Asymptotic formulas for probabilities of large deviations of ladder heights
Asymptotic formulas for large-deviation probabilities of a ladder height in a random walk generated by a sequence of sums of i.i.d. random variables are deduced. Two cases are considered: a) the distribution F(x) of summands is normal with a zero mean. b) F(x) belongs to the domain of the normal...
Gespeichert in:
| Datum: | 2008 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2008
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/4541 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Asymptotic formulas for probabilities of large deviations of ladder heights / S.V. Nagaev // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 100–116. — Бібліогр.: 17 назв.— англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-4541 |
|---|---|
| record_format |
dspace |
| spelling |
Nagaev, S.V. 2009-11-25T11:07:02Z 2009-11-25T11:07:02Z 2008 Asymptotic formulas for probabilities of large deviations of ladder heights / S.V. Nagaev // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 100–116. — Бібліогр.: 17 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4541 519.21 Asymptotic formulas for large-deviation probabilities of a ladder height in a random walk generated by a sequence of sums of i.i.d. random variables are deduced. Two cases are considered: a) the distribution F(x) of summands is normal with a zero mean. b) F(x) belongs to the domain of the normal attraction of a stable law with the exponent 0 < α < 1. The method of Laplace transforms is applied in proofs. This article was partially supported by the Russian Foundation for Basic Research (grant 06-01-00069) en Інститут математики НАН України Asymptotic formulas for probabilities of large deviations of ladder heights Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Asymptotic formulas for probabilities of large deviations of ladder heights |
| spellingShingle |
Asymptotic formulas for probabilities of large deviations of ladder heights Nagaev, S.V. |
| title_short |
Asymptotic formulas for probabilities of large deviations of ladder heights |
| title_full |
Asymptotic formulas for probabilities of large deviations of ladder heights |
| title_fullStr |
Asymptotic formulas for probabilities of large deviations of ladder heights |
| title_full_unstemmed |
Asymptotic formulas for probabilities of large deviations of ladder heights |
| title_sort |
asymptotic formulas for probabilities of large deviations of ladder heights |
| author |
Nagaev, S.V. |
| author_facet |
Nagaev, S.V. |
| publishDate |
2008 |
| language |
English |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
Asymptotic formulas for large-deviation probabilities of a ladder height in a random
walk generated by a sequence of sums of i.i.d. random variables are deduced.
Two cases are considered:
a) the distribution F(x) of summands is normal with a zero mean.
b) F(x) belongs to the domain of the normal attraction of a stable law with
the exponent 0 < α < 1.
The method of Laplace transforms is applied in proofs.
|
| issn |
0321-3900 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/4541 |
| citation_txt |
Asymptotic formulas for probabilities of large deviations of ladder heights / S.V. Nagaev // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 100–116. — Бібліогр.: 17 назв.— англ. |
| work_keys_str_mv |
AT nagaevsv asymptoticformulasforprobabilitiesoflargedeviationsofladderheights |
| first_indexed |
2025-11-24T03:25:34Z |
| last_indexed |
2025-11-24T03:25:34Z |
| _version_ |
1850839477441789952 |
| fulltext |
Theory of Stochastic Processes
Vol. 14 (30), no. 1, 2008, pp. 100–116
UDC 519.21
SERGEY V. NAGAEV
ASYMPTOTIC FORMULAS FOR PROBABILITIES
OF LARGE DEVIATIONS OF LADDER HEIGHTS
Asymptotic formulas for large-deviation probabilities of a ladder height in a random
walk generated by a sequence of sums of i.i.d. random variables are deduced.
Two cases are considered:
a) the distribution F (x) of summands is normal with a zero mean.
b) F (x) belongs to the domain of the normal attraction of a stable law with
the exponent 0 < α < 1.
The method of Laplace transforms is applied in proofs.
1. Introduction
Let X,X1, X2, . . . , Xn, . . . be i.i.d. variables with the distribution function F (x), not
degenerate at zero. Put
Sn =
n∑
i=1
Xi, Fn(D) = P(Sn ∈ D).
Introduce the notation
N+ = min
{
n : Sn > 0
}
, N− = min
{
n : Sn ≤ 0
}
.
Let Z+ := SN+ , Z− := SN− be respectively ascending and descending ladder heights,
and
F+(x) = P(Z+ < x), F−(x) = P(Z− > x).
Denote, by H+(x), the renewal function corresponding to the distribution F+ of the
ascending ladder height,
H+(x) =
∞∑
n=0
F+
n (x),
where F+
n , n ≥ 1, is the n-th convolution of F+, F0 is the degenerate distribution
concentrated at zero. Similarly, the renewal function H−(x) is defined by F−.
Notice that
H+(x) = F0(x) +
∞∑
n=1
P
(
min
0<k≤n−1
Sk > 0, 0 < Sn < x
)
,
H−(x) = F0(x) +
∞∑
n=1
P
(
max
0≤k≤n−1
Sk ≤ 0, x < Sn ≤ 0
)
(see [1], Ch.12, § 2).
2000 AMS Mathematics Subject Classification. Primary 60F10.
Key words and phrases. Characteristic function, harmonic renewal measure, Karamata’s criterion,
ladder height, Laplace transform, slowly varying function, Spitzer series, Tauberian theorem.
This article was partially supported by the Russian Foundation for Basic Research (grant 06-01-00069)
100
PROBABILITIES OF LARGE DEVIATIONS OF LADDER HEIGHTS 101
Hence,
P(Z+ > x) = −
∫ 0+
−∞
P(X > x− y)dH−(y) =
∫ ∞
x
H−(x− y)dF (y), (1)
if x > 0, and
P(Z− > x) =
∫ ∞
0+
P(X < x− y)dH+(y) =
∫ x
−∞
H+(x− y)dF (y), (2)
if x < 0.
