Еxact rates in the Davis – Gut law of iterated logarithm for the first moment convergence of independent identically distributed random variables
Let $\{X, X_n, n \geq 1\}$ be a sequence of independent identically distributed random variables and let $S_n = \sum^n_{i=1} X_i$, $M_n = \max_{1\leq k\leq n} |S_k|$. For $r > 0$, let $a_n(\varepsilon)$ be a function of $\varepsilon$ such that $a_n(\varepsilon ) \mathrm{l}\mathrm{o}\mathr...
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| Мова: | Англійська |
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Institute of Mathematics, NAS of Ukraine
2017
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507524939120640 |
|---|---|
| author | Xiao, X.-Y. Yin, H.-W. Сяо, Х.-І. Інь, Х.-В. |
| author_facet | Xiao, X.-Y. Yin, H.-W. Сяо, Х.-І. Інь, Х.-В. |
| author_sort | Xiao, X.-Y. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:23:56Z |
| description | Let $\{X, X_n, n \geq 1\}$ be a sequence of independent identically distributed random variables and let $S_n = \sum^n_{i=1} X_i$,
$M_n = \max_{1\leq k\leq n} |S_k|$.
For $r > 0$, let $a_n(\varepsilon)$ be a function of $\varepsilon$ such that $a_n(\varepsilon ) \mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} n \rightarrow \tau$ as $n \rightarrow \infty$ and $\varepsilon \searrow
\surd r$.
If $EX^2I\{|X| \geq t\} = o(\text{log}\text{log}t)^{-1})$ as $t \rightarrow \infty$ , then, by using the strong approximation, we show that
$$\lim_{\varepsilon \searrow \surd r} \frac 1{-\text{log}(\varepsilon^2 - r)} \sum ^{\infty}_{n=1}\frac{(\text{log} n)^{r-1}}{n^{3/2}}E \Bigl\{ M_n - (\varepsilon + a_n(\varepsilon ))\sigma \sqrt{2n \text{log log} n} \Bigr\}_{+} = \frac{2\sigma \varepsilon^{-2\tau \sqrt{r}}}{\sqrt{2\pi}r}$$
holds if and only if $EX = 0, EX^2 = \sigma^2$, and $EX = 0, EX^2 = \sigma^2$ та $EX^2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )^{r-1}(\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} | X| )^{-\frac 12} < \infty$. |
| first_indexed | 2026-03-24T02:10:41Z |
| format | Article |
| fulltext |
UDC 519.21
X.-Y. Xiao, H.-W. Yin (Nanchang Univ., China)
EXACT RATES IN THE DAVIS – GUT LAW OF ITERATED LOGARITHM
FOR THE FIRST MOMENT CONVERGENCE OF INDEPENDENT
IDENTICALLY DISTRIBUTED RANDOM VARIABLES*
ТОЧНI ШВИДКОСТI В ЗАКОНI ПОВТОРНОГО ЛОГАРИФМА
ДЕВIСА – ГУТА ДЛЯ ЗБIЖНОСТI ПЕРШОГО МОМЕНТУ НЕЗАЛЕЖНИХ
ОДНАКОВО РОЗПОДIЛЕНИХ ВИПАДКОВИХ ВЕЛИЧИН
Let \{ X,Xn, n \geq 1\} be a sequence of independent identically distributed random variables and let Sn =
\sum n
i=1 Xi,
Mn = \mathrm{m}\mathrm{a}\mathrm{x}1\leq k\leq n | Sk| . For r > 0, let an(\varepsilon ) be a function of \varepsilon such that an(\varepsilon ) \mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g}n \rightarrow \tau as n \rightarrow \infty and \varepsilon \searrow
\surd
r.
If \BbbE X2I\{ | X| \geq t\} = o((\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} t) - 1) as t \rightarrow \infty , then, by using the strong approximation, we show that
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g}n)r - 1
n3/2
\BbbE
\Bigl\{
Mn - (\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n \mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} n
\Bigr\}
+
=
2\sigma e - 2\tau
\surd
r
\surd
2\pi r
holds if and only if \BbbE X = 0, \BbbE X2 = \sigma 2, and \BbbE X2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )r - 1(\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} | X| ) -
1
2 < \infty .
