Precise asymptotics over a small parameter for a series of large deviation probabilities
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| Цитувати: | Precise asymptotics over a small parameter for a series of large deviation probabilities / V.V. Buldygin, O.I. Klesov, J.G. Steinebach // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 44-56. — Бібліогр.: 27 назв.— англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859739623885897728 |
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| author | Buldygin, V.V. Klesov, O.I. Steinebach, J.G. |
| author_facet | Buldygin, V.V. Klesov, O.I. Steinebach, J.G. |
| citation_txt | Precise asymptotics over a small parameter for a series of large deviation probabilities / V.V. Buldygin, O.I. Klesov, J.G. Steinebach // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 44-56. — Бібліогр.: 27 назв.— англ. |
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Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.44-56
V. V. BULDYGIN, O. I. KLESOV, AND J. G. STEINEBACH
PRECISE ASYMPTOTICS OVER A SMALL
PARAMETER FOR A SERIES OF LARGE
DEVIATION PROBABILITIES
We obtain the asymptotics of the series
∞∑
k=1
wkP (|Sk| ≥ εϕk)
as ε ↓ 0, where Sk are partial sums of independent and identically
distributed random variables in the domain of attraction of a non-
degenerate stable law, w and ϕ are regularly varying functions (in
Karamata’s sense).
1. Introduction
Let X, {Xn, n ≥ 1} be independent, identically distributed (i.i.d.) ran-
dom variables with distribution function F and let {Sn, n ≥ 1} denote the
sequence of their partial sums.
Let w and ϕ be two given positive functions and put wk = w(k) and ϕk =
ϕ(k). We study the convergence and asymptotics over a small parameter of
the series
Q(ε) =
∞∑
k=1
wkP(|Sk| ≥ εϕk), ε > 0. (1)
We deal with the case of regularly varying functions w and ϕ in this paper.
The definitions of and necessary results for regularly varying functions can
be found in [24] or [4]. We use the notation f ∈ RV (a) to say that f is a
measurable regularly varying function of order a. We treat the case of large
deviation probabilities in (1) (in the sense that Sk/ϕk → 0 in probability as
Invited lecture.
Partially supported by DFG grants UKR 113/41/0 and UKR 113/68/0
2000 Mathematics Subject Classifications. 60F05, 60E07
Key words and phrases. Spitzer series, large deviations, stable laws, regularly varying
functions
44
PRECISE ASYMPTOTICS OVER A SMALL PARAMETER 45
k → ∞), however there is a number of other papers devoted to moderate
and small deviation probabilities.
Series (1) is useful for various applications of limit theorems in prob-
ability theory. We mention only one of them related to the topic of the
paper.
Hsu and Robbins [17] introduced the so-called complete convergence
of sequences of random variables. According to the definition in [17], the
sequence Sn/n converges completely to zero if
∞∑
k=1
P
(∣∣∣∣Sk
k
∣∣∣∣ ≥ ε
)
< ∞ for all ε > 0.
Obviously the Hsu–Robbins series is nothing else but Q(ε) with wk = 1 and
ϕk = k, k ≥ 1. It was proved in [17] that Q(ε) < ∞ for all ε > 0 if
EX = 0, EX2 < ∞. (2)
Erdös [8] was able to show that the converse is also true for the Hsu–Robbins
series. After the papers of Hsu and Robbins [17] and Erdös [8], many results
have been obtained concerning the series (1) for various functions w and ϕ.
Below is a list of cases studied in earlier papers:
c1) wk = 1, ϕk = k (Hsu and Robbins [17] and Erdös [8]);
c2) wk = 1/k, ϕk = k (Spitzer [26]);
c3) wk = kr, ϕk = k1/p (Katz [20] and Baum and Katz [3]);
c4) w ∈ RV (r), w(t)/tr is increasing, ϕk = k1/p (Heyde and Rohatgi [16]).
The above list is incomplete; it contains only the cases we touch in this
paper.
The Spitzer series c2) is remarkable, since it corresponds to the “bound-
ary” case in the range of sequences wk = kr: namely, the series
∑
wk con-
verges if r < −1, while it diverges otherwise. The necessary and sufficient
condition for the convergence of the Spitzer series c2) is
EX = 0 (3)
(see [26]).
