arXiv:1704.02137v1 [math.PR] 7 Apr 2017
Modern Stochastics: Theory and Applications 4 (2017) 65–78
DOI: 10.15559/17-VMSTA74
Randomly stopped maximum and maximum of sums
with consistently varying distributions
Ieva Marija Andrulytė, Martynas Manstavičius, Jonas Šiaulys∗
Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24,
Vilnius LT-03225, Lithuania
[email protected] (I.M. Andrulytė), [email protected] (M. Manstavičius),
[email protected] (J. Šiaulys)
Received: 10 December 2016, Revised: 23 January 2017, Accepted: 27 January 2017,
Published online: 6 March 2017
Abstract Let {ξ1 , ξ2 , . . .} be a sequence of independent random variables, and η be a counting random variable independent of this sequence. In addition, let S0 := 0 and Sn := ξ1 +
ξ2 + · · · + ξn for n > 1. We consider conditions for random variables {ξ1 , ξ2 , . . .} and η under
which the distribution functions of the random maximum ξ(η) := max{0, ξ1 , ξ2 , . . . , ξη } and
of the random maximum of sums S(η) := max{S0 , S1 , S2 , . . . , Sη } belong to the class of
consistently varying distributions. In our consideration the random variables {ξ1 , ξ2 , . . .} are
not necessarily identically distributed.
Keywords Heavy tail, consistently varying tail, randomly stopped maximum, randomly
stopped maximum of sums, closure property
2010 MSC 62E20, 60E05, 60F10, 44A35
1
Introduction
Let {ξ1 , ξ2 , . . .} be a sequence of independent random variables (r.v.s) with distribution functions (d.f.s) {Fξ1 , Fξ2 , . . .}, and let η be a counting r.v., that is, an integervalued, nonnegative, and nondegenerate at zero r.v. In addition, suppose that the r.v.
η and r.v.s {ξ1 , ξ2 , . . .} are independent.
∗ Corresponding
author.
© 2017 The Author(s). Published by VTeX. Open access article under the CC BY license.
www.i-journals.org/vmsta
66
I.M. Andrulytė et al.
Let S0 = 0, Sn = ξ1 + ξ2 + · · · + ξn for n ∈ N, and let
Sη =
η
X
ξk
k=1
be the randomly stopped sum of r.v.s {ξ1 , ξ2 , . . .}.
Similarly, let ξ(n) = max{0, ξ1 , ξ2 , . . . , ξn } for n ∈ N, ξ(0) = 0, and let
ξ(η) =
(
max{0, ξ1 , ξ2 , . . . , ξη } if η > 1,
0
if η = 0
be the randomly stopped maximum of r.v.s {ξ1 , ξ2 , . . .}.
Finally, let S(n) := max{S0 , S1 , S2 , . . . , Sn } for n > 0, and let
S(η) := max{S0 , S1 , S2 , . . . , Sη }
be the randomly stopped maximum of sums S0 , S1 , S2 , . . ..
We denote the distribution functions (d.f.s) of Sη , ξ(η) , and S(η) by FSη , Fξ(η) , and
FS(η) respectively, together with their tails F Sη , F ξ(η) , and F S(η) . For any positive x,
we have
F Sη (x) =
F ξ(η) (x) =
F S(η) (x) =
∞
X
n=1
∞
X
n=1
∞
X
P(η = n)P(Sn > x),
P(η = n)P(ξ(n) > x),
P(η = n)P(S(n) > x).
n=1
In [14], conditions were found for the d.f.s FSη to belong to the class of consistently varying distributions. In this paper, we are interested in sufficient conditions
under which the d.f.s Fξ(η) and FS(η) have consistently varying tails.
Throughout this paper, for two vanishing (at infinity) functions f and g,
f (x) = o(g(x)) means that limx→∞ f (x)/g(x) = 0, and f (x) ∼ g(x) means that
limx→∞ f (x)/g(x) = 1. Also, we denote the support of a counting r.v. η by
supp(η) := n ∈ Z+ : P(η = n) > 0 .
