Strong invariance principle for renewal and randomly stopped processes
The strong invariance principle for renewal process and randomly stopped sums when summands belong to the domain of attraction of an α-stable law is presented
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irk-123456789-45272009-11-25T12:00:32Z Strong invariance principle for renewal and randomly stopped processes Zinchenko, N. The strong invariance principle for renewal process and randomly stopped sums when summands belong to the domain of attraction of an α-stable law is presented 2007 Article Strong invariance principle for renewal and randomly stopped processes / N. Zinchenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 233–246. — Бібліогр.: 36 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4527 en Інститут математики НАН України |
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The strong invariance principle for renewal process and randomly stopped sums when summands belong to the domain of attraction of an α-stable law is presented |
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Zinchenko, N. Strong invariance principle for renewal and randomly stopped processes |
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Strong invariance principle for renewal and randomly stopped processes |
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Strong invariance principle for renewal and randomly stopped processes |
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Strong invariance principle for renewal and randomly stopped processes |
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Strong invariance principle for renewal and randomly stopped processes |
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Strong invariance principle for renewal and randomly stopped processes |
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strong invariance principle for renewal and randomly stopped processes |
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Strong invariance principle for renewal and randomly stopped processes / N. Zinchenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 233–246. — Бібліогр.: 36 назв.— англ. |
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Theory of Stochastic Processes
Vol.13 (29), no.4, 2007, pp.233–246
NADIIA ZINCHENKO
STRONG INVARIANCE PRINCIPLE FOR
RENEWAL AND RANDOMLY STOPPED
PROCESSES
The strong invariance principle for renewal process and randomly
stopped sums when summands belong to the domain of attraction of
an α-stable law is presented
1. Introduction
Let {X, Xi, i ≥ 1} be independent identically distributed random vari-
ables (i.i.d.r.v) with common distribution function (d.f.) F (x) and char-
acteristic function (ch.f.) ϕ(t). Suppose that EX = m if E|X| < ∞ and
V ar(X) = 1 if E|X|2 < ∞. Put
S(n) =
n∑
i=1
Xi, S(0) = 0, S(x) = S([x]), (1)
where [a] is entire of a > 0.
Let {z, zi, i ≥ 1} be another sequence of i.i.d.r.v. independent of
{Xi, i ≥ 1} with d.f. F1(x) and ch.f. ϕ1(t), Ez = 1/λ if E|z| < ∞ and
V ar(X) = τ 2 if E|Z|2 < ∞.
Denote
Z(n) =
n∑
i=1
zi, Z(0) = 0, Z(x) = Z([x])
and define the renewal counting process as
N(t) = inf{x ≥ 0 : Z(x) > t}. (2)
Invited lecture.
2000 Mathematics Subject Classifications 60F17, 60F15, 60G52, 60G50.
Key words and phrases. Lévy processes, stable processes, invariance principle, domain
of attraction,renewal process, randomly stopped process, risk models.
233
234 NADIIA ZINCHENKO
We shall consider also the randomly stopped sum process (i.e. the super-
position of random processes S(n) and N(t))
D(t) = S(N(t)) =
N(t)∑
i=1
Xi, (3)
where renewal process N(t) is defined by (2).
The main task of this paper is to study the asymptotic behavior of the
random processes D(t) and N(t) when F (x) and F1(x) are heavy tailed.
This problem has a deep relation with investigations of risk process U(T )
and approximation of ruin probabilities in Sparre Anderssen collective risk
model
U(t) = u + ct −
N(t)∑
i=1
Xi, (4)
where: u ≥ 0 denotes the initial capital; c > 0 stands for the premium in-
come rate; i.i.d.r.v {Xi, i ≥ 1} are interpreted as claim sizes; N(t) describes
the claim arrival process and stands for the number of claims until time t;
{zi, i ≥ 1} being the inter-arrival times.
In such model S(N(t)) is interpreted as total claim amount process and
is a stochastic part of risk process.