C. C. Heyde initiated the investigations of the asymptotics of Q(ε) as
ε ↓ 0. He proved in [14] that if (2) holds, then
lim
ε↓0
ε2Q(ε) = σ2 (4)
in the case of the Hsu–Robbins series c1) where σ2 = var [X].
46 V. V. BULDYGIN, O. I. KLESOV, AND J. G. STEINEBACH
The asymptotic behavior as ε ↓ 0 of the Spitzer series c2) is studied in
Chow and Lai [6]. In particular, they proved for the case of c2) that
lim
ε↓0
Q(ε)
ln(1/ε)
= 2 (5)
if (2) is satisfied. Recall that condition (3) is sufficient in this case for the
convergence of Q(ε) for all ε > 0, so that condition (2) seems to be too
strong for the asymptotics (5).
Spatăru [25] used another assumption to study the asymptotics of the
Spitzer series c2), namely
F belongs to the domain of attraction of a nondegenerate
α-stable law with 1 < α ≤ 2.
(6)
Spatăru [25] proved under conditions (3) and (6) that
lim
ε↓0
Q(ε)
ln(1/ε)
=
α
α − 1
. (7)
If (2) holds, then the distribution function F belongs to the domain of
attraction of the Gaussian law, that is α = 2 and (5) and (7) coincide in
this case. Note further that (6) implies E|X|η < ∞ for all 1 < η < α and
this is weaker than (2) but still is stronger than (3), so that condition (6)
seems also to be too strong for the asymptotics.
In fact, the assumption that the limit law is nondegenerate is missed
in [25]. This, however, is an important restriction, since if the limit law
is concentrated at the origin, then we consider the random variable X = 0
which obviously is attracted to that stable law. In this case, P(|Sk| ≥ εk) =
0 for all ε > 0 and asymptotics (7) fails.
The Baum–Katz series c3) (including the case of r = −1) is studied by
Gut and Spatăru [11] under condition (6). Gut and Steinebach [12] showed
that one can take 0 < α < 1 in assumption (6), too.
More general results concerning the asymptotics of series (1) under con-
dition (6) are obtained by Rozovskĭı [22] for α �= 2.
A weaker assumption than (6) is used by Scheffler [23] for the Spitzer
series c2) and for Baum–Katz series c3), namely
F belongs to the domain of semistable attraction
of a semistable law with index α > 0.
(8)
Further asymptotics of Q(ε) as ε ↓ 0 have been obtained by Chen [5] for
the Baum–Katz series c3) with r ≥ 0 (this case includes Heyde’s result (4)).
The moment condition used in Chen [5] to obtain the asymptotics of Q(ε)
is the same as that for its convergence.
PRECISE ASYMPTOTICS OVER A SMALL PARAMETER 47
The proof in Chow and Lai [6] has its roots in the area of the central
limit theorem and uses the truncation, symmetrization, Berry–Esseen esti-
mate, and desymmetrization. The proof in Chen [5] follows the lines of the
proof of Heyde [14], which consists of first obtaining the result for normally
distributed summands and then approximating the general case with the
Gaussian one. Spatăru [25] adapted the Heyde [14] method for the case of
attraction to a stable law. Scheffler [23] used special properties of distribu-
tions partially attracted to semi-stable laws. Rozovskĭı [22] applied a large
deviation principle in the case of attraction to stable laws (this principle is
due to Heyde [15] but Rozovskĭı [22] does not cite that paper; instead he
refers to another his own paper).
We should however mention that the proof in Spatăru [25] is not com-
plete.
There is a remarkable difference between the series c2) and c3) with
r > −1, namely the Heyde [14] result (4) says that the asymptotics of
the Hsu–Robbins series c1) (as a “representative” of series c3)) is “almost”
independent of the distribution function F : it depends on F only via its
variance σ2. Roughly speaking, the result is independent of the distribution
up to a scale parameter. This is not the case for the Spitzer series c2) whose
asymptotic behavior depends on F via the index α of the stable law to which
F is attracted (we discuss the case of a simpler condition (6)). Therefore
the limit result is irrelevant of other properties of the distribution function
F , since the index α does not completely determine the distribution.