Before discussing the properties of Fξ(η) and FS(η) , we recall the definitions of
some classes of heavy-tailed d.f.s. For a d.f. F , we denote F (x) = 1 − F (x) for
real x.
• A d.f. F is heavy-tailed (F ∈ H) if for every δ > 0,
lim F (x)eδx = ∞.
x→∞
Randomly stopped maximum and maximum of sums with consistently varying distributions
67
• A d.f. F is long-tailed (F ∈ L) if for every y (equivalently, for some y > 0),
F (x + y) ∼ F (x).
• A d.f. F has dominatingly varying tail (F ∈ D) if for every y ∈ (0, 1) (equivalently, for some y ∈ (0, 1)),
lim sup
x→∞
F (xy)
< ∞.
F (x)
• A d.f. F has consistently varying tail (F ∈ C) if
lim lim sup
y↑1
x→∞
F (xy)
= 1.
F (x)
• A d.f. F has regularly varying tail (F ∈ R) if there is α > 0 such that
F (xy)
= y −α ,
x→∞ F (x)
lim
for all y > 0.
• A d.f. F supported on the interval [0, ∞) is subexponential (F ∈ S) if
lim
x→∞
F ∗ F (x)
= 2.
F (x)
(1)
If a d.f. G is supported on R, then we say that G is subexponential (G ∈ S) if
the d.f. F (x) = G(x)1[0,∞) (x) satisfies relation (1).
It is known (see, e.g., [5, 12, 15], and Chapters 1.4 and A3 in [10]) that these
classes satisfy the following inclusions:
R ⊂ C ⊂ L ∩ D ⊂ S ⊂ L ⊂ H,
D ⊂ H.
These inclusions are depicted in Fig. 1 borrowed from the paper [14]. In this figure,
the class C of distributions having consistently varying tails is highlighted.
It should be noted that the subject of the paper is partially motivated by the closure problem of the random convolution. In the case of independent and identically
distributed (i.i.d.) r.v.s {ξ, ξ1 , ξ2 , . . .}, we say that a class K of d.f.s is closed with
respect to the random convolution if the condition Fξ ∈ K implies FSη ∈ K. The
first result on the convolution closure of subexponential distributions was obtained
by Embrechts and Goldie (see Thm. 4.2 in [11]) and by Cline (see Thm. 2.13 in [6]).
Theorem 1. Let {ξ1 , ξ2 , . . .} be independent copies of a nonnegative r.v. ξ with
subexponential d.f. Fξ . Let η be a counting r.v. independent of {ξ1 , ξ2 , . . .}. If E(1 +
δ)η < ∞ for some δ > 0, then the d.f. FSη ∈ S.
The random closure results for class D can be found in [7, 16], and for the class
L, in [1, 16, 19, 20]. The random closure results for the class C can be derived from
the results of [14]. We note that in [7, 14, 20], the case of not necessarily identically
68
I.M. Andrulytė et al.
H
L
S
D
C
R
Fig. 1. Classes of heavy-tailed distributions
distributed r.v.s {ξ1 , ξ2 , . . .} was considered. We further recall two known results.
First, in Theorem 2, we give conditions for FSη to belong to the class D. Its proof can
be found in [7, Thm. 2.1]. Recall that a d.f. F belongs to the class D if and only if its
upper Matuszewska index JF+ < ∞, where by definition
1
F (xy)
log lim inf
.
x→∞ F (x)
y→∞ log y
JF+ = − lim
Theorem 2. Let r.v.s {ξ1 , ξ2 , . . .} be nonnegative independent but not necessary identically distributed, and η be a counting r.v. independent of {ξ1 , ξ2 , . . .}. Then the d.f.
FSη belongs to the class D if the following three conditions are satisfied:
(i)
(ii)
(iii)
Fξκ ∈ D for some κ ∈ supp(η),
n
X
1
F ξi (x) < ∞,
lim sup sup
x→∞ n>κ nF ξκ (x)
i=1
Eη p+1 < ∞ for some p > JF+ξ .