The measure
ν(D) =
∞∑
n=1
Fn(D)
n
(3)
is called the harmonic renewal measure.
Since
Fn(x+ l)− Fn(x) < c(F )
l + 1√
n
(4)
(see, e.g., [2]), the measure ν is σ-finite. Harmonic renewal measures were studied in
[3–7].
For every x > 0, put G+(x) = ν((0, x)). Evidently,
G+(x) =
∞∑
n=1
n−1[Fn(x) − Fn(0+)] <∞.
Similarly, define G−(x) = ν((x, 0]) for x ≤ 0. Harmonic renewal measures are of interest
for us, first of all, because∫ ∞
0
e−sxdH+(x) = exp
{
−
∫ ∞
0+
e−sxdG+(x)
}
(5)
and ∫ 0
−∞
esxdH−(x) = − exp
{
−
∫ 0+
−∞
esxdG−(x)
}
, (6)
which provides the possibility of studying the asymptotic behaviour of H±(±x) as x→
∞.
Proposition. Let
a := EX = 0, 0 < σ2 := EX2 <∞. (7)
Then
lim
s↓0
(∫ ∞
0−
e−sxdG+(x) + ln s
)
= Q− 1
2
ln
σ2
2
, (8)
where
Q =
∞∑
n=1
n−1
[
P(Sn ≥ 0)− 1
2
]
is the Spitzer series, and
lim
s↓0
(
−
∫ 0+
−∞
esxdG−(x) + ln s
)
= −Q− 1
2
ln
σ2
2
. (9)
Hence, by using the refinement of Karamata’s Tauberian theorem given in [8], we
immediately obtain
102 SERGEY V. NAGAEV
Corollary 1. If conditions (7) hold, then
lim
x→±∞
(
G±(x) − ln |x|
)
= C0 ±Q− 1
2
ln
σ2
2
, (10)
where C0 is the Euler constant.
This result is obtained in [7] by using direct probabilistic arguments. Laplace trans-
forms are used in [3,4], however, in the case where the distribution F is concentrated on
a semiaxis or stable. In paper [5], the representation for ν([−x, x]) is obtained under the
condition that EX = 0, E|X |3 < ∞, and some convolution of F (x) has an absolutely
continuous component. In that representation, the Spitzer series Q is absent, which is
quite explainable since, by (8) and (9),∫ ∞
−∞
e−h|x|ν(dx) = −
(
ln s+ ln
σ2
2
)
.
Instead of Laplace transforms, the generalized Fourier transforms are used in [5].
Combining (5) and (8) and then (5) and (9), we obtain
Corollary 2. If conditions (7) are fulfilled, then
lim
s↓0
s
∫ ∞
0−
e−sxdH+(x) =
√
2
σ
eQ (11)
and
lim
s↓0
s
∫ 0+
−∞
e−sxdH−(x) = −
√
2
σ
e−Q. (12)
The Karamata’s Tauberian theorem makes it possible to obtain the asymptotics of
H+(x) for x→∞, namely,
Corollary 3. If conditions (7) hold, then
lim
x→±∞ |x|
−1H±(x) =
√
2
σ
e±Q. (13)
It is known that, under conditions (7) and (8), EZ+ < ∞, EZ− < −∞ (see [9]).
Therefore, by the renewal theorem,
lim
|x|→∞
|x|−1H±(x) =
1
|m±| , (14)
where m± = EZ±. Comparing (13) and (14), we conclude that
m± = ± σ√
2
e−Q. (15)
We say that the distribution F has a long right tail if, for any l > 0,
lim
x→∞
F (x+ l)− F (x)
F (x)
= 0. (16)
Respectively F has a long left tail if
lim
x→−∞
F (x+ l)− F (x)
F (x)
= 0. (17)
PROBABILITIES OF LARGE DEVIATIONS OF LADDER HEIGHTS 103
Theorem 1. If conditions (7) and (17) hold, then, for x→ −∞,
P(Z− < x) ∼ ω−
∫ x
−∞
F (y)dy, (18)
where ω− =
√
2
σ eQ.
If conditions (7) and (16) hold, then, for x→∞,
P(Z+ > x) ∼ ω+
∫ ∞
x
(1− F (y))dy, (19)
where ω+ =
√
2
σ e−Q.
Here and below, a(x) ∼ b(x) means that lim
x→∞ a(x)/b(x) = 1. We use c, c(·), c(·, ·)
to denote constants which may be different in different contexts.
An asymptotics of large deviation probabilities is studied in [10–13]. The formula
P(Z+ > x) ∼ 1
m−
∫ ∞
x
(1 − F (y))dy (20)
is obtained, in particular, in [13], and is valid if
A :=
∞∑
n=1
1
n
P(Sn ≤ x) =∞, E|Z−| <∞.
As we have already noticed above, E|Z−| <∞ under conditions (7) and (8). In addition,
A = ∞ in this case. On the other hand, as it is shown above (see (15)), m− = σ√
2
e−Q.
The additional information which is contained in (19) as compared with (20) consists
namely in this fact. Notice also that (19) is deduced by the quite different method by
comparison with (20).
Theorem 2. Let, for x→∞,
F (−x) ∼ q
xα
, 1− F (x) ∼ p
xα
, (21)
where 0 < α < 1, p ≥ 0, q ≥ 0. Then there exists the slowly varying function L(x), x > 0,
such that, for x→ −∞,
H−(x) ∼ |x|γL(|x|), (22)
where
γ =
α
2
− c(α, β)
π
, c(α, β) = arctan
(
β tan
πα
2
)
, β =
p− q
p+ q
.