Нехай \{ X,Xn, n \geq 1\} — множина незалежних однаково розподiлених випадкових величин та Sn =
\sum n
i=1 Xi,
Mn = \mathrm{m}\mathrm{a}\mathrm{x}1\leq k\leq n | Sk| . Крiм того, для r > 0 нехай an(\varepsilon ) — функцiя \varepsilon така, що an(\varepsilon ) \mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g}n \rightarrow \tau при n \rightarrow \infty та
\varepsilon \searrow
\surd
r. У випадку \BbbE X2I\{ | X| \geq t\} = o((\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} t) - 1) при t \rightarrow \infty за допомогою сильної апроксимацiї доведено,
що спiввiдношення
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g}n)r - 1
n3/2
\BbbE
\Bigl\{
Mn - (\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n \mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} n
\Bigr\}
+
=
2\sigma e - 2\tau
\surd
r
\surd
2\pi r
виконуються тодi i тiльки тодi, коли \BbbE X = 0, \BbbE X2 = \sigma 2 та \BbbE X2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )r - 1(\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} | X| ) -
1
2 < \infty .
1.1. Introduction and main result. Throughout the paper, we suppose that \{ X,Xn, n \geq 1\} is
a sequence of independent identically distributed (i.i.d.) random variables and let Sn =
\sum n
i=1Xi,
Mn = \mathrm{m}\mathrm{a}\mathrm{x}1\leq k\leq n | Sk| , for n \geq 1. Let \{ W (t); t \geq 0\} be a standard Wiener process, and N
be the standard normal random variable. We denote by C a positive constant which may vary
from line to line, and define \lfloor x\rfloor = \mathrm{s}\mathrm{u}\mathrm{p}\{ m : m \leq x,m \in \BbbZ +\} . Let \mathrm{l}\mathrm{o}\mathrm{g} x = \mathrm{l}\mathrm{n}(x \vee e) and
\varphi (x) := \mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} x = \mathrm{l}\mathrm{n}(\mathrm{l}\mathrm{n}(x \vee ee)). The notation an \sim bn means that an/bn \rightarrow 1 as n \rightarrow \infty .
It is well known that Hsu and Robbins [8] and Erdös [4, 5] first introduced the concept of complete
convergence. So far many authors have considered various extensions. Li, Wang and Rao [11], Gut
and Spǎtaru [7] showed the precise rates in the law of the iterated logarithm (LIL) that
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow 0
\varepsilon 2
\infty \sum
n=1
1
n \mathrm{l}\mathrm{o}\mathrm{g} n
\BbbP
\Bigl(
| Sn| \geq \varepsilon
\sqrt{}
n\varphi (n)
\Bigr)
= \sigma 2
holds if and only if \BbbE X = 0 and \BbbE X2 = \sigma 2. Jiang and Zhang [10] extended it for moment
convergence. They proved, for an = O(1/\varphi (n)) and b > - 1, that
* This work was supported by the Natural Science Foundation of Jiangxi Province of China (Grant
No. 20151BAB201021) and the National Natural Science Foundation of China (Grant No. 61563033 and No. 11401293).
c\bigcirc X.-Y. XIAO, H.-W. YIN, 2017
240 ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
EXACT RATES IN THE DAVIS – GUT LAW OF ITERATED LOGARITHM . . . 241
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow 0
\varepsilon 2(b+1)
\infty \sum
n=1
\varphi (n)b
n \mathrm{l}\mathrm{o}\mathrm{g} n
\BbbE
\Bigl\{
| Sn| - \sigma (\varepsilon + an)
\sqrt{}
2n\varphi (n)
\Bigr\}
+
=
\sigma 2 - b - 1\BbbE | N | 2b+3
(b+ 1)(2b+ 3)
holds if and only if \BbbE X = 0 and \BbbE X2 = \sigma 2.
On the other hand, Davis [3] and Gut [6] showed that
\infty \sum
n=3
1
n
\BbbP
\Bigl(
| Sn| \geq
\sqrt{}
(2 + \varepsilon )n\varphi (n)
\Bigr) \Biggl\{ < \infty , if \varepsilon > 0,
= \infty , if - 2 < \varepsilon < 0,
holds if and only if \BbbE X = 0 and \BbbE X2 = 1. Gut and Spǎtaru [7] obtained the precise rates in the LIL
that for an = O(
\surd
n(\varphi (n)) - \gamma ) with \gamma > 1/2,
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow \sigma
\surd
2
\sqrt{}
\varepsilon 2 - 2\sigma 2
\infty \sum
n=3
1
n
\BbbP
\Bigl(
| Sn| \geq \varepsilon
\sqrt{}
n\varphi (n) + an
\Bigr)
= \sigma
\surd
2
holds as long as \BbbE X = 0, \BbbE X2 = \sigma 2 and \BbbE X2(\varphi (| X| ))1+\delta < \infty for some \delta > 0. They also pointed
out that
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow 2\sigma
\sqrt{}
\varepsilon 2 - 4\sigma 2
\infty \sum
n=3
\mathrm{l}\mathrm{o}\mathrm{g} n
n
\BbbP
\Bigl(
| Sn| \geq \varepsilon
\sqrt{}
n\varphi (n)
\Bigr)
= \sigma
holds if and only if \BbbE X = 0, \BbbE X2 = \sigma 2 and \BbbE X2 \mathrm{l}\mathrm{o}\mathrm{g} | X| (\varphi (| X| )) - 1 < \infty .