Limit theorems for risk process such as (weak) invariance principle which
constitute the weak convergence of U(t) to the Wiener process W (t) with
the drift (when EX2 < ∞, Ez2 < ∞) or to the α-stable Lévy process Yα(t)
(when EX2 = ∞, Ez2 < ∞) lead to useful approximations of the ruin
probability
ψ(u) = P{inf
t>0
U(t) < 0}. (5)
Thus, in the case EX2 < ∞, Ez2 < ∞ one obtains the ”diffusion ap-
proximation” for ψ(u) as a distribution of infimum of the Wiener process
( Iglehart (1969), Grandell (1991), Embrechts et al.(1997)) and in the case
EX2 = ∞, Ez2 < ∞ the ruin probability ψ(u) is approximated by the dis-
tribution of infimum of the corresponding α-stable process ( Furrur, Michna
and Weron (1997), Furrur (1998))
2. Strong invariance principle for the partial sums
Strong invariance principle (almost sure approximation) is an umbrella
name for the class of limit theorems which ensure the possibility to construct
{Xi, i ≥ 1} and Lévy process Y (t), t ≥ 0 on the same probability space in
such a way that with probability 1
|S(t) − mt − Y (t)| = o(r(t)), as t → ∞ (6)
STRONG INVARIANCE PRINCIPLE 235
or
|S(t) − mt − Y (t)| = O(r(t)), as t → ∞ (7)
were approximation error (rate) r(.) is a non-random function depending
only on assumptions posed on X.
Additional assumptions on X clear up the type of Y (t) and form of
r(·). Note that the complete solution of the problem of a.s. approximation
depends not only on the distribution of {Xi, i ≥ 1} but also on a structure
of the probability space, and (possibly) requires a “richer” probability space
and equivalent r.v.{X ′
i , i ≥ 1}. However, for brevity we do not distinguish
between r.v.{Xi} and{X ′
i} as well as between their sums S(n) and S
′
(n) =∑n
i=1 X
′
i .
We also use the concept of a.s. approximation in a wider sense, and say
that a random process ξ(t) admits the a.s. approximation by the random
process η(t) if ξ(t) (or stochastically equivalent {ξ′
(t), t ≥ 0}) can be con-
structed on the rich enough probability space together with η(t), t ≥ 0 in
such a way that a.s.
|ξ(t) − η(t)| = o(r1(t)) ∨ O(r1(t)),
where r1(.) is again a non-random function.
The origin of this topic in the theory of limit theorems goes back to the
famous ”Skorokhod representation” and ”Skorokhod embedding scheme”
( Skorokhod (1961). Skorokhod representation allows one to study a se-
quence of values of the Wiener process W (Tn), where Tn, n ≥ 1, are some
stopping times, instead of partial sums S(n). Based on Skorokhod embed-
ding scheme Strassen (1964, 1965) proved the first variant of the strong
invariance principle.
In 1970 – 1995 the further investigations were carried out by a number
of authors, among them: Kiefer, M.Csörgő, Révész, Komlós, Major, Tus-
nady, Berkes, Horváth (quantile Hungerian method), Stout, Phillip, Berkes
(relationship between the strong invariance principle and convergence in
Prokhorov metrics), Horváth ( inverse processes).
The wide bibliography which covers the period between 1961 and 1980
is presented in M.Csörgő, P.Révész (1981); more recent results in M.Csörgő,
L. Horváth, (1993), see also Zinchenko (2000).
Summarizing all mentioned above results we have
Theorem A1. It is possible to define partial sum process S(t), t ≥ 0 and a
standard Wiener process W (t), t ≥ 0 in such a way that a.s.
|S(t) − mt − W (t)| = o(r(t)), (8)
with: r(t) = t1/p if and only if E|X|p < ∞, p > 2, r(t) = (t log log(t))1/2 if
236 NADIIA ZINCHENKO
and only if E|X|2 < ∞; while (8) can be changed on O(log t) if and only if
E exp(tX) < ∞ for some t > 0.
3. Strong invariance principle for the sums of r.v.
attracted to the stable law
Suppose that EX2 = ∞; more precise we assume that {X, Xi, i ≥ 1}
belong to the domain of normal attraction of the stable law Gα,β ( notation
{Xi} ∈ DNA(Gα,β)).
Here Gα,β(.) is a d.f. of the stable law with parameters 0 < α < 2,
|β| ≤ 1 and ch.f.
gα,β(u) = exp(−K(u)), (10)
where
K(u) = Kα,β(u) − |u|(1 − iβ(u/|u|)
(u, α)), (11)
(u, α) = tan(πα/2) if 1 < α < 2,
(u, α) = −(2/π) log |u| if α = 1.
