08:13Z 2009-11-25T11:08:13Z 2008 Convergence in Skorokhod J-topology for compositions of stochastic processes / D.S. Silvestrov // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 126–143. — Бібліогр.: 75 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4543 519.21 A survey on functional limit theorems for compositions of stochastic processes is presented. Applications to stochastic processes with random scaling of time, random sums, extremes with random sample size, generalised exceeding processes, sum- and max-processes with renewal stopping, and shock processes are discussed. en Інститут математики НАН України Convergence in Skorokhod J-topology for compositions of stochastic processes Article published earlier |
| spellingShingle | Convergence in Skorokhod J-topology for compositions of stochastic processes Silvestrov, D.S. |
| title | Convergence in Skorokhod J-topology for compositions of stochastic processes |
| title_full | Convergence in Skorokhod J-topology for compositions of stochastic processes |
| title_fullStr | Convergence in Skorokhod J-topology for compositions of stochastic processes |
| title_full_unstemmed | Convergence in Skorokhod J-topology for compositions of stochastic processes |
| title_short | Convergence in Skorokhod J-topology for compositions of stochastic processes |
| title_sort | convergence in skorokhod j-topology for compositions of stochastic processes |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4543 |
| work_keys_str_mv | AT silvestrovds convergenceinskorokhodjtopologyforcompositionsofstochasticprocesses |