κ
A similar result for r.v.s having d.f.s with consistently varying tails can be found
in [14, Thm. 6].
Theorem 3. Let {ξ1 , ξ2 , . . .} be independent real-valued r.v.s, and η be a counting r.v.
independent of {ξ1 , ξ2 , . . .}. Then the d.f. FSη belongs to the class C if the following
conditions are satisfied:
(a)
Fξ1 ∈ C,
(b)
for each k > 2, either Fξk ∈ C or F ξk (x) = o F ξ1 (x) ,
n
X
1
F ξi (x) < ∞,
lim sup sup
x→∞ n>1 nF ξ1 (x)
i=1
(c)
(d)
Eη p+1 < ∞ for some p > JF+ξ .
1
In this work, we consider the randomly stopped maximum ξ(η) and randomly
stopped maximum of sums S(η) of independent but not necessarily identically dis-
Randomly stopped maximum and maximum of sums with consistently varying distributions
69
tributed r.v.s. As was noted before, we restrict our consideration to the class C but
extend it to the real-valued r.v.s as in Theorem 3.
If r.v.s {ξ1 , ξ2 , . . .} are not identically distributed, then different collections of
conditions on r.v.s {ξ1 , ξ2 , . . .} and η imply that Fξ(η) ∈ C or FS(η) ∈ C. We suppose
that some r.v.s from {ξ1 , ξ2 , . . .} have distributions belonging to the class C, and we
find conditions for r.v.s {ξ1 , ξ2 , . . .} and η such that the distribution of the randomly
stopped maximum or the randomly stopped maximum of sums remains in the same
class. The results presented and their proofs are closely related to the results of the
papers [7, 8, 14].
It is worth noting that the closure properties for d.f.s FS(η) in the case of i.i.d. r.v.s
can be derived, for instance, from the asymptotic formulas obtained in [9, 13, 17, 18,
21]. Unfortunately, in the case of nonidentically distributed r.v.s, similar asymptotic
formulas do not exist. Therefore we have to use other methods to prove our main
results.
The rest of the paper is organized as follows. In Section 2, we present our main
results together with a few examples of randomly stopped maximum ξ(η) and randomly stopped maximum of sums S(η) with d.f.s having consistently varying tails.
Section 3 is a collection of auxiliary lemmas, and the proofs of the main results are
presented in Section 4.
2
Main results
In this section, we present four statements, two theorems and two corollaries. Theorem 4 and Corollary 1 deal with the belonging of the d.f. Fξ(η) to the class C.
Theorem 4. Let {ξ1 , ξ2 , . . . } be a sequence of independent real-valued r.v.s, and let
η be a counting r.v. independent of {ξ1 , ξ2 , . . . }. The d.f. of the randomly stopped
maximum Fξ(η) belongs to the class C if the following conditions hold:
(a) Fξκ ∈ C for some κ ∈ supp(η),
(b) for each k 6= κ, either Fξk ∈ C or F ξk (x) = o F ξκ (x) ,
n
X
1
(c) lim sup sup
F ξk (x) < ∞,
x→∞ n>1 ϕ(n)F ξκ (x)
k=1
where {ϕ(n)}∞
n=1 is a positive sequence such that E(ϕ(η)1[1,∞) (η)) < ∞.
If r.v.s {ξ1 , ξ2 , . . .} are identically distributed with common d.f. Fξ ∈ C, then conditions (a), (b), and (c) are satisfied with ϕ(n) = n for n ∈ N. Hence, the following
statement immediately follows from Theorem 4.
Corollary 1. Let {ξ1 , ξ2 , . . . } be a sequence of i.i.d. real-valued r.v.s with common
d.f. Fξ ∈ C, and let η be a counting r.v. independent of {ξ1 , ξ2 , . . . }. The d.f. of the
randomly stopped maximum Fξ(η) belongs to the class C if Eη is finite.
In Theorem 5 and Corollary 2, we present conditions under which the d.f. of the
randomly stopped maximum of sums S(η) has consistently varying tail.