We recall that i.i.d.r.v. {Xi} ∈ DNA(Gα,β) if for normalized and cen-
tered sums S∗
n there is a weak convergence
S∗
n = n−1/α(S(n) − an) ⇒ Gα,β, (12)
where an = nEX = mn if 1 < α < 2, an = O if 0 < α < 1 and an =
(2/π)β log n if α = 1.
For the r.v. X ∈ DNA(Gα,β) (as well as for the α-stable r.v.)
E|X|p < ∞ ∀p < α, but E|X|p = ∞ ∀p > α.
Denote by Y (t) = Yα(t) = Yα,β(t), t ≥ 0, the α-stable Lévy process with
ch.f.
gα(t; u) = gα,β(t; u) = exp(tKα,β(u)), (13)
where Kα,β(u) is defined in (11), Yα(0) = 0. In what follows we omit index
β if it is not essential.
Strong invariance principle for {Xi} ∈ DNA(Gα,β) when approximating
process is α-stable Lévy process (or partial sum process with stable sum-
mands) was studied by Stout (1979), Mijnheer ( 1983, 1995), Zinchenko
(1984), Berkes, Dabrowski, Dehling, Philipp (1986), Berkes and Dehling
(1989) in the case of symmetric stable law (β = 0) and Zinchenko (1985,
1989, 1997) without any restriction on parameters α and β.
The fact that {Xi} ∈ DNA(Gα,β) is not enough to obtain ”good” error
term in (6), thus, certain additional assumptions are needed. We formulate
them in terms of ch.f.
Assumption (C) :
there are a1 > 0, a2 > 0 and l > α such that for |u| < a1
|f(u) − gα,β(u)| < a2|u|l (14)
STRONG INVARIANCE PRINCIPLE 237
where f(u) = e−itmϕ(t) is a ch.f. of (X−EX) if 1 < α < 2 and f(u) = ϕ(t),
i.e. ch.f of X if 0 < α ≤ 1.
Put
A = [max{α + 1, 2α(2 + α + 1/α)/(l − α)}] + 1, if 0 < α < 1,
A = [max{α(α + 1), 2α(2α + 1)/(l − α)}] + 1, if 1 < α < 2,
A = {[10/(l − 1)] + 1, 6}], if α = 1,
where [a] is entire of a > 0.
Theorem A2. Put m = EX for 1 < α < 2 and m = 0 for 0 < α ≤ 1.
Under assumption (C) it is possible to define α-stable process Yα,β(t), t ≥ 0
such that a.s.
sup
0≤t≤T
|S(t) − mt − Yα,β(t)| = o(T 1/α−ρ), (15)
for some ρ = ρ(α, l) ∈ (0, 1/α(A + 1)).
In the case EX = 0 Theorem A2 was proved by Zinchenko (1987, 1997),
obvious centering when 1 < α < 2 provides (15). If α
= 1 detail analysis of
the proof in Zinchenko (1997) shows that it is possible to obtain a shaper
estimate for ρ and establish that a.s.
sup
0≤t≤T
|S([t]) − mt − Yα,β(t)| = O(T 1/α−ρ1), (16)
where
ρ1 = 1/4α(A + 1). (17)
Thus, (15) holds for any ρ ∈ (0, 1/4α(A + 1)).
It worth mentioning that unlike Theorem A1 (α = 2) Theorem A2
presents only sufficient condition for strong invariance principle and tells
nothing about optimality of the error term.
4. Asymptotic behaviour of the renewal process. Auxiliary
results
Let N(t) = inf{x > 0 : Z(x) > t} be renewal(counting) process asso-
ciated with sum process Z(n) =
∑n
i=1 zi with 0 < Ez1 = 1/λ < ∞. For
applications it is often convenient to suppose that zi are non-negative (non-
zero) r.v. It is clear that N(t) is the generalized right-continuous inverse of
right-continuous process Z(t).
Following auxiliary results will be useful for further investigations.
238 NADIIA ZINCHENKO
Lemma 1 (Csörgő, Horvách (1993))Let 0 < λ < ∞, then a.s.
lim sup
t→∞
N(t)/t ≤ λ. (18)
Order of magnitude of N(t) is described by following theorem which in-
cludes strong law of large numbers (SLLN), Marcinkiewich-Zygmund SLLN
and law of iterated logarithm for renewal process.