A “complete” description of a distribution function F attracted to an
α-stable law can be given in terms of the normalizing sequence in the corre-
sponding attraction to a stable law. Namely, it follows from (6) that there
are two sequences of real numbers {an, n ≥ 1} and {cn, n ≥ 1} such that the
distributions of Sn/cn − an converge weakly to a stable law of index α > 1.
Assumption (3) reduces the consideration to the case of an = 0, n ≥ 1. Nev-
ertheless, the sequence {cn, n ≥ 1} is still present in the weak convergence
and one expects that its properties should somehow be reflected in (7), too.
It is known that cn = nαh(n) for some slowly varying function h. While
α is involved in (7), h is not. In other words, there is no difference in (7)
between the case of normal attraction (h(x) ≡ const) and the general case
of non-normal attraction (h(x) �≡ const).
The latter observation raises two questions: fix w ∈ RV (−1);
• is there any difference in the asymptotics of Q(ε) between the cases
of normal attraction (where cn = cn1/α, c > 0) and general attraction
(where cn = nαh(n), h a slowly varying function)?
• If this phenomenon is not a general law, why does it occur in the case
of wk = 1/k, k ≥ 1?
48 V. V. BULDYGIN, O. I. KLESOV, AND J. G. STEINEBACH
We answer both questions in Remark 6 below by applying our Theo-
rem 1. It turns out that the phenomenon mentioned above appears only in
the case of the Spitzer series.
Our main aim in this paper is to obtain the asymptotic of the series (1)
where w ∈ RV (r), r ≥ −1, and ϕ ∈ RV (1/p), 0 < p < α (note that this
case is even wider than the case of series c4)).
We keep condition (6) and show that it applies not only for power func-
tions but also for the general case of regularly varying functions.
The proof of the main result consists in two standard steps: first we get
the asymptotic behavior for the limiting stable law, and then we approxi-
mate the general case by this particular one. Nevertheless, we develop new
methods for both steps. Earlier papers make use of the Euler–MacLaurin
summation formula for the first step, while we apply an approach based on
some Abelian theorems for slowly varying functions due to Aljanĉić et al. [1].
Our approach works for stable laws but for other distribution functions, too.
The proof for the second step is based on some new large deviation
results for distributions in the domain of attraction to a stable law. The
cases α �= 2 and α = 2 are different and require different tools in the
proof. The case of α = 1 is also obtained: it differs from the other cases by
centering constants explicitely involved into the result.
The proof of the large deviation result is an extension of the method used
in Heyde [15] but our result holds for the domain x ≥ 0 instead of x ≥ xn
as in Heyde [15] where a sequence {xn} is such that xn → ∞, n → ∞. The
price we pay for this generalization is that we obtain an upper bound and
do not get the precise asymptotics as in Heyde [15].
Rozovskĭı [22] also used an asymptotic result similar to the Heyde [15]
large deviation principle and dealt with series like (1) but he restricted
himself to the case of α �= 2.
The paper is organized as follows. In Section 2, our main result, The-
orem 1, is stated together with some corollaries exhibiting the possible
range of asymptotics in the simple cases of either w(x) = xr lnq(n), or
ϕ(x) = x1/p lnq(x), or b(x) = x1/α lnq(x), or a mixture of these cases for
various r and q. We also give the negative solution of the conjecture of [11]
that the asymptotics for the normal and non-normal attractions coincide if
r > −1.
The proof of Theorem 1 will be given elsewhere. Nevertheless we prove
all other results, since they are of their own interest.
2. Notation, main result, and corollaries
Throughout the paper we assume that X, {Xn, n ≥ 1} are independent
and identically distributed random variables and let Sn be the partial sums
of Xk’s.
PRECISE ASYMPTOTICS OVER A SMALL PARAMETER 49
We write X ∈ DA (α, {cn}, {an}) if there exists a nondegenerate α-
stable random variable Zα such that Sn/cn − an ⇒ Zα, i.e. if X belongs
to the domain of attraction of the nondegenerate random variable Zα with
normalizing sequence {cn} and centering sequence {an}. We write X ∈
DA (α, {cn}, {0n}) if an = 0, n ≥ 1.