Theorem 5. Let {ξ1 , ξ2 , . . . } be independent real-valued r.v.s, and let η be a counting
r.v. independent of {ξ1 , ξ2 , . . . }. The d.f. FS(η) belongs to the class C if the following
conditions hold:
70
I.M. Andrulytė et al.
(a) Fξk ∈ C for each k ∈ N,
n
X
1
F ξk (x) < ∞,
(b) lim sup sup
x→∞ n>1 nF ξ1 (x)
k=1
(c) Eη p+1 < ∞ for some p > JF+ξ .
1
Similarly to Corollary 1, we can state the following corollary. We note that in the
i.i.d. case condition (b) is obviously satisfied if F ξ1 (x) > 0 for all x ∈ R.
Corollary 2. Let {ξ1 , ξ2 , . . . } be i.i.d. real-valued r.v.s with common d.f. Fξ ∈ C, and
let η is a counting r.v. independent of {ξ1 , ξ2 , . . .}. The d.f. FS(η) belongs to the class
C if Eη p+1 < ∞ for some p > JF+ξ .
Further in this section, we present two examples of r.v.s {ξ1 , ξ2 , . . .} and η showing the applicability of our theorems.
Example 1. Let {ξ1 , ξ2 , . . .} be independent r.v.s such that:
• r.v.s ξk are exponentially distributed for k ≡ 0 mod 3, that is,
F ξk (x) = 1(−∞,0) (x) + e−x 1[0,∞) (x),
k ∈ {3, 6, 9, . . .};
• r.v.s ξk are degenerate at zero for all k ≡ 2 mod 3 and k ≡ 1 mod 3, k > 4,
that is,
Fξk (x) = 1[0,∞) (x),
k ∈ {2, 5, 8, . . .} ∪ {4, 7, 10, . . .};
• ξ1 = (1+U)2G , where r.v.s U and G are independent, U is uniformly distributed
on the interval [0, 1], and G is geometrically distributed with parameter q ∈
(0, 1), that is,
P(G = l) = (1 − q)q l ,
l ∈ {0, 1, 2, . . .}.
In addition, let η be a counting r.v. independent of {ξ1 , ξ2 , . . .}.
Theorem 4 implies that the d.f. of the randomly stopped maximum ξ(η) belongs
to the class C if 1 ∈ supp(η) and Eη < ∞ because:
/ R due to considerations in pp. 122–123 of [2],
• Fξ1 ∈ C, but Fξ1 ∈
• F ξk (x) = o(F ξ1 (x)) for each k 6= 1,
• sup nF 1 (x)
n>1
ξ1
n
P
F ξk (x) 6 2 for sufficiently large x.
k=1
Example 2. Let {ξ1 , ξ2 , . . .} be independent r.v.s distributed according to Pareto-type
laws, namely,
1
1[0,∞) (x) if k ∈ {1, 3, 5, . . .},
(1 + x)3
1
F ξk (x) = 1(−∞,−1) (x) +
1[−1,∞) (x) if k ∈ {2, 4, 6, . . .}.
(2 + x)3
F ξk (x) = 1(−∞,0) (x) +
Randomly stopped maximum and maximum of sums with consistently varying distributions
71
Further, let η be a counting r.v. independent of {ξ1 , ξ2 , . . .} that has the zeta distribution with parameter 6, that is,
P(η = m) =
1
1
,
ζ(6) (m + 1)6
m ∈ {0, 1, 2, . . .},
where ζ denotes the Riemann zeta function.
Theorem 5 implies that the d.f. of the randomly stopped maximum of sums S(η)
belongs to the class C because:
• Fξk ∈ R ⊂ C for each k ∈ N;
• sup nF 1 (x)
n>1
ξ1
n
P
F ξk (x) = 1 for positive x;
k=1
• JF+ξ = 3 and Eη 4.5 < ∞.
1
3
Auxiliary lemmas
In this section, we give auxiliary lemmas, which we use in the proof of Theorem 5.