The analogous result takes place for H+(x).
We see that the greatest value γ = α and the least γ = 0 are achieved, respectively, for
β = −1 and for β = 1. These values of β correspond to the extreme types of stable laws
with the exponent α which F is attracted to. If β = 0, then evidently γ = α
2 . Letting
α = 2 in the last equality, we obtain the value 1 for γ.
It is not improbable that the function L(x) in Theorem 2 is in fact constant. This is
the case if the distribution F is concentrated on the negative semiaxis (see [3]).
The analysis of the proof of Theorem 2 shows that L(x) = const in the case of
symmetric F .
Theorem 3. If conditions (16) and (21) are fulfilled, then there exists the slowly varying
function l(x) such that, for x→∞,
1− F+(x) ∼ xγ−αl(x), (23)
where
γ =
α
2
− c(α, β)
π
, c(α, β) = arctan
(
β tan
πα
2
)
.
104 SERGEY V. NAGAEV
Notice that L(x) and l(x), generally speaking, do not satisfy the condition L(x) ∼
cl(x). However, if L(x) equals a constant, then it is true for l(x) as well. The result
similar to Theorem 3 is also valid for F−(x).
2. Proof of Proposition
Denote, by f(t), the characteristic function of the random variable X . The starting
point is the next formula deduced in [14] (see also [15])
∞∑
n=1
1
n
∫ ∞
0+
e−hxdFn(x) +
1
2
∞∑
n=1
(Fn(0+)− Fn(0))n−1
= −h
π
∫ ∞
0
ln |1− f(t)|
h2 + t2
dt− 1
π
∫ ∞
0
t arg (1− f(t))
h2 + t2
dt (24)
with
h
π
∫ ∞
0
ln |1− f(t)|
h2 + t2
dt = −1
2
∫
|x|�=0
e−h|x|dG(x), (25)
1
π
∫ ∞
0
t arg (1− f(t))
h2 + t2
dt =
1
2
∫
x<0
ehxdG(x) − 1
2
∫
x>0
e−hxdG(x). (26)
First, we show that the right-hand side of equality (26) goes to
1
2
∞∑
n=1
(P(Sn < 0)−P(Sn > 0))
as h ↓ 0.
We need several lemmas to proof it.
Lemma 2.1. If a = 0 and σ2 <∞, then
P(Sn > 0)−E
{
e−hSn ;Sn > 0
}
≤ hσ√n, (27)
and
P(Sn < 0)−E
{
ehSn ;Sn < 0
}
< hσ
√
n. (28)
Proof. Applying the inequalities 1− e−x < x, x > 0, and E|Sn| ≤ σ√n, we have
P(Sn > 0)−E
{
e−hSn ;Sn > 0
}
= E
{
1− e−hSn ;Sn > 0
}
< hE
{
Sn;Sn > 0
}
< hσ
√
n.
In just the same way, (28) is proved. �
Put
Δn(h) =
∫
x>0
e−hxdFn(x)−
∫
x<0
ehxdFn(x) + P(Sn < 0)−P(Sn > 0). (29)
Lemma 2.2. Under conditions of Lemma 2.1,
lim
n→∞ sup
h>0
∣∣Δn(h)
∣∣ = 0. (30)
Proof. Integrating by parts on the right-hand side of formula (29), we find that
Δn(h) = h
∫ ∞
0
(1 − Fn(x) − Fn(−x))e−hxdx.
Consequently,
|Δn(h)| < h sup
x
∣∣∣1− Fn(x)− Fn(−x)
∣∣∣ ∫ ∞
0
e−hxdx.
By CLT,
lim
n→∞
(
1− Fn(x)− Fn(−x)
)
= 0.
PROBABILITIES OF LARGE DEVIATIONS OF LADDER HEIGHTS 105
The assertion of the lemma follows from two previous relations. �
Lemma 2.3. If the distribution F is not degenerate at zero, then
E
{
e−hSn ;Sn > 0
}
<
c(F )
h
√
n
, (31)
E
{
e−hSn ;Sn < 0
}
<
c(F )
h
√
n
. (32)
Proof. We restrict ourselves to proving (31). Obviously,∫ ∞
0+
e−hxdFn(x) <
∞∑
k=0
e−hkP(k < Sn ≤ k + 1).
Since, by (4),
P(k < Sn ≤ k + 1) <
c(F )√
n
,∫ ∞
0+
e−hxdFn(x) <
c(F )√
n
∞∑
k=0
e−hk.
On the other hand,
∞∑
k=0
e−hk =
1
1− e−h
<
1
h
.
The desired result follows from these two inequalities. �
Consider the series
Σ(h) =
∞∑
n=1
n−1
∣∣∣Δn(h)
∣∣∣
=
∑
n< ε2
h2
n−1
∣∣∣Δn(h)
∣∣∣+ ∑
ε2
h2 <n≤ 1
h2ε2
n−1
∣∣∣Δn(h)
∣∣∣+ n−1∑
n> 1
h2ε2
n−1
∣∣∣Δn(h)
∣∣∣ =∑
1
+
∑
2
+
∑
3
.
(33)
Applying Lemma 2.1, we find that∑
1
< h
∑
n≤ ε2
h2
1√
n
< 3ε. (34)
Further, ∑
2
< sup
ε2
h2 <n≤ 1
h2ε2
Δn(h)
(
h2
ε2
+
∫ 1
ε2h2
ε2
h2
dx
x
)
.