We consider weight functions for the series (1) such that
w ∈ RV (r) , wk = w(k). (9)
The normalizing function for the large deviation probabilities in the series
(1) is such that
ϕ ∈ RV (1/p) , ϕk = ϕ(k). (10)
If X ∈ DA (α, {cn}, {an}) and p < α, then there exists a mormalizing
sequence {bn} for which X ∈ DA (α, {bn}, {an}) and
b(x) = x1/αh(x), bk = b(k), (11)
where h is a slowly varying function such that ϕ(x)/b(x) is continuous and
increasing. In what follows we assume that {bn, n ≥ 1} is chosen according
to (11), so that all the inverse functions considered below exist.
Put
ψ(x) = ϕ(x)/b(x), ψk = ψ(k). (12)
Note that the inverse ψ−1 exists and ψ−1 ∈ RV (αp/(α − p)) (see, e.g.,
Seneta [24], Section 1.5, Proposition 5◦). With W (t) =
∫ t
1
w(s) ds, we set
U(x) = W
(
ψ−1(x)
)
. (13)
It is easy to see that U ∈ RV ((r + 1)αp/(α − p)).
In what follows we use at least one of the following conditions:
X ∈ DA (α, {bn}, {an}) (14)
or
X ∈ DA (α, {bn}, {0n}) . (15)
Recall that each of them means, in particular, that the limit law is nonde-
generate.
Now we are ready to state our main result.
Theorem 1. Let X, {Xn, n ≥ 1} be independent identically distributed
random variables. Assume that condition (14) holds. In addition, we as-
sume (15) if α = 1 or (3) if α > 1. Denote by Zα an α-stable random
variable to which X is attracted. Let conditions (9) and (10) hold with
r ≥ −1 and 0 < p < α. Let the functions U and Q be as defined in (13)
and (1), respectively. If the series
∑
wk diverges and α > p(r + 2), then
lim
ε↓0
1
U (1/ε)
∞∑
k=1
wkP (|Sk| ≥ εϕk) = E|Zα|(r+1)αp/(α−p). (16)
50 V. V. BULDYGIN, O. I. KLESOV, AND J. G. STEINEBACH
Remark 1. As seen from conditions of Theorem 1, the cases α = 1
and α �= 1 are different, since we assume (15) if α = 1 instead of (14)
if α �= 1. The reason is that (15) holds automatically if α < 1; it also
holds for α > 1 under condition (3). We prefer to deal with a nicer se-
ries
∑
wkP(|Sk| ≥ εϕk), thus we assume (15) if α = 1. By the way, if
EX = μ �= 0 for α > 1, then the asymptotics is evaluated for the series∑
wkP(|Sk − kμ| ≥ εϕk).
Remark 2. Note that (r + 1)αp/(α − p) < α if α > p(r + 2), so that the
right hand side of (16) is finite. If α �= 2 and α ≤ p(r + 2), then the limit
value in (16) becomes infinite, so that the function U does not appropriately
describes the asymptotics in this case.
Remark 3. If α < p(r + 2) and α �= 2, then not only the right hand side
of (16) is infinite but the series on the left hand side of (16) diverges for
all ε > 0, so that there is no nice asymptotics in this case. The divergence
of the series on the left hand side of (16) can easily be shown for α <
min{2, p(r + 2)}. Indeed, fix ε > 0 and put xk = ϕk/bk. Since p < α,
xk → ∞ as k → ∞ and we apply the Heyde [14] result
P(|Sk| ≥ εϕk) ∼ kP(|X1| ≥ εϕk), k → ∞,
to prove that the convergence of
∑
kwkP(|Sk| ≥ εϕk) is equivalent to the
convergence of
∑
kwkP(|X1| ≥ εϕk). The latter series converges if and only
if ∞∑
k=1
kwkg(εϕk)
ϕα
k
< ∞.
This condition fails if α < p(r + 2), since g is a slowly varying function. In
the boundary case of α = p(r + 2), both convergence and divergence of the
series is possible (see [22]).