The first lemma was proved in [17, Thm. 2.1]; a more general case can be found in
[3, Thm. 2.1]. We recall only that C ⊂ L. Hence, the statement of the next lemma
holds for d.f.s with consistently varying tails.
Lemma 1. Let {X1 , X2 , . . . , Xn } be independent real-valued r.v.s such that FXk ∈
L for all k ∈ N. Then
!
!
n
k
X
X
Xi > x .
Xi > x
∼ P
P max
16k6n
x→∞
i=1
i=1
The second auxiliary lemma was proved in [14, Lemma 3]. It describes the situation where the d.f. of sums of independent r.v.s belongs to the class C.
Lemma 2. Let {X1 , X2 , . . . , Xn } be independent real-valued r.v.s. The d.f. of the
sum Σn := X1 + X2 + · · ·+ Xn belongs to the class C if the following two conditions
are satisfied:
(a) FX1 ∈ C,
(b) for each k ∈ {2, . . . , n}, either FXk ∈ C or F Xk (x) = o F X1 (x) .
The following statement was proved in [7, Lemma 3.2]. It gives an upper estimate
for the tail of the sum of r.v.s from the class D.
Lemma 3. Let {X1 , X2 , . . . } be nonnegative independent r.v.s, FXν ∈ D for some
ν > 1 such that
lim sup sup
x→∞ n>ν
1
nF Xν (x)
n
X
i=1
F Xi (x) < ∞.
72
I.M. Andrulytė et al.
Then, for each p > JF+Xν , there exists a positive constant c1 such that
F Σn (x) 6 c1 np+1 F Xν (x),
for all n > ν and x > 0, where FΣn is the d.f. of the sum Σn = X1 + · · · + Xn .
The last useful lemma is an obvious conclusion from Theorem 3.1 in [4].
Lemma 4. Let {X1 , X2 , . . . Xn } be independent real-valued r.v.s. If FXk ∈ C for
each k ∈ {1, 2, . . . , n}, then
!
n
n
X
X
F Xi (x).
Xi > x ∼
P
i=1
i=1
4
Proofs of the main results
Proof of Theorem 4. It suffices to prove that
F ξ(η) (xy)
lim sup lim sup
F ξ(η) (x)
x→∞
y↑1
(2)
6 1.
For all x > 0 and K ∈ N, we have
F ξ(η) (x) =
∞
X
F ξ(n) (x)P(η = n)
n=1
K
X
=
+
n=1
∞
X
!
P
n=K+1
n
[
!
{ξk > x} P(η = n).
k=1
Therefore,
PK
Sn
> xy} P(η = n)
=
P(ξ(η) > x)
F ξ(η) (x)
Sn
P∞
k=1 {ξk > xy} P(η = n)
n=K+1 P
+
P(ξ(η) > x)
F ξ(η) (xy)
n=1
P
k=1 {ξk
=: J1 + J2 ,
(3)
for all x > 0 and y ∈ (0, 1).
Denote
K := k ∈ N : Fξk ∈
/ C and F ξk (x) = o F ξκ (x) .
If x > 0, 1/2 6 y < 1, and K > κ, then
J1 6
1
K X
n
X
F ξ(η) (x) n=1
k=1
k∈K
/
F ξk (xy)P(η = n) +
K X
n
X
F ξk (x/2)
P(η = n).
F ξ(η) (x)
n=1 k=1
k∈K
(4)
73
Randomly stopped maximum and maximum of sums with consistently varying distributions
If k ∈ K, then
(5)
F ξk (x/2) = o F ξ(η) (x) ,
because
F ξk (x/2)
F ξk (x/2)
F ξk (x/2)
F ξκ (x/2)
6
6
,
F ξ(η) (x)
F ξ(κ) (x)P(η = κ)
F ξκ (x/2)P (η = κ) F ξκ (x)
and Fξκ ∈ C ⊂ D.
The obtained asymptotic relation (5) and estimate (4) imply
lim sup J1 6 lim sup
x→∞
x→∞
6 lim sup
x→∞
K X
n
X
1
F ξ(η) (x) n=1
max
16k6K
k∈K
/
F ξk (xy)P(η = n)
k=1
k∈K
/
!