Jiang, Zhang and Pang [9] provided the precise rates in the law of logarithm for moment conver-
gence that for r > 1,
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r - 1
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - (r - 1))
\infty \sum
n=1
nr - 2 - 1/2\BbbE
\Bigl\{
Mn - \varepsilon \sigma
\sqrt{}
2n \mathrm{l}\mathrm{o}\mathrm{g} n
\Bigr\}
+
=
2\sigma
(r - 1)
\surd
2\pi
holds if and only if \BbbE X = 0, \BbbE X2 = \sigma 2 and \BbbE (X2r/(\mathrm{l}\mathrm{o}\mathrm{g} | X| )r) < \infty .
Inspired by [9], we consider the precise rates in the LIL extended from Davis [3] and Gut [6] for
the first moment of Sn and Mn. We obtain the following results.
Theorem 1.1. For r > 0, let an(\varepsilon ) be a function of \varepsilon such that
an(\varepsilon )\varphi (n) \rightarrow \tau , as n \rightarrow \infty and \varepsilon \searrow
\surd
r. (1.1)
Let \{ X,Xn;n \geq 1\} be a sequence of i.i.d. random variables with
\BbbE X = 0, \BbbE X2 = \sigma 2, and \BbbE X2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )r - 1(\varphi (| X| )) - 1/2 < \infty . (1.2)
Suppose that
\BbbE X2I\{ | X| \geq t\} = o((\varphi (t)) - 1), as t \rightarrow \infty . (1.3)
Then we have
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\BbbE
\Bigl\{
| Sn| - (\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n)
\Bigr\}
+
=
\sigma e - 2\tau
\surd
r
\surd
2\pi r
(1.4)
and
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
242 X.-Y. XIAO, H.-W. YIN
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\BbbE
\Bigl\{
Mn - (\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n)
\Bigr\}
+
=
2\sigma e - 2\tau
\surd
r
\surd
2\pi r
. (1.5)
Conversely, for an(\varepsilon ) satisfying (1.1), if either (1.4) or (1.5) holds for r > 0 and some \sigma > 0, then
(1.2) holds and
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
t\rightarrow \infty
\varphi (t)\BbbE X2I\{ | X| \geq t\} = 0. (1.6)
Remark 1.1. The condition (1.3) is sharp. A sufficient condition for it is given by \BbbE X2\varphi (| X| ) <
< \infty . Obviously, when r > 1, condition (1.3) is implied by (1.2).
Corollary 1.1. Under conditions of (1.1) and (1.2), for any \varepsilon >
\surd
r > 0, we have
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\BbbE
\Bigl\{
| Sn| - (\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n)
\Bigr\}
+
< \infty ,
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\BbbE
\Bigl\{
Mn - (\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n)
\Bigr\}
+
< \infty
and
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
\sqrt{}
\varphi (n)
n
\BbbP
\Bigl\{
| Sn| \geq (\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n)
\Bigr\}
< \infty ,
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
\sqrt{}
\varphi (n)
n
\BbbP
\Bigl\{
Mn \geq (\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n)
\Bigr\}
< \infty .
2. Some lemmas.
Lemma 2.1 [1, p. 79, 80]. For any x > 0,
\mathrm{P}
\biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
0\leq t\leq 1
| W (t)| \geq x
\biggr\}
= 1 -
\infty \sum
k= - \infty
( - 1)k\mathrm{P}\{ (2k - 1)x \leq N \leq (2k + 1)x\} =
= 4
\infty \sum
k=0
( - 1)k\mathrm{P}\{ N \geq (2k + 1)x\} = 2
\infty \sum
k=0
( - 1)k\mathrm{P}\{ | N | \geq (2k + 1)x\} .