Remark 4. The convergence of series (1) for α > p(r + 2) and all ε > 0
can be proved in a similar manner. To avoid the repetition and to consider
the case of α = 2 together with α �= 2 we use another method based on the
result of Baum and Katz [3]. Indeed, choose r′ > r and p′ > p such that
α > p′(r′ +2). Then wk ≤ const kr′ and ϕk ≥ const k1/p′ for all k ≥ 1. Thus
the convergence of series (1) for all ε > 0 follows from the convergence of
the series ∞∑
k=1
kr′P
(
|Sk| ≥ εk1/p′
)
PRECISE ASYMPTOTICS OVER A SMALL PARAMETER 51
for all ε > 0. The latter series converges for all ε > 0 if and only if
E|X|p′(r′+2) < ∞ (see [3]). This moment is finite. This reasoning makes it
clear that we assumed more in Theorem 1 than what is needed for just the
convergence of series (1).
Remark 5. Note also that our restriction α > p(r + 2) coincides with
that used in Theorem 1 in [11] (the difference between the restrictions is a
matter of the different notation here and in [11]).
Remark 6. One can answer both questions posed in the Introduction by
using Theorem 1. To be more specific, let p = 1. Then, Theorem 1 in the
case of wk = 1/k, k ≥ 1, implies that Q(ε) is equivalent to ln (ψ−1(1/ε))
as ε ↓ 0 where ψ−1 is the inverse to the function defined by (12). The
function h in (11) determines the non-normal attraction (if h(t) ≡ c, then
the attraction is normal).
It is a general fact about regularly varying functions that ln (f(t)) ∼
β ln(t) if f ∈ RV (β) (see, e.g,̇ Proposition 2◦, p. 18, in Seneta [24]). Since
the logarithm “kills” any slowly varying function in the asymptotic sense
and ψ−1 ∈ RV (α/(α − 1)), the function h disappears in the asymptotics
(7) and only the part corresponding to the normal attraction is left there.
This answers the second question in the Introduction.
To answer the first question we again consider the simplest case of ϕ(t) =
t. Put β = α/(α − 1) and consider the function w such that w(t) = 1 for
0 ≤ t < 2 and
w(t) =
eln1/2(t)
t ln1/2(t)
for t ≥ 2.
Then W (t) ∼ 2eln1/2(t) as t → ∞, where W (t) =
∫ t
1
w(s) ds. Note that
w ∈ RV (−1) and thus W ∈ RV (0). In the case of normal attraction,
b(t) = ct1/α for some c > 0, whence ψ−1(t) = (ct)β and Q(ε) is equivalent
to U(1/ε) ∼ 2eβ1/2 ln1/2(1/ε) as ε ↓ 0 by Theorem 1.
Now consider a special case of non-normal attraction, where ψ−1(t) =
tβelna(t), 0 < a < 1. (Formally, the case of normal attraction corresponds
to the case of a = 0.) Then ψ−1 ∈ RV (β) and the normalization b is easy
to evaluate from ψ−1. Theorem 1 implies that the asymptotics of Q(ε) as
ε ↓ 0 is given by 2e(β ln(1/ε)+lna(1/ε))1/2
. To distinguish between the functions
U for the normal attraction and non-normal attraction we use the notation
U0 and Ua, respectively. Then it is easy to see that
Ua(x)
U0(x)
∼ exp
{
lna(x)√
β ln(x) + lna(x) +
√
β ln(x)
}
and thus U0 = o(Ua) if a > 1/2, while Ua ∼ U0 if 0 < a < 1/2 and
Ua ∼ e1/2
√
βU0 if a = 1/2. The cas eof a > 1/2 means that the asymptotics
52 V. V. BULDYGIN, O. I. KLESOV, AND J. G. STEINEBACH
for the normal and non-normal attractions do not coincide in general. This
answers the first question above.
One can notice that Ua “dominates” U0 in this example. This could
produce an impression that this is a general fact. The example of ψ−1(t) =
tβe− lna(t), 1
2
< a < 1, shows that Ua = o(U0) and the impression is wrong.
Now we show how Theorem 1 can be applied in particular cases. We
start with the case of r > −1.
Corollary 1. Let X, {Xn, n ≥ 1} be independent identically distributed
random variables. Assume that condition (14) holds for bn = cn1/α, c > 0
(thus we deal with the normal attraction). In addition, we assume (15)
if α = 1 or (3) if α > 1. Denote by Zα an α-stable random variable to
which X is attracted. Let r > −1, 0 < p < α, and α > p(r + 2). Set
ν = (r + 1)αp/(α − p). Then
∞∑
k=1
krP
(|Sk| ≥ εk1/p
) ∼
(
1
ε
)ν
· cν
r + 1
·E|Zα|ν .