Since ∫ 1
ε2h2
ε2
h2
dx
x
= −4 ln ε,
we have, by (30),
lim
h↓0
∑
2
= 0. (35)
Notice that, by Lemma 2.3, ∣∣∣Δn(h)
∣∣∣ < c(F )
h
√
n
+ |2Fn(0)− 1|.
106 SERGEY V. NAGAEV
Therefore, ∑
3
<
c(F )
h
∑
n> 1
ε2h2
1
n3/2
+
∑
n> 1
(εh)2
n−1|2Fn(0)− 1|.
Further, ∑
n> 1
(εh)2
1
n3/2
< h3ε3 +
∫ ∞
u>(εh)−2
du
u3/2
< h3ε3 + 2εh.
The series ∞∑
1
n−1(2Fn(0)− 1)
absolutely converges (see [16]). Thus,
lim
h↓0
∑
3
< 2ε. (36)
It follows from (33) - (36) that
lim
h↓0
Σ(h) < 5ε.
It means by (29) that
lim
h↓0
(∫
x>0
e−hxdG(x) −
∫
x<0
ehxdG(x)
)
=
∞∑
n=1
n−1
(
P(Sn > 0)−P(Sn < 0)
)
. (37)
Proceed now to the left-hand side of equality (25). Choose δ in the partition∫ ∞
0
ln |1− f(t)|
t2 + h2
dt =
(∫ δ
0
+
∫ ∞
δ
) ln |1− f(t)|
t2 + h2
dt = I1(h) + I2(h) (38)
in such a way as the function f(t) �= 1 in the interval (0, δ). Put
f1(t) = 2
1− f(t)
σ2t2
.
Then
I1(h) =
∫ δ
0
ln f1(t)
t2 + h2
dt
+ ln
σ2
2
∫ δ
0
dt
t2 + h2
+ 2
∫ δ
0
ln t
t2 + h2
dt = I11(h) + ln
(σ2
2
)
I12(h) + I13(h). (39)
Since f1(0) = 1,
lim
h↓0
hI11(h) = 0. (40)
Further,
lim
h↓0
hI12(h) =
π
2
. (41)
It is easily seen that
lim
h↓0
h
∫ ∞
δ
ln t
t2 + h2
dt = 0.
Consequently, for h ↓ 0,
hI13(h) = h
∫ ∞
0
ln t
t2 + h2
dt+ o(1).
On the other hand,∫ ∞
0
ln t
t2 + h2
dt =
lnh
h
∫ ∞
0
dt
1 + t2
+
1
h
∫ ∞
0
ln t
1 + t2
dt =
π
2h
lnh. (42)
PROBABILITIES OF LARGE DEVIATIONS OF LADDER HEIGHTS 107
We use the equality ∫ ∞
0
ln t
1 + t2
dt = 0
(see [17], p. 546, Section 4.231, formula 8). Thus, for h ↓ 0,
hI13(h) =
π
2
lnh+ o(1). (43)
It follows from (39) - (43) that
hI1(h) = π
(
lnh+
1
2
ln
σ2
2
)
+ o(1). (44)
Estimate now I2(h). First of all,∣∣∣ln |1− f(t)|
∣∣∣ < ∞∑
n=1
n−1|fn(t)|.
Consequently,
I2(h) <
∞∑
n=1
1
n
∫ ∞
δ
|f(t)|n
t2
dt.
The next bound ∫
|t−v|≤0.65�σ2(L)
�β3(L)
|fn(t)| dt ≤ c
σ̃(L)
√
n
(45)
holds, where σ̃2(L) =
∫
|x|≤L x
2 dF̃ (x), β̃3(L) =
∫
|x|≤L |x|3 dF̃ (x), c < 7.61579, F̃ (x)
is the symmetrization of X (see [14], Lemma 2.1.)
Splitting the interval (δ,∞) into intervals of the length 1.3σ2(L)
β3(L) and applying bound
(45), it is not hard to show that ∫ ∞
δ
|f(t)|n
t2
dt <
c(F )√
n
.
Consequently, uniformly in h > 0,
I2(h) < c(F ). (46)
Returning now to (38) and taking (44) and (45) into account, we obtain that, for h ↓ 0
h
π
∫ ∞
0
ln |1− f(t)|
t2 + h2
dt = lnh+
1
2
ln
σ2
2
+ o(1). (47)
Combining (24), (37), and (47), we arrive at formula (8). Formula (9) is deduced in the
same way. �
3. Proof of Theorem 1
Without loss of generality, we may assume that F (x) is continuous. By (13) for any
x < x(ε) < 0,
−(1− ε) < x
σ√
2
H−(x) < (1 + ε)x. (48)
Using (1), we have
P(Z+x) =
∫ ∞
x−x(ε)
H−(x− y)dF (y) +
∫ x−x(ε)
x
H−(x− y)dF (y) = I1(x) + I2(x). (49)
Obviously, by (48),
(1− ε) σ√
2
∫ ∞
x−x(ε)
(y − x)dF (y) < I1(x) <
σ√
2
∫ ∞
x−x(ε)
(y − x)dF (y)(1 + ε). (50)
108 SERGEY V. NAGAEV
Further,
I2(x) < H−(x(ε))(F (x(ε)) − F (x)).
Hence, by condition (16) for x→∞,
I2(x) = o(1 − F (x)). (51)
By the same reason for x→∞,∫ x−x(ε)
x
dF (y) = o(1− F (x)). (52)
Put
F (x) =
∫ ∞
x
(y − x)dF (y) =
∫ ∞
x
(1− F (y))dy.