In particular,
\mathrm{P}
\biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
0\leq t\leq 1
W (t) \geq x
\biggr\}
= 2\mathrm{P}\{ N \geq x\} \sim 2\surd
2\pi x
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
- x2
2
\biggr)
, as x \rightarrow \infty .
Lemma 2.2 (Lemma 1.1.1 of [2]). For any \varepsilon > 0,
\BbbP
\Biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
0\leq s\leq 1 - h
\mathrm{s}\mathrm{u}\mathrm{p}
0\leq t\leq h
\bigm| \bigm| W (s+ t) - W (s)
\bigm| \bigm| \geq \nu
\surd
h
\Biggr)
\leq C
h
e
- \nu 2
2+\varepsilon
holds for every \nu > 0 and 0 < h < 1.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
EXACT RATES IN THE DAVIS – GUT LAW OF ITERATED LOGARITHM . . . 243
Lemma 2.3 (Lemma 2.4 of [9], see also [13, p. 78]). For q \geq 2, let \{ \xi k; 1 \leq k \leq n\} be a
sequence of independent random variables with \BbbE \xi k = 0 and \BbbE | \xi k| q < \infty . Then, for any y > 0,
\BbbP
\Biggl(
\mathrm{m}\mathrm{a}\mathrm{x}
k\leq n
\bigm| \bigm| \bigm| k\sum
i=1
\xi i
\bigm| \bigm| \bigm| \geq y
\Biggr)
\leq 2 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- y2
8
\sum n
k=1 \BbbE \xi 2k
\biggr\}
+ (2Aq)qy - q
n\sum
i=1
\BbbE | \xi i| q,
where A is a universal constant.
Lemma 2.4. There exists N0 > 0 such that when n \geq N0, we have
n\BbbP
\Bigl(
| X| \geq 4\varepsilon
\sqrt{}
n\varphi (n)
\Bigr)
\leq 4\BbbP
\Bigl(
| Sn| \geq \varepsilon
\sqrt{}
n\varphi (n)
\Bigr)
for \varepsilon > 0.
Proof. The proof is similar to that of Lemma 2.3 in [12] in which we only need to replace
\surd
\mathrm{l}\mathrm{o}\mathrm{g} n
by
\sqrt{}
\varphi (n).
3. Proof of Theorem 1.1 and Corollary 1.1. Theorem 1.1 is based on the following propositions.
Proposition 3.1. For r > 0, let an(\varepsilon ) be a function of \varepsilon satisfying (1.1), we have
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n
\BbbE
\Bigl\{
| N | - (\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n)
\Bigr\}
+
=
e - 2\tau
\surd
r
\surd
2\pi r
(3.1)
and
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n
\BbbE
\biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
0\leq s\leq 1
| W (s)| - (\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n)
\biggr\}
+
=
2e - 2\tau
\surd
r
\surd
2\pi r
. (3.2)
Proof. By Lemma 2.1 and condition (1.1), uniformly with respect to all x \geq 0, we have
\BbbP
\biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
0\leq s\leq 1
| W (s)| \geq x+ (\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n)
\biggr\}
\sim
\sim 2\BbbP
\Bigl\{
| N | \geq x+ (\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n)
\Bigr\}
=
= 4\BbbP
\Bigl\{
N \geq x+ (\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n)
\Bigr\}
\sim
\sim 4
\surd
2\pi
\Bigl(
x+ (\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n)
\Bigr) \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- 1
2
\Bigl(
x+ (\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n)
\Bigr) 2\biggr\}
\sim
\sim 4
\surd
2\pi
\Bigl(
x+ \varepsilon
\sqrt{}
2\varphi (n)
\Bigr) \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- 1
2
\Bigl(
x+ \varepsilon
\sqrt{}
2\varphi (n)
\Bigr) 2
- an(\varepsilon )
\sqrt{}
2\varphi (n)
\Bigl(
x+ \varepsilon
\sqrt{}
2\varphi (n)
\Bigr) \biggr\}
\sim
\sim 4
\surd
2\pi
\Bigl(
x+ \varepsilon
\sqrt{}
2\varphi (n)
\Bigr) \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- 1
2
\Bigl(
x+ \varepsilon
\sqrt{}
2\varphi (n)
\Bigr) 2\biggr\}
\mathrm{e}\mathrm{x}\mathrm{p}\{ - 2\varepsilon \tau \} ,
as n \rightarrow \infty and \varepsilon \searrow
\surd
r.