Corollary 1 is proved in [11], Theorem 1. Note however that the con-
stant c(r+1)αp/(α−p) is missed in [11]. The proof of Corollary 1 is sim-
ple: ψ(t) = t(α−p)/αp/c in this case, whence ψ−1(t) = (ct)αp/(α−p). Since
W (t) ∼ tr+1/(r + 1), we derive Corollary 1 from Theorem 1.
Further we give a negative solution of the conjecture in [11] that the
asymptotics obtained in Corollary 1 for the normal attraction holds also
in the case of the non-normal attraction. In fact, any slowly varying func-
tion may appear in the asymptotic of the underlying series. Below is the
corresponding result for the function lnq(t).
Corollary 2. Let X, {Xn, n ≥ 1} be independent identically distributed
random variables. Assume that condition (14) holds for b(t) = t1/α lnq(t).
In addition, we assume (15) if α = 1 or (3) if α > 1. Denote by Zα an
α-stable random variable to which X is attracted. Let r > −1, 0 < p < α,
and α > p(r + 2). Set ν = (r + 1)αp/(α − p). Then
∞∑
k=1
krP
(|Sk| ≥ εk1/p
) ∼
(
lnq(1/ε)
ε
)ν
·
(
αp
α − p
)νq
1
r + 1
· E|Zα|ν.
First we note that the normalization b(t) = t1/α lnq(t) appears in the
attraction to an α-stable law for the following distribution
P(|X| ≥ x) ∼ const
(
lnq(x)
x
)α
.
PRECISE ASYMPTOTICS OVER A SMALL PARAMETER 53
Corollary 2 easily follows from Theorem 1. Indeed, ψ(t) = t
α−p
αp ln−q(t)
in this case, whence ψ−1(t) ∼
(
αp
α−p
)qαp/(α−p)
tαp/(α−p) lnqαp/(α−p)(t). Using
the asymptotics of W found in the proof of Corollary 1 we complete the
proof of Corollary 2.
Now we turn to the case of r = −1.
Corollary 3. Let X, {Xn, n ≥ 1} be a sequence of independent identically
distributed random variables. Assume that condition (14) holds. In addition,
we assume (15) if α = 1 or (3) if α > 1. If 0 < p < α, then
∞∑
k=1
1
k
P(|Sk| ≥ εk1/p) ∼ ln(1/ε) · αp
α − p
.
Corollary 3 is proved in [11] (also see [23]). Corollary 3 easily follows
from Theorem 1. Indeed, W (t) ∼ ln(t) and ψ−1 ∈ RV (αp/(α − p)) in
this case. Therefore, U (x) ∼ αp
α−p
ln(x) (see, Proposition 2◦ in Seneta [24],
p. 18).
Below we give two more corollaries corresponding to the case of r = −1.
Corollary 4. Let X, {Xn, n ≥ 1} be a sequence of independent identically
distributed random variables. Assume that condition (14) holds. In addition,
we assume (15) if α = 1 or (3) if α > 1. If 0 < p < α, then, for q > −1,
∞∑
k=2
lnq(k)
k
P
(|Sk| ≥ εk1/p
) ∼ lnq+1(1/ε) · 1
q + 1
(
αp
α − p
)q+1
.
Indeed, W (t) ∼ 1
q+1
lnq+1(t) and ψ ∈ RV ((α − p)/αp) in this case. Thus
ψ−1 ∈ RV (αp/(α − p)), whence Corollary 4 follows.
For q = −1, the right-hand side of the preceding equality becomes mean-
ingless. It turns out that the asymptotic behavior of the series changes in
this case.
Corollary 5. Let X, {Xn, n ≥ 1} be a sequence of independent identically
distributed random variables. Assume that condition (14) holds. In addition,
we assume (15) if α = 1 or (3) if α > 1. If 0 < p < α, then
∞∑
k=2
1
k ln(k)
P
(|Sk| ≥ εk1/p
) ∼ ln ln(1/ε).
Indeed, W (t) ∼ ln ln(t) in this case, whence Corollary 5 follows in the
same way as above.