It is easily seen that
1− F (x) = o(F (x)). (53)
It follows from (51) and (53) that
I2(x) = o(F (x)), (54)
and from (52), (53) ∫ ∞
x−x(ε)
(y − x)dF (y) ∼ F (x). (55)
By (50) and (55),
σ√
2
(1 − ε) ≤ lim
x→∞ inf I1(x)/F (x) ≤ lim
x→∞ sup I2(x)/F (x) ≤ σ√
2
(1 + ε). (56)
Combining (49), (54), and (56), we get the desired result. �
4. Proof of Theorem 2
Previously, we prove several lemmas.
Lemma 4.1. Let, for x→∞, 1− F (x) ∼ c
xα , 0 < α < 1. Then for t ↓ 0∫ ∞
0
sin(tx)dF (x) ∼ ctα
∫ ∞
0
cosx
xα
dx. (57)
Proof. Clearly,∫ ∞
0
sin(tx)dF (x) =
∫ M/t
0
sin(tx)dF (x) +
∫ ∞
M/t
sin(tx)dF (x). (58)
There exists the constant K such that
1− F (x) <
Kc
xα
. (59)
Therefore, for any M > 0,∣∣∣ ∫ ∞
M/t
sin(tx)dF (x)
∣∣∣ < 1− F (M/t) <
Kc
Mα
tα. (60)
Integrating by parts, we have∫ M/t
0
sin(tx)dF (x) = t
∫ M/t
0
(1− F (x)) cos(tx)dx + (1− F (M/t)) sin(M/t).
Hence, by (59),∣∣∣ ∫ M/t
0
sin(tx)dF (x) − t
∫ M/t
0
(1− F (x)) cos(tx)dx
∣∣∣ < Kc
Mα
tα. (61)
PROBABILITIES OF LARGE DEVIATIONS OF LADDER HEIGHTS 109
Further,∫ M/t
0
(1 − F (x)) cos(tx)dx =
∫ M/t
ε/t
(1 − F (x)) cos(tx)dx +
∫ ε/t
0
(1− F (x)) cos(tx)dx.
Hence, by (59),∣∣∣ ∫ ε/t
0
(1 − F (x)) cos(tx)dx
∣∣∣ < Kc
∫ ε/t
0
dx
xα
<
Kcε1−α
1− α tα−1.
For M and ε fixed,∫ M/t
ε/t
(1− F (x)) cos(tx)dx =
c
t1−α
(∫ M
ε
cosx
xα
dx+ o(1)
)
.
As a result, we obtain, for every fixed M and ε,∫ M/t
0
(1−F (x)) cos(tx)dx = ctα−1
(∫ M
ε
cosx
xα
dx+o(1)
)
+θ
Kcε1−α
1− α tα−1, |θ| ≤ 1. (62)
Returning now to (61), we conclude that∫ M/t
0
sin(tx)dF (x) = ctα
(∫ M
ε
cosx
xα
dx+ o(1)
)
+ θKCtα
(
M−α +
ε1−α
1− α
)
. (63)
The desired result follows from (58), (60), and (62). �
Lemma 4.2. Under conditions of Lemma 4.1 for t→ 0,∫ ∞
0
(1 − cos(tx))dF (x) ∼ c|t|α
∫ ∞
0
sinx
xα
dx. (64)
Proof. Without loss of generality, we may assume t > 0. Obviously,∫ ∞
0
(1− cos(tx))dF (x) =
∫ M/t
0
(1− cos(tx))dF (x) +
∫ ∞
M/t
(1− cos(tx))dF (x). (65)
By (59) for M > 0,
1− F (M/t) <
Kc
Mα
tα. (66)
Hence, ∫ ∞
M/t
(1 − cos(tx))dF (x) <
Kc
Mα
tα. (67)
Integrating by parts, we have∫ M/t
0
(1− cos(tx))dF (x) = t
∫ M/t
0
(1 − F (x)) sin(tx)dx + (1− F (M/t))(1 − cosM).
Hence, by (67),∣∣∣ ∫ M/t
0
(1 − cos(tx))dF (x) − t
∫ M/t
0
(1− F (x)) sin(tx)dx
∣∣∣ < Kc
Mα
tα. (68)
Further,∫ M/t
0
(1− F (x)) sin(tx)dx =
∫ ε/t
0
(1 − F (x)) sin(tx)dx +
∫ M/t
ε/t
(1− F (x)) sin(tx)dx.
110 SERGEY V. NAGAEV
By (59), ∣∣∣∫ ε/t
0
(1− F (x)) sin(tx)dx
∣∣∣ < Kcε1−α
1− α tα−1.
For M and ε fixed,∫ M/t
ε/t
(1− F (x)) sin(tx)dx =
c
t1−α
(∫ M
ε
sinx
xα
dx+ o(1)
)
.
It follows from two last formulas that
t
∫ M/t
0
(1− F (x)) sin(tx)dx = ctα
(∫ M
ε
sinx
xα
dx+ o(1)
)
+ θ
Kctα
1− αε
1−α, |θ| ≤ 1. (69)
Combining (65)–(69), we obtain the assertion of Lemma 4.2. �
Lemma 4.3. For any 0 < α < 1,∫ ∞
0
eix
xα
dx = i1−αΓ(1− α). (70)
Proof. By changing the contour of integration in accordance with the change of the
variable x = iy, we find∫ ∞
0
eix
xα
dx = i1−α
∫ ∞
0
e−yy−αdy = i1−αΓ(1− α). �
Lemma 4.4. Let F (x) satisfy conditions (21). Then, for t→ 0,
1−
∫ ∞
−∞
eitxdF (x) ∼ |t|α
(
(p+ q) cos
πα
2
+ i(q − p) t|t| sin
πα
2
)
Γ(1− α). (71)
Proof. Obviously,
1−
∫ ∞
−∞
eitxdF (x) =
∫ ∞
−∞
(1− cos(tx))dF (x) − i
∫ ∞
−∞
sin(tx)dF (x).