Therefore, we obtain
\BbbE
\biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
0\leq s\leq 1
| W (s)| - (\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n)
\biggr\}
+
\sim 2\BbbE
\Bigl\{
| N | - (\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n)
\Bigr\}
+
\sim
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
244 X.-Y. XIAO, H.-W. YIN
\sim 4e - 2\varepsilon \tau
\infty \int
0
1
\surd
2\pi
\Bigl(
x+ \varepsilon
\sqrt{}
2\varphi (n)
\Bigr) \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- 1
2
\Bigl(
x+ \varepsilon
\sqrt{}
2\varphi (n)
\Bigr) 2\biggr\}
dx,
as \varepsilon \searrow
\surd
r and n \rightarrow \infty . Next we only need to prove (3.1).
Since the limit in (3.1) does not depend on any finite terms of the infinite series, it follows that
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n
\BbbE
\Bigl\{
| N | - (\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n)
\Bigr\}
+
=
= \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n
2e - 2\varepsilon \tau
\infty \int
0
1
\surd
2\pi
\Bigl(
x+ \varepsilon
\sqrt{}
2\varphi (n)
\Bigr) \times
\times \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- 1
2
\Bigl(
x+ \varepsilon
\sqrt{}
2\varphi (n)
\Bigr) 2\biggr\}
dx =
= \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
2e - 2\varepsilon \tau
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \int
ee
(\mathrm{l}\mathrm{o}\mathrm{g} y)r - 1
y
dy
\infty \int
0
1
\surd
2\pi
\Bigl(
x+ \varepsilon
\sqrt{}
2\varphi (y)
\Bigr) \times
\times \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- 1
2
\Bigl(
x+ \varepsilon
\sqrt{}
2\varphi (y)
\Bigr) 2\biggr\}
dx =
= \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
2e - 2\varepsilon \tau
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \int
ee
(\mathrm{l}\mathrm{o}\mathrm{g} y)r - 1
y
dy
\infty \int
\varepsilon
\surd
2\varphi (y)
1
z
\surd
2\pi
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- z2
2
\biggr\}
dz =
= \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
2e - 2\varepsilon \tau
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \int
\surd
2\varepsilon
t
\varepsilon 2
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
rt2
2\varepsilon 2
\biggr\}
dt
\infty \int
t
1
z
\surd
2\pi
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- z2
2
\biggr\}
dz =
= \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
2e - 2\varepsilon \tau
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \int
\surd
2\varepsilon
1
z
\surd
2\pi
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- z2
2
\biggr\}
dz
z\int
\surd
2\varepsilon
t
\varepsilon 2
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
rt2
2\varepsilon 2
\biggr\}
dt =
= \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
2e - 2\varepsilon \tau
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \int
\surd
2\varepsilon
1
z
\surd
2\pi
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- z2
2
\biggr\}
1
r
\biggl(
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
rz2
2\varepsilon 2
\biggr\}
- er
\biggr)
dz =
=
1\surd
2\pi r
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
2e - 2\varepsilon \tau
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \int
\surd
2\varepsilon
1
z
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- z2
2
+
rz2
2\varepsilon 2
\biggr\}
dz =
=
1\surd
2\pi r
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
e - 2\varepsilon \tau
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \int
\varepsilon 2 - r
1
s
e - sds
\biggl(
s =
z2
2
\Bigl(
1 - r
\varepsilon 2
\Bigr) \biggr)
=
=
e - 2\tau
\surd
r
\surd
2\pi r
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
1\int
\varepsilon 2 - r
1
s
e - sds =
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
EXACT RATES IN THE DAVIS – GUT LAW OF ITERATED LOGARITHM . . . 245
=
e - 2\tau
\surd
r
\surd
2\pi r
.
Proposition 3.1 is proved.