Theorem A3.
(i) If 0 < Ez = 1/λ < ∞, then a.s.
N(t)/t → λ, (19)
(ii) if E|z|p < ∞ for some p ∈ (1, 2) then a.s.
t−1/p(N(t) − λt) → 0, (20)
(iii) if τ 2 = V ar(X) < ∞ then
lim sup
t→∞
(2t log log t)−1/2|N(t) − λt| = τλ3/2, (21)
while the for the moments we have
EN(t) ∼ λt, V ar(N(t)) ∼ τλ3/2.
The sketch of the proof is presented in Embrechts et al. (1977), see also
A.Gut (1988). Original Marcinkiewich-Zygmund SLLN for partial sums of
i.i.d.r.v. can be find in Loéve (1978).
Weak convergence, particularly, weak invariance principle for renewal
process is in details presented in the book by Whitt (2002).
Next two simple lemmas from Csörgő, Horvách (1993) deals with the
properties of the inverse step functions.
Here a function θ(t), t ∈ [0,∞), is called a right-continuous step function
if there is a decomposition of [0,∞) =
∞⋃
i=1
[ti, ti+1) such that 0 = t1 < t2 < . . .
and θ(t) = qi for t ∈ [ti, ti+1), qi ∈ R1, q1 = 0. The right-continuous inverse
of θ is defined by
ψ(x) = inf{t ≥ 0 : θ(t) > x}, 0 ≤ u < ∞, inf ∅ = ∞
Lemma 2. For any T ≥ 0
sup
0≤x≤T
|ψ(x) − x| ≤ sup
0≤t≤ψ(T )
|θ(t) − t|. (22)
STRONG INVARIANCE PRINCIPLE 239
Lemma 3. For any T ≥ 0
t − ψ(t) = θ(ψ(t)) − ψ(t) − (θ(ψ(t)) − t), (23)
sup
0≤t≤T
|θ(ψ(t)) − t| ≤ sup
0≤t≤T
|θ(t) − θ(t−)|. (24)
The growth rate of α-stable Lèvy process Yα(t) when t → ∞ is described
by the following statement.
Lemma 4. If Yα(t) is an α-stable Lèvy process with 0 < α < 2 then a.s.
Yα(t) = o(t1/α+ε), ∀ε > 0.
This fact follows immediately from the integral test for upper/lower func-
tions of Lévy process (Gikhman and Skorokhod (1973, ch.4).
Keeping in mind these facts and equivalence in weak convergence for
S(n) and associated N(t) it is natural to ask about a.s. approximation of
N(t).
5. Strong invariance principle for renewal process
5.a. Assumptions: Ez2 < ∞, 0 < Ez = 1/λ < ∞.
During 1984 - 2000 strong approximation of the counting process N(t)
associated with partial sum process Z(x) =
∑[x]
i=1 zi in the case E|z|p < ∞
for p ≥ 2 ( or more general moment conditions) was investigated by a
number of authors, among them Horvách, M. Csörgő, Steinebach, Aalex,
Deheuvels, Mason, van Zwet. They studied a.s. approximation of the type
|λt − N(t) − λW (λt)| = o(r(t)) ∨ O(r(t)). (25)
For instance, M. Csörgő, Horvách and Steinebach (1986, 1987) obtained
the best possible approximations of N(t). It turned out that conditions
which provide (25) and corresponding optimal errors in the case of non-
negative r.v. {zi} are just the same as for partial sums Z(n) (see Theo-
rem A1).
5.b. Assumptions: {zi} ∈ NDA(Gα,β) with 1 < α < 2.
Theorem 1. Let {zi} satisfy (C) with 1 < α < 2 and 0 < Ez = 1/λ < ∞
then a.s.
|tλ − N(t) − λYα,β(λt)| = o(r(t)), (26)
where r(t) = t1/α+δ for any δ > 0.