By (57) and (64) for t→ 0,∫ ∞
−∞
(1 − cos(tx))dF (x) ∼ (p+ q)|t|α
∫ ∞
0
sinx
xα
dx
and ∫ ∞
−∞
sin(tx)dF (x) ∼ (p− q)|t|α t
|t|
∫ ∞
0
cosx
xα
dx.
Thus,
1−
∫ ∞
−∞
eitxdF (x) ∼ |t|α
(
(p+ q)
∫ ∞
0
sinx
xα
dx + i(q − p) t|t|
∫ ∞
0
cosx
xα
dx
)
.
According to Lemma 4.3,∫ ∞
0
cosx
xα
dx = Γ(1− α)Re i1−α = sin
πα
2
,∫ ∞
0
sinx
xα
dx = Γ(1− α)Im i1−α = cos
πα
2
.
Substituting these values into the previous equality, we obtain the desired result. �
Considering as before that F (x) satisfies conditions (21),we study the Laplace trans-
form of the projection of the harmonic renewal measure (3) on the semiaxis (−∞, 0].
PROBABILITIES OF LARGE DEVIATIONS OF LADDER HEIGHTS 111
It is easily seen that ∫ 0−
−∞
ehxdG−(x) = −
∫ ∞
0+
e−hxdG−(−x). (72)
Hence, ∫ 0+
−∞
e−hxdG−(x) =
∫ ∞
0+
ehxdG−(x) + ν({0}). (73)
By (24),∫ ∞
0+
e−hxdG−(−x) +
1
2
ν({0}) = −h
π
∫ ∞
0
ln |1− f(−t)|
h2 + t2
dt− 1
π
∫ ∞
0
t arg (1 − f(−t))
h2 + t2
dt.
(74)
Using Lemma 4.4, it is easy to verify that, for t→ 0,
|1− f(−t)| = |t|αΓ(1− α)(p2 + q2 − 2pq sinπα)1/2 + o(1).
Consequently, for t→ 0,
ln |1− f(−t)| = α ln |t|+ c(α, p, q) + o(1),
where
c(α, p, q) =
1
2
ln(p2 + q2 − 2pq sinπα) + ln Γ(1− α).
Hence, by (42),
h
π
∫ ∞
0
ln |1− f(−t)|
h2 + t2
dt =
αh
π
∫ ∞
0
ln t
h2 + t2
dt+ ψ(h) =
α
2
lnh+ ψ(h), (75)
where limh→0 ψ(h) = c(α, p, q).
According to (71),
arg (1 − f(−t)) =
t
|t| arctan
(
β tg
πα
2
)
+ ϕ(t), (76)
where β = (p− q)/(p+ q), ϕ(t)→ 0 as t→ 0.
Lemma 4.5. The function
W (h) = exp
{∫ 1
h
tϕ(t)
h2 + t2
dt
}
, (77)
where ϕ(t)→ 0 for t→ 0, is slowly varying as h ↓ 0, i.e. for any c > 0,
lim
h↓0
W (ch)
W (h)
= 1.
Proof. Changing the variable t = u−1, we have
I(h) :=
∫ 1
h
tϕ(t)
h2 + t2
dt =
∫ 1/h
1
ϕ(1/u)
1 + h2u2
du
u
=
∫ 1/h
1
ϕ(1/u)
u
du − h2
∫ 1/h
1
uϕ(1/u)
1 + h2u2
du = I1(h) + I2(h).
It is easily seen that∣∣∣ ∫ 1/h
1
uϕ(1/u)
1 + h2u2
du
∣∣∣ < ∫ 1/h
1
|ϕ(1/u)|udu = o(h−2).
Consequently,
lim
h→0
I2(h) = 0.
112 SERGEY V. NAGAEV
According to Karamata’s criterion, the function
Z(x) = exp
{∫ x
1
ϕ(1/u)
u
du
}
is slowly varying as x→∞. Hence, Z(1/h) is slowly varying as h ↓ 0. Since
W (h) = Z(1/h) exp{I2(h)},
the function W (h) has the same property as well. �
Lemma 4.6. Let a function ϕ(t) be continuous, and ϕ(0) = 0. Then
lim
h↓0
∫ h
−h
tϕ(t)
t2 + h2
dt = 0.
Proof. The conclusion of the lemma follows from the inequalities∣∣∣ ∫ h
−h
tϕ(t)
t2 + h2
dt
∣∣∣ < 2 sup
|t|≤h
|ϕ(t)|
∫ h
0
t
t2 + h2
dt < sup
|t|≤h
|ϕ(t)|. �
Lemma 4.7. For h ↓ 0,∫ 1
0
t arg(1− f(−t))
t2 + h2
dt = c(α, β) ln
1
h
+ lnW (h) + o(1) (78)
Proof. Based on formula (76) and Lemmas 4.5 and 4.6, we can state that∫ 1
0
t arg (1 − f(−t))
t2 + h2
dt = c(α, β)
∫ 1
0
t
t2 + h2
dt+ lnW (h) + o(1),
where
c(α, β) = arctan (β tan
πα
2
).
Obviously, ∫ 1
0
t
t2 + h2
dt =
1
2
ln(t2 + h2)
∣∣∣1
0
= ln
1
h
+
1
2
ln(1 + h2).