K X
n
X
F ξk (xy)
1
F ξk (x)P(η = n)
F ξk (x) F ξ(η) (x) n=1 k=1
k∈K
/
F ξ (xy)
6 max lim sup k
16k6K
x→∞ F ξk (x)
k∈K
/
(
)
K X
n
X
1
× lim sup
F ξk (x)P(η = n) .
F ξ(η) (x) n=1 k=1
x→∞
(6)
For each 1 6 n 6 K, we have
F ξ(n) (x) = P
n
[
{ξk > x}
k=1
!
>
n
X
F ξk (x) 1 −
k=1
K
X
k=1
!
F ξk (x) ,
due to the Bonferroni inequality. This implies that, for an arbitrary ε > 0,
n
X
F ξk (x) 6 (1 + ε)F ξ(n) (x),
k=1
if x is sufficiently large.
Substituting the last estimate into inequality (6), we get
F ξk (xy)
lim sup lim sup J1 6 (1 + ε) max lim sup lim sup
16k6K
x→∞ F ξk (x)
x→∞
y↑1
y↑1
k∈K
/
= 1 + ε,
because Fξk ∈ C for each k ∈
/ K.
Since ε > 0 is arbitrary, the last estimate implies that
lim sup lim sup J1 6 1.
y↑1
x→∞
Since
P(ξ(η) > x) > P(ξ(κ) > x)P(η = κ) > P(ξκ > x)P(η = κ),
(7)
74
I.M. Andrulytė et al.
we obtain
P∞
n=K+1
J2 6
Pn
k=1
F ξk (xy)P(η = n)
F ξκ (x)P(η = κ)
,
for all x > 0 and y ∈ (0, 1).
Condition (c) of the theorem implies that, for some constant c2 > 0,
n
X
F ξk (x) 6 c2 ϕ(n)F ξκ (x),
k=1
for all sufficiently large x and all n > 1.
Consequently,
lim sup lim sup J2
x→∞
y↑1
6 lim sup lim sup
c2 F ξκ (xy)
P∞
n=K+1
ϕ(n)P(η = n)
F ξκ (x)P(η = κ)
X
∞
c2
F ξ (xy)
lim sup lim sup κ
=
ϕ(n)P(η = n).
P(η = κ)
x→∞ F ξκ (x)
y↑1
n=K+1
y↑1
x→∞
(8)
Relations (3), (7), and (8) imply that
lim sup lim sup
x→∞
y↑1
F ξ(η) (xy)
61+
F ξ(η) (x)
c2
E ϕ(η)1[K+1,∞) (η) ,
P(η = κ)
for arbitrary K > κ.
The desired inequality (2) now follows from the last estimate because
E(ϕ(η)1[1,∞) (η)) is finite due to the conditions of the theorem. Theorem 4 is proved.
Proof of Theorem 5. Similarly as in the proof of Theorem 4, it suffices to show that
lim sup lim sup
x→∞
y↑1
F S(η) (xy)
F S(η) (x)
6 1.
If K ∈ N and x > 0, then
P(S(η) > x) =
K
X
+
n=1
∞
X
!
P(S(n) > x)P(η = n).
n=K+1
Therefore,
P(S(η) > xy)
=
P(S(η) > x)
PK
n=1
+
P∞
P(S(n) > xy)P(η = n)
P(S(η) > x)
n=K+1
P(S(n) > xy)P(η = n)
P(S(η) > x)
(9)
Randomly stopped maximum and maximum of sums with consistently varying distributions
75
(10)
=: I1 + I2 ,
for all x > 0 and y ∈ (0, 1).
The r.v. η is not degenerate at zero. Therefore, there exists a ∈ N such that P(η =
a) > 0. If K > a, then
PK
P(S(n) > xy)P(η = n)
I1 6 Pn=1
K
n=1
P(S(n) > x)P(η = n)
6
max
16n6K
n∈supp(η)
P(S(n) > xy)
,
P(S(n) > x)
due to the inequality
a1 + a2 + · · · + am
am
a1 a2
,
6 max
, ,...,
b1 + b2 + · · · + bm
b1 b2
bm
provided for all ai > 0, bi > 0, i ∈ {1, 2, . . . , m}, and m ∈ N.