Proof. We use the idea of M. Csörgő, L.Horvách and Steinebach about the
240 NADIIA ZINCHENKO
correspondence between a.s.approximation of Z(n) and associated counting
process N(t). Consider
Z1(x) = λZ(x), N1 = inf{x : Z1(x) > t} = inf{x : Z(x) > t/λ}. (27)
Thus, N1(t) = N(t/λ) and (18) or (19) yields
lim sup
t→∞
N1(t)/t ≤ 1. (28)
As far as condition (C) is concerned, Theorem A2 ensure the possibility to
define α-stable process Yα(t) = Yα,β(t) such that a.s.
|Z(t) − tλ − Yα(t)| = O(T 1/α−ρ1), (29)
for some ρ1 = ρ(α, l) > 0.
Thus,
|Z1(t) − t − λYα(t)| = O(T 1/α−ρ1). (30)
By Lemma 3 and definition of Z1(t), N(t)
t − N1(t) = Z1(N1(t)) − N1(t) + A1(t) (31)
where
sup
0≤t≤T
|A1(t)| ≤ sup
0≤t≤N1(T )
|Z1(t) − Z1(t−)| ≤ λ max
0≤t≤N1(T )
|zi| (32)
Since r.v.{zi} ∈ NDA(Gα,β) with 1 < α < 2 have finite moments
E|Zi|p < ∞ for any p < α, Marcinkiewich-Zygmund SLLN for Z(n) yields
a.s.
max
0≤i≤n
|zi| = o(n1/p), ∀p ∈ (1, α) (33)
From (28) and (33) we conclude that a.s.
sup
0≤t≤T
|A1(t)| ≤ λ max
0≤t≤N1(T )
|zi| = o(T 1/α+ε) ∀ε > 0. (34)
Therefore, (31) and (34) implies that
L(T ) = sup
0≤t≤T
|t − N1(t) − λYα(t)| ≤
≤ sup
0≤t≤T
|Z1(N1(t)) − N1(t) − Yα(N1(t))|+
+ sup
0≤t≤T
|Yα(N1(t)) − λYα(t)| + sup
0≤t≤T
|A1(t)|. (35)
STRONG INVARIANCE PRINCIPLE 241
Next by (30) and (28) a.s.
sup
0≤t≤T
|Z1(N1(t)) − N1(t) − Yα(N1(t))| = O(T 1/α−ρ1), (36)
for some ρ1 = ρ1(α, l) > 0.
Lemma 4, which provides an upper function for Yα(t), implies
sup
0≤t≤T
|Yα(N1(t)) − Yα(t)| = o(T 1/α+ε), ∀ε > 0. (37)
Hence, (35)-(37) provide
L(T ) = o(T 1/α+ε), ∀ε > 0. (38)
Recalling that N1(t) = N(t/λ), we immediately derive (26) from (38). �
6. Strong invariance principle for randomly stopped
processes
Let {X, Xi,≥ 1}, {z, zi,≥ 1}, S(n), Z(n),N(t) be as in Introduction,
EX = m, Ez = 1/λ > 0. Put
D(t) = S(N(t)) =
N(t)∑
i=1
Xi.
Weak invariance principle for D(t) was studied in a lot of works; we mention
only fundamental monographs: Billingsley (1968), Gut (1988), Gnedenko
and Korolev (1996), Whitt (2002), Silvestrov (1974, 2004), for applications
of such topic to risk theory see also Embrechts et al.(1997), Korolev, Bening
and Shorgin (2007).
Strong invariance principle for S(N(t)) when EX2 < ∞ and EY 2
1 < ∞
(and may satisfy stronger moment conditions) was studied by M.Csörgő,
Horváth, Steinebach, Deheuvels, for detail bibliography see already cited
monograph by Csörgő and Horváth (1994), as well as survey article by
Aalex and Steinebach (1994).
In forthcoming we focus on the case E|X|2 = ∞ when {X, Xi,≥ 1}
belong to DNA(Gα1,β), 1 < α1 < 2, while {z, zi,≥ 1} can be attracted
to the normal law (α = 2, V ar(z) = τ 2 < ∞) or to the α2-stable law,
1 < α2 < 2.
Our approach is close to the methods presented in Csörgő and Horváth
(1993).
Theorem 2. Let {Xi, i ≥ 1} satisfy (C) with 1 < α < 2 and Ez2 < ∞
then a.s.
|D(N(t)) − mλt − Yα,β(λt)| = o(t1/α−�2), ρ2 ∈ (0, ρ0), (39)
242 NADIIA ZINCHENKO
for some �0 = �0(α, l) > 0.