The conclusion of the lemma follows from last two formulas. �
Consider the integral
I(h) =
1
π
∫ ∞
1
t arg (1− f(−t))
t2 + h2
dt.
By (24) and (78), I(h) <∞ for h > 0.
Lemma 4.8. For every distribution F , there exists the finite limit
lim
h→0
I(h) = I0.
Proof. Arguing in the same way as in deducing formula (2.20) in [14], we are sure that
1
π
∫ ∞
1
t(1− arg f(−t))
t2 + h2
dt = −
∞∑
n−1
n−1
∫ ∞
1
tImfn(−t)
t2 + h2
dt. (79)
Evidently, ∫ ∞
1
tImfn(−t)
t2 + h2
dt = −
∫ ∞
1
tImfn(t)
t2 + h2
dt.
Further,∫ ∞
1
tImfn(t)
t2 + h2
dt =
∫ ∞
1
t
t2 + h2
∫ ∞
−∞
sin(tx)dFn(x) =
∫ ∞
−∞
dFn(x)
∫ ∞
1
t sin (tx)
t2 + h2
dx.
PROBABILITIES OF LARGE DEVIATIONS OF LADDER HEIGHTS 113
By Lemma 1.1 in [14] for α < h ≤ 1,
∣∣∣∫ ∞
1
t sin (tx)
t2 + h2
dt
∣∣∣ < { 2/|x|, |x| ≥ 1,
3, |x| < 1.
It follows from last two relations that∣∣∣∫ ∞
1
tIm fn(t)
t2 + h2
dt
∣∣∣ < 3
∫
|x|<n1/4
dFn(x) + 2
∫
|x|>n1/4
dFn(x)
|x| <
c(F )
n1/4
.
We have applied here a bound for the concentration function (4). Thus, series (79)
converges uniformly in the interval (0, 1].
On the other hand, for any n,
lim
h↓0
∫ ∞
1
tIm fn(t)
t2 + h2
dt =
∫ ∞
1
Im fn(t)
t
dt.
Consequently,
lim
h↓0
I(h) =
1
π
∞∑
n=1
∫ ∞
1
Im fn(t)
t
dt = I0. �
It follows from Lemmas 4.7 and 4.8 that, for h ↓ 0,∫ ∞
0
t arg (1− f(−t))
t2 + h2
dt = c(α, β) ln
1
h
+ lnW (h) + I0 + o(1). (80)
Combining (72)–(75) and (80), we conclude that
−
∫ 0+
−∞
ehxdG−(x) =
∫ ∞
0+
e−hxdG−(−x) + ν({0})
= γ ln
1
h
− π−1(lnW (h) + c0(F )) +
1
2
ν({0}),
where
c0(F ) = c(α, p, q) + I0, γ =
α
2
− c(α, β)
π
.
Applying now the Baxter identity (see, e.g., [1], Ch. 18, § 3), we find that∫ ∞
0+
e−hxdH−(−x) = exp
{∫ ∞
0+
e−hxdG−(−x)
}
∼ h−γ exp
{
− 1
π
(
lnW (h) + c0(F )
)
+
1
2
ν({0})
}
.
(81)
Using the Tauberian theorem for the Laplace transform (see [1], Ch. 13, § 5), we have
H−(−x) ∼ xγL(x), (82)
where
L(x) =
1
Γ(1 + γ)
exp
{
− 1
π
(
lnW (x−1) + c0(F )
)
+
1
2
ν({0})
}
, x > 0,
which is equivalent to the assertion of the theorem. �
114 SERGEY V. NAGAEV
5. Proof of Theorem 3
Using formula (1), we have
P(Z+ > x) ∼ p
∫ 0+
−∞
(x− y)−αdH−(y) = pα
∫ 0
−∞
(x− y)−α−1H−(y)dy. (83)
We need several lemmas to find the asymptotics of the last integral.
Lemma 5.1. For any x > 0,∫ 0
−√
x
(x− y)−α−1H−(y)dy < c(ε)xγ/2−α+ε, (84)
where ε is as small as one likes.
Proof. By (82), there exists a constant c such that, for every x > 0,∫ 0
−√
x
(x− y)−α−1H−(y)dy < c
∫ 0
−√
x
(x+ |y|)−α−1|y|γL(|y|)dy
= cxγ−α
∫ 0
− 1√
x
(1 + |y|)−α−1|y|γL(x|y|)dy < c(ε)xγ/2−α+ε/2 (85)
since
L(x|y|) < c(ε)(x|y|)ε. �
Lemma 5.2. As x→∞,
Γ(1 + γ)
∫ −√
x
−∞
(x − y)−α−1H−(y)dy ∼ xγ−α−1
∫ −√
x
−∞
(1 + |y|)−α−1|y|γL(x|y|)dy. (86)
Proof. The assertion of the lemma follows from asymptotics (82). �
Lemma 5.3. As x→∞,
Γ(1 + γ)
∫ 0
−∞
(x− y)−α−1H−(y)dy ∼ xγ−α
∫ 0
−∞
(1 + |y|)−α−1|y|γL(|y|)dy. (87)
Proof. Obviously,∫ −√
x
−∞
(x− y)−α−1H−(y)dy =
∫ −√
x
−∞
+
∫ 0
−√
x
= I1 + I2.
By Lemma 5.2,
I1 > c(ε)xγ−α−ε. (88)
It follows from (84) and (88) that
I2 = o(I1).
Hence, by (86), ∫ 0
−∞
(x − y)−α−1H−(y)dy
∼
∫ −√
x
−∞
(x− y)−α−1H−(y)dy ∼ xγ−α
Γ(1 + γ)
∫ −√
x
−∞
(1 + |y|)−α−1|y|γL(x|y|)dy.