Next, without loss of generality, we assume that \sigma = 1 throughout the proof of the direct part of
our main results. For each n and 1 \leq j \leq n, 1/2 < p \leq 2, let
X \prime
nj = XjI\{ | Xj | \leq
\surd
n/(\varphi (n))p\} , X
(1)
nj = X \prime
nj - \BbbE X \prime
nj , Bn =
n\sum
j=1
\mathrm{V}\mathrm{a}\mathrm{r}X
(1)
nj ,
S
(1)
nk =
k\sum
j=1
X
(1)
nj , M (1)
n = \mathrm{m}\mathrm{a}\mathrm{x}
1\leq k\leq n
| S(1)
nk | , \Delta n = \mathrm{m}\mathrm{a}\mathrm{x}
1\leq k\leq n
| Sk - S
(1)
nk | (3.3)
and
X \prime \prime
nj = XjI\{
\surd
n/(\varphi (n))p < | Xj | \leq
\sqrt{}
n\varphi (n)\} , X
(2)
nj = X \prime \prime
nj - \BbbE X \prime \prime
nj ,
X \prime \prime \prime
nj = XjI\{ | Xj | >
\sqrt{}
n\varphi (n)\} , X
(3)
nj = X \prime \prime \prime
nj - \BbbE X \prime \prime \prime
nj .
Also we define S
(2)
nj , S
(3)
nj , M
(2)
n , M
(3)
n similarly.
Proposition 3.2. Let 1/2 < p\prime < p \leq 2. For any \varepsilon > 0 and x > 0 there exists a sequence of
positive numbers pn such that
\sqrt{}
Bn\BbbE
\biggl\{
| N | - x - 1
(\varphi (n))p\prime
\biggr\}
+
- pn \leq \BbbE
\left\{ \bigm| \bigm| \bigm|
n\sum
j=1
X
(1)
nj
\bigm| \bigm| \bigm| - x
\sqrt{}
Bn
\right\}
+
\leq
\leq
\sqrt{}
Bn\BbbE
\biggl\{
| N | - x+
1
(\varphi (n))p\prime
\biggr\}
+
+ pn (3.4)
and \sqrt{}
Bn\BbbE
\biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
0\leq s\leq 1
| W (s)| - x - 1
(\varphi (n))p\prime
\biggr\}
+
- pn \leq \BbbE
\Bigl\{
M (1)
n - x
\sqrt{}
Bn
\Bigr\}
+
\leq
\leq
\sqrt{}
Bn\BbbE
\biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
0\leq s\leq 1
| W (s)| - x+
1
(\varphi (n))p\prime
\biggr\}
+
+ pn, (3.5)
where
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
pn < \infty . (3.6)
Proof. We can define a Wiener process
Wn(tBn) =
\lfloor nt\rfloor \sum
i=1
\xi i + (nt - \lfloor nt\rfloor )\xi \lfloor nt\rfloor +1,
where \{ \xi n;n \geq 1\} is a sequence of independent normal variables with \BbbE \xi j = 0 and \BbbE \xi 2j = \mathrm{V}\mathrm{a}\mathrm{r}\mathrm{X}\prime
nj .
The proofs of (3.4) and (3.5) are similar to that of Proposition 2.2 in [9], and we have
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
246 X.-Y. XIAO, H.-W. YIN
pn \leq Cn(3 - q)/2(\varphi (n))p
\prime (q - 1)\BbbE | X| qI
\biggl\{
| X| \leq
\surd
n
(\varphi (n))p
\biggr\}
+
+
\surd
n\BbbP
\biggl(
\mathrm{m}\mathrm{a}\mathrm{x}
0\leq s\leq 1
\bigm| \bigm| \bigm| Wn(s) - Wn
\biggl(
\lfloor ns\rfloor
n
\biggr) \bigm| \bigm| \bigm| \geq 1
2(\varphi (n))p\prime
\biggr)
+
+
\surd
n
\infty \int
1
\BbbP
\biggl(
\mathrm{m}\mathrm{a}\mathrm{x}
0\leq s\leq 1
\bigm| \bigm| \bigm| Wn(s) - Wn
\biggl(
\lfloor ns\rfloor
n
\biggr) \bigm| \bigm| \bigm| \geq x
\biggr)
dx =:
=: pn1 +Dn1 +Dn2.