54 V. V. BULDYGIN, O. I. KLESOV, AND J. G. STEINEBACH
The case of q < −1 is special, since
∑
lnq(k)/k converges and thus
lim
ε↓0
∞∑
k=2
lnq(k)
k
P(|Sk| ≥ εk1/p) =
∞∑
k=2
lnq(k)
k
P(Sk �= 0)
by the dominated convergence theorem.
The above corollaries dealt with the case of ϕ(x) = x1/p. The results
below show how does the change of the function ϕ influence the asymptotics
of Q.
Corollary 6. Let X, {Xn, n ≥ 1} be a sequence of independent identically
distributed random variables. Assume that condition (14) holds with bn =
cn1/α for some c > 0 (thus we assume the normal attraction). In addition,
we assume (15) if α = 1 or (3) if α > 1. Put ν = (r + 1)αp/(α − p). If
r > −1, p > 0, and α > p(r + 2), then
∞∑
k=1
krP
(|Sk| ≥ εk1/p lnq(k)
) ∼
(
1
ε lnq(1/ε)
)ν
·
[
c
(
α − p
αp
)q]ν
1
r + 1
·E|Zα|ν .
Indeed, W (t) ∼ 1
r+1
tr+1 and ψ(t) = 1
c
x(α−p)/αp lnq(t) in this case. Thus
ψ−1(t) ∼ c′tαp/(α−p) ln−qαp/(α−p)(t) for c′ =
[
c
(
α−p
αp
)q]αp/(α−p)
, whence Corol-
lary 6 follows.
Corollary 7. Let X, {Xn, n ≥ 1} be a sequence of independent identically
distributed random variables. Assume that condition (14) holds. In addition,
we assume (15) if α = 1 or (3) if α > 1. If 0 < p < α, then
∞∑
k=2
1
k
P
(|Sk| ≥ εk1/p lnq(k)
) ∼ ln(1/ε) · αp
α − p
.
Indeed, W (t) ∼ ln(t) in this case and Corollary 7 follows in the same
way as above.
A different asymptotic behavior of the series (1) appears if p = α. One
of possible results is obtained by Scheffler [23] in this case. We will consider
the case of p = α in detail elsewhere.
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Department of Mathematical Analysis and Probability Theory, Na-
tional Technical University of Ukraine (KPI), pr. Peremogy, 37,
Kyiv 03056, Ukraine.
E-mail address: valbuld@comsys.ntu-kpi.kiev.ua
Department of Mathematical Analysis and Probability Theory, Na-
tional Technical University of Ukraine (KPI), pr. Peremogy, 37,
Kyiv 03056, Ukraine
E-mail address: tbimc@ln.ua
Universität zu Köln, Mathematisches Institut, Weyertal 86–90, D–
50931 Köln, Germany
E-mail address: jost@math.uni-koeln.de
|
| id | nasplib_isofts_kiev_ua-123456789-4477 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-01T16:06:46Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
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| spelling | Buldygin, V.V. Klesov, O.I. Steinebach, J.G. 2009-11-19T10:10:34Z 2009-11-19T10:10:34Z 2007 Precise asymptotics over a small parameter for a series of large deviation probabilities / V.V. Buldygin, O.I. Klesov, J.G. Steinebach // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 44-56. — Бібліогр.: 27 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4477 Partially supported by DFG grants UKR 113/41/0 and UKR 113/68/0 en Інститут математики НАН України Precise asymptotics over a small parameter for a series of large deviation probabilities Article published earlier |
| spellingShingle | Precise asymptotics over a small parameter for a series of large deviation probabilities Buldygin, V.V. Klesov, O.I. Steinebach, J.G. |
| title | Precise asymptotics over a small parameter for a series of large deviation probabilities |
| title_full | Precise asymptotics over a small parameter for a series of large deviation probabilities |
| title_fullStr | Precise asymptotics over a small parameter for a series of large deviation probabilities |
| title_full_unstemmed | Precise asymptotics over a small parameter for a series of large deviation probabilities |
| title_short | Precise asymptotics over a small parameter for a series of large deviation probabilities |
| title_sort | precise asymptotics over a small parameter for a series of large deviation probabilities |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4477 |
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