Since C ⊂ L, using Lemma 1, we obtain
lim sup lim sup I1
y↑1
x→∞
6 lim sup lim sup
=
F S(n) (xy)
max
16n6K
n∈supp(η)
y↑1
x→∞
max
lim sup lim sup
16n6K
n∈supp(η)
y↑1
x→∞
6 max lim sup lim sup
16n6K
y↑1
x→∞
F S(n) (x)
F S(n) (xy) F Sn (xy) F Sn (x)
F Sn (xy) F Sn (x) F S(n) (x)
F Sn (xy)
.
F Sn (x)
According to Lemma 2, the d.f. FSn belongs to the class C for each fixed n.
Therefore,
lim sup lim sup I1 6 1.
y↑1
(11)
x→∞
Let us now consider the term I2 from expression (10). If x > 0, then the numerator of I2 can be estimated as follows:
∞
X
P(S(n) > xy)P(η = n)
n=K+1
=
6
=
∞
X
n=K+1
∞
X
n=K+1
∞
X
n=K+1
P max{S1 , . . . , Sn } > xy P(η = n)
(+)
P max S1 , . . . , Sn(+) > xy P(η = n)
P Sn(+) > xy P(η = n),
76
I.M. Andrulytė et al.
(+)
where Sk := ξ1+ + · · · + ξk+ and ξ + := ξ1[0,∞) (ξ).
We can apply Lemma 3 for the last sum because of condition (b) of the theorem
and the fact that Fξ1 ∈ C ⇒ Fξ+ ∈ D. Using this lemma, we get
1
∞
X
∞
X
P(S(n) > xy)P(η = n) 6 c3 F ξ1 (xy)
np+1 P(η = n),
(12)
n=K+1
n=K+1
for some positive constant c3 and for all x > 0 and y ∈ (0, 1).
On the other hand, for the denominator of I2 , we have
P(S(η) > x) =
∞
X
P(S(n) > x)P(η = n)
n=1
> P max{S1 , . . . , Sa } > x P(η = a)
> P(Sa > x)P(η = a).
According to the conditions of the theorem, the d.f. Fξk belongs to the class C for
each fixed index k ∈ N. Hence, using Lemma 4, we get
lim inf
x→∞
Pa
P(Sa > x)
F ξ (x)
P(Sa > x)
> lim inf Pa
lim inf i=1 i
x→∞
x→∞
(x)
F ξ1 (x)
F
i=1 ξi
F ξ1 (x)
Pa
F ξ (x)
F ξ1 (x)
> lim inf
= lim inf i=1 i
= 1.
x→∞ F ξ (x)
x→∞
1
F ξ1 (x)
The last two estimates imply that
P(S(η) > x) >
1
F ξ (x)P(η = a),
2 1
(13)
for sufficiently large x.
Therefore, by inequalities (12) and (13) we have
lim sup lim sup I2
y↑1
x→∞
X
∞
2c3
F ξ1 (xy)
6
np+1 P(η = n).
lim sup lim sup
P(η = a)
x→∞ F ξ1 (x)
y↑1
(14)
n=K+1
Finally, substituting estimates (11) and (14) into (10), we obtain that
lim sup lim sup
y↑1
x→∞
P(S(η) > xy)
2c3
61+
E η p+1 1[K+1,∞) (η) ,
P(S(η) > x)
P(η = a)
for an arbitrary K > a.
This inequality implies the desired relation (9) due to condition (c) of the theorem.
Theorem 5 is proved.
Randomly stopped maximum and maximum of sums with consistently varying distributions
77
Acknowledgments
We would like to thank the anonymous referees for the detailed and helpful comments
on the first version of the manuscript and especially for the proposed new way of
proving Theorem 4.
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