Proof. The key moment in the proof is an expression
Δ(T ) = sup
0≤t≤T
|S(N(t)) − mλt − Yα(λt)|
≤ sup
0≤t≤T
|S(N(t)) − mN(t) − Yα(N(t))|
+ sup
0≤t≤T
|m(N(t) − λt)| + sup
0≤t≤T
|Yα(N(t)) − Yα(λt)|
≤ Δ1(T ) + Δ2(T ) + Δ3(T ). (40)
Now we estimate separately Δi, i = 1, 2, 3. Condition (C), Theorem A2
and (28) ensure the possibility to define Yα(t) such that a.s. for certain �1
Δ1 = O((N(T ))1/α−ρ1) = O(T 1/α−ρ1). (41)
The LIL for renewal process N(t) (see (21)) yields
Δ2(T ) = O((T log log T )1/2). (42)
Using the stationary of increments of the stable process, Lemma 4 and (21)
we obtain a.s.
Δ3(T ) = o((T log log T )1/2α+ε2), ∀ε2 > 0. (43)
Thus, Δ3(T ) can be made o(T 1/α−ρ1) by choosing an appropriate ε2.
Hence, combining (40) – (43) we obtain
Δ(T ) = o(T 1/α−ρ2) ∀ρ2 ∈ (0, ρ0)
for 1 < α < 2 and ρ0 = min(ρ1, (2 − α)/2α). �
Corollary. Theorem 2 holds if N(t) is a Poisson process.
In this case D(t) can be interpreted as total claims until moment t in
classic risk model.
Developing such approach we proved rather general result concerning a.s.
approximation of the randomly stopped process (not obligatory connected
with the partial sum processes).
Let Z∗(t), D∗(t) be two real-valued random processes, N∗ – the inverse
of Z∗(t) is defined by
N∗(t) = inf{t > 0 : Z∗(x) > t}, 0 ≤ t < ∞,
STRONG INVARIANCE PRINCIPLE 243
Theorem 3. Suppose that for some constants m, γ, a > 0, σ > 0, τ > 0 a.s.
sup
0≤t≤T
|σ−1(Z(t) − at) − W1(t)| = O(r(T )), (44)
where W1(t) is a Wiener process, r(t) ↑ ∞, r(t)/t ↓ 0 as t → ∞ and
sup
0≤t≤T
|D∗(t) − mt − Yα(t)| = O(q(T )), (45)
Yα(t) being α-stable process independent of W1(t), q(t) ↑ ∞, q(t)/t ↓ 0 as
t → ∞, then ∀ε > 0 a.s.
|D(N∗(t)) − (m/a)t − (Yα(t/a) − (mσ/a)W2(t/a))| =
= O(q(t)) + O(r(T ) + log t)
+O((r(t) + (t log log t)1/2)1/(α−ε)), (46)
where W2(t) is a Wiener process independent of Yα(t).
Proof. The essential point of the proof is to apply the inequality
|D(N∗(t)) − (m/a)t − (Yα(t/a) − (mσ/a)W2(t/a))|
≤ |D(N∗(t)) − mN∗(t) − Yα(N∗(t))|
+|Yα(N∗(t)) − Yα(t/a)|
+|m(N∗(t) − t/a + (σ/a)W2(t/a))|
≤ Δ∗
1(t) + Δ∗
2(t) + Δ∗
3(t)
and estimate each Δ∗
i (t) using a.s.approximation for D∗(t), N∗(t) and
growth rate for stable and Wiener processes.�
In the case of partial sum processes S(t) and Z(t) with Ez2 < ∞, Xi
satisfying (C), N∗(t) = N(t) is counting renewal process, q(t) = T 1/α−�1 ,
�1 > 0, the worst estimate for r(t) is (t log log t)1/2. These facts lead to
statement of the Theorem 2.
The same approach provides
Theorem 4. Let {Xi, i ≥ 1} satisfy (C) with 1 < α1 < 2, and {zi} satisfy
(C) with 1 < α2 < 2,
α1 ≤ α2
then a.s.
|S(N(t)) − mλt − Yα1,β(λt)| = o(t1/α1−�3)
for some �3 = �3(α1, l) > 0.
244 NADIIA ZINCHENKO
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Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: znm@univ.kiev.ua
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