It remains to remark that, by (85) and (87),∫ −√
x
−∞
(1 + |y|)−α−1|y|γL(x|y|)dy ∼
∫ 0
−∞
(1 + |y|)−α−1|y|γL(x|y|)dy. � (89)
PROBABILITIES OF LARGE DEVIATIONS OF LADDER HEIGHTS 115
Lemma 5.4. The function
h(x) :=
∫ 0
−∞
(1 + |y|)−α−1|y|γL(x|y|)dy (90)
is slowly varying.
Proof. By (89) for x→∞,
h(cx)
h(x)
∼
∫ −√
x
−∞ (1 + |y|)−α−1|y|γL(cx|y|)dy∫ −√
x
−∞ (1 + |y|)−α−1|y|γL(x|y|)dy
∼ 1. �
It follows from Lemmas 5.3 and 5.4 that∫ 0
−∞
(x− y)−α−1H−(y)dy ∼ xγ−α
Γ(1 + γ)
h(x), (91)
where h(x) is a slowly varying function. Comparing (83) and (91), we find that
P(Z+ > x) ∼ pα
Γ(1 + γ)
xγh(x).
Hence, letting
l(x) =
pα
Γ(1 + γ)
,
we obtain the conclusion of Theorem 3. �
In conclusion, we remark that if the integral∫ 1
0
ϕ(t)
t
dt,
where ϕ is defined by (76), is finite, then W (h) in (77) converges to some constant as
h ↓ 0.
Indeed, in this case for any η > h,∣∣∣∣∣
∫ η
h
tϕ(t)
t2 + h2
dt−
∫ η
h
ϕ(t)
t
dt
∣∣∣∣∣ ≤ sup
0≤t≤η
|ϕ(t)|
∫ η
h
(
t
t2 + h2
− 1
t
)
dt.
Obviously, ∫ η
h
(
t
t2 + h2
− 1
t
)
dt < h2
∫ η
h
dt
t3
<
1
3
.
On the other hand, for every 0 < η < 1,
lim
h↓0
∫ 1
η
tϕ(t)
t2 + h2
dt =
∫ 1
η
ϕ(t)
t
dt.
Thus,
lim
h↓0
∫ 1
h
tϕ(t)
t2 + h2
dt =
∫ 1
0
ϕ(t)
t
dt.
Further, if there exists the finite limit lim
h↓0
W (h), then the same is true for L(x) in (82)
as x→∞. But then, by (89) and (90),
h(x) ∼ c(α, γ)L(x),
where
c(α, γ) =
∫ ∞
0
(1 + y)−α−1yγdy.
116 SERGEY V. NAGAEV
Bibliography
1. W.Feller, An Introduction to Probability Theory and Its Applications II, Wiley, New York,
1967.
2. S.V.Nagaev, S.S.Khodzhabagjan, On an estimate of the concentration function of sums of
independent random variables, Theory Probab. Appl. 41 (1996), no. 3, 560-568.
3. P.Greenwood, I.Omey and J.L.Teugels, Harmonic renewal measures, Z. Warsch. Verw. Gebiete
59 (1982), 391-409.
4. P.Greenwood, I.Omey and J.L.Teugels, Harmonic renewal measures and bivariate domains of
attractions in fluctuation theory, Z. Warsch. Verw. Gebiete 61 (1982), 527-539.
5. R.J.Grubel, On harmonic renewal measures, Probab. Theory Rel. Fields 71 (1986), 393-403.
6. R.J.Grubel, Harmonic renewal sequences and the first positive sum, J. London Math. Soc. 38
(1988), no. 2, 179-192.
7. A.J.Stam, Some theorems on harmonic renewal measures, Stochastic processes and their Appl.
39 (1991), 277-285.
8. L.de Haan, An Abel–Tauber theorem for Laplace transforms, J. London Math. Soc. 13 (1976),
no. 3, 537-542.
9. R.A.Doney, Moments of ladder heights in random walks, J. Appl. Probab. 17 (1980), 248-252.
10. B.A.Rogosin, On the distribution of the first jump, Theory Probab. Appl. 9 (1964), 450-465.
11. N.Veraverbeke, Asymptotic behaviour of Wiener-Hopf factors of a random walk, Stochastic
processes and their Appl. 5 (1977), 27-37.
12. P.Embrechts, C.M.Goldie, and N.Vereverbeke, Subexponentiality and infinite divisibility, Z.
Warsch. Verw. Gebiete 49 (1979), 335-347.
13. R.Grubel, Tail behaviour of ladder–height distibulions in random walks, J. Appl. Probab. 22
(1985), 705-709.
14. S.V. Nagaev, Exact expressions for moments of ladder heigsts, Preprint 2007/192. IM SO RAN,
Novosibirsk, 2007 (In Russian).
15. S.V.Nagaev, The formula for the Laplace transform of the projection of a distribution on the
positive semiaxis and some its applications, Dokl. Math. 416 (2007), no. 5.
16. B.Rosen, On the asymptotic distribution of sums of independent indentically distributed random
variables, Ark.Mat. 4 (1962), no. 4, 323-332.
17. I.S.Gradshtein, I.M.Ryzhik, Tables of integrals, sums, series and products, Fiz. Mat. Giz.,
Moscow, 1963.
� ' ��% ����
����
��������
��� )� & ����� /�+� � % �
'
��� (, 8 � -���
�
E-mail : nagaev@math.nsc.ru, nagaevs@hotmail.com
|