Next we only prove (3.6). On one hand, we get
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
pn1 \leq C
\infty \sum
n=1
n - q/2(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1(\varphi (n))p
\prime (q - 1)\BbbE | X| qI
\biggl\{
| X| \leq
\surd
n
(\varphi (n))p
\biggr\}
\leq
\leq C
\infty \sum
n=1
(\varphi (n))p
\prime (q - 1)
nq/2(\mathrm{l}\mathrm{o}\mathrm{g} n)1 - r
n\sum
j=1
\BbbE | X| qI
\biggl\{ \surd
j - 1
(\varphi (j - 1))p
< | X| \leq
\surd
j
(\varphi (j))p
\biggr\}
=
= C
\infty \sum
j=1
\BbbE | X| qI
\biggl\{ \surd
j - 1
(\varphi (j - 1))p
< | X| \leq
\surd
j
(\varphi (j))p
\biggr\} \infty \sum
n=j
(\varphi (n))p
\prime (q - 1)
nq/2(\mathrm{l}\mathrm{o}\mathrm{g} n)1 - r
\leq
\leq C
\infty \sum
j=1
j1 - q/2 (\varphi (j))
p\prime (q - 1)
(\mathrm{l}\mathrm{o}\mathrm{g} j)1 - r
\BbbE | X| qI
\biggl\{ \surd
j - 1
(\varphi (j - 1))p
< | X| \leq
\surd
j
(\varphi (j))p
\biggr\}
\leq
\leq C\BbbE X2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )r - 1(\varphi (| X| ))p\prime - (p - p\prime )(q - 2) \leq
\leq C\BbbE X2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )r - 1(\varphi (| X| )) - 1/2 < \infty ,
if p\prime - (p - p\prime )(q - 2) \leq - 1/2, or equivalently q \geq 2 + (p\prime + 1/2)/(p - p\prime ).
On the other hand, by Lemma 2.2 and the basic inequality that e - x \leq 2x - 2 for x > 0, it holds
that
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
Dn1 =
=
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n
\BbbP
\Biggl(
\mathrm{m}\mathrm{a}\mathrm{x}
0\leq s\leq 1
\bigm| \bigm| \bigm| Wn(s) - Wn
\biggl(
\lfloor ns\rfloor
n
\biggr) \bigm| \bigm| \bigm| \geq \sqrt{} 1
n
\surd
n
2(\varphi (n))p\prime
\Biggr)
\leq
\leq C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- n
12(\varphi (n))2p\prime
\biggr\}
\leq C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1(\varphi (n))4p
\prime
n2
< \infty .
Similarly, we obtain
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
Dn2 =
=
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n
\infty \int
1
\BbbP
\Biggl(
\mathrm{m}\mathrm{a}\mathrm{x}
0\leq s\leq 1
\bigm| \bigm| \bigm| Wn(s) - Wn
\biggl(
\lfloor ns\rfloor
n
\biggr) \bigm| \bigm| \bigm| \leq \sqrt{} 1
n
\surd
nx
\Biggr)
dx \leq
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
EXACT RATES IN THE DAVIS – GUT LAW OF ITERATED LOGARITHM . . . 247
\leq C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
\infty \int
1
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- nx2
3
\biggr\}
dx \leq
\leq C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n2
\infty \int
1
x - 4dx \leq C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n2
< \infty .
Then we complete the proof of this proposition.
Proposition 3.3. For any \lambda > 0, we have
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\Pi n1 < \infty , (3.7)
where
\Pi n1 =
\infty \int
0
\BbbP (M (1)
n \geq \lambda
\sqrt{}
2n\varphi (n) + x, \Delta n >
\surd
n/(\varphi (n))p)dx,
\Delta n is defined in (3.3).
Proof. Let \beta n = n\BbbE | X| I\{ | X| >
\surd
n/(\varphi (n))p\} , then | \BbbE
\sum k
i=1X
\prime
ni| \leq \beta n, 1 \leq k \leq n. Set
H = \{ n : \beta n \leq
\surd
n/(8(\varphi (n))2)\} , then we get\biggl\{
\Delta n >
\surd
n
(\varphi (n))p
\biggr\}
\subset
n\bigcup
j=1
\{ Xj \not = X \prime
nj\} , n \in H.
Paralleling the proof of Proposition 2.3 in [9] and then using Lemma 2.3, we obtain
\sum
n\in H
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\infty \int
0
\BbbP
\biggl(
M (1)
n \geq \lambda
\sqrt{}
2n\varphi (n) + x,\Delta n >
\surd
n
(\varphi (n))p
\biggr)
dx \leq
\leq
\sum
n\in H
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
n\BbbP
\biggl(
| X| >
\surd
n
(\varphi (n))p
\biggr) \infty \int
0
\BbbP
\biggl(
M (1)
n \geq \lambda
2
\sqrt{}
2n\varphi (n) + x
\biggr)
dx \leq
\leq
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
\surd
n
\BbbP
\biggl(
| X| >
\surd
n
(\varphi (n))p
\biggr) \infty \int
0
\biggl(
\mathrm{e}\mathrm{x}\mathrm{p}
\Biggl\{
-
(\lambda
\sqrt{}
2n\varphi (n)/2 + x)2
8Bn
\Biggr\}
+
+C
\biggl(
x+
\lambda
2
\sqrt{}
2n\varphi (n)
\biggr) - q n\sum
j=1
\BbbE | X(1)
nj |
q
\biggr)
dx =:
=:
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
\surd
n
\BbbP
\biggl(
| X| >
\surd
n
(\varphi (n))p
\biggr)
(Fn1 + Fn2).
On one hand, since
\int \infty
x
1\surd
2\pi
e - t2/2dt \leq e - x2/2/2 holds for x \geq 0, it follows that
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
\surd
n
\BbbP
\biggl(
| X| >
\surd
n
(\varphi (n))p
\biggr)
Fn1 =
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
248 X.-Y. XIAO, H.-W. YIN
=
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
\surd
n
\BbbP
\biggl(
| X| >
\surd
n
(\varphi (n))p
\biggr)
2
\sqrt{}
Bn
\infty \int
\lambda
\surd
2n\varphi (n)/(4
\surd
Bn)
\mathrm{e}\mathrm{x}\mathrm{p}\{ - y2/2\} dy \leq
\leq C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- \lambda 2n\varphi (n)
16Bn
\biggr\}
\BbbE I
\biggl\{
| X| >
\surd
n
(\varphi (n))p
\biggr\}
\leq
\leq C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1 - \lambda 2/16
\infty \sum
j=n
\BbbE I
\biggl\{ \surd
j
(\varphi (j))p
< | X| \leq
\surd
j + 1
(\varphi (j + 1))p
\biggr\}
=
= C
\infty \sum
j=1
\BbbE I
\biggl\{ \surd
j
(\varphi (j))p
< | X| \leq
\surd
j + 1
(\varphi (j + 1))p
\biggr\} j\sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1 - \lambda 2/16 \leq
\leq C
\infty \sum
j=1
\BbbE I
\biggl\{ \surd
j
(\varphi (j))p
< | X| \leq
\surd
j + 1
(\varphi (j + 1))p
\biggr\}
j(\mathrm{l}\mathrm{o}\mathrm{g} j)r - 1 - \lambda 2/16 \leq
\leq C\BbbE X2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )r - 1 - \lambda 2/16(\varphi (| X| ))2p < \infty .
On the other hand, applying Markov’s inequality, we have
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
\surd
n
\BbbP
\biggl(
| X| >
\surd
n
(\varphi (n))p
\biggr)
Fn2 \leq
\leq C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
\surd
n
(\varphi (n))2p\BbbE X2
n
\infty \int
\lambda
\surd
2n\varphi (n)/2
x - q
n\sum
j=1
\BbbE | X(1)
nj |
qdx \leq
\leq C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1(\varphi (n))2p
n3/2
(n\varphi (n))(1 - q)/2n\BbbE | X| qI
\biggl\{
| X| \leq
\surd
n
(\varphi (n))p
\biggr\}
\leq
\leq C
\infty \sum
n=1
(\varphi (n))2p+(1 - q)/2
nq/2(\mathrm{l}\mathrm{o}\mathrm{g} n)1 - r
n\sum
j=1
\BbbE | X| qI
\biggl\{ \surd
j - 1
(\varphi (j - 1))p
< | X| \leq
\surd
j
(\varphi (j))p
\biggr\}
=
= C
\infty \sum
j=1
\BbbE | X| qI
\biggl\{ \surd
j - 1
(\varphi (j - 1))p
< | X| \leq
\surd
j
(\varphi (j))p
\biggr\} \infty \sum
n=j
(\varphi (n))2p+(1 - q)/2
nq/2(\mathrm{l}\mathrm{o}\mathrm{g} n)1 - r
\leq
\leq C
\infty \sum
j=1
\BbbE | X| qI
\biggl\{ \surd
j - 1
(\varphi (j - 1))p
< | X| \leq
\surd
j
(\varphi (j))p
\biggr\}
j1 - q/2(\varphi (j))2p+(1 - q)/2
(\mathrm{l}\mathrm{o}\mathrm{g} j)1 - r
\leq
\leq C\BbbE X2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )r - 1(\varphi (| X| ))2p - (q - 2)(p+1/2) - 1/2 < \infty ,
if 2p - (q - 2)(p+ 1/2) \leq 0, or equivalently q \geq 2 + 4p/(2p+ 1). Then we get\sum
n\in H
n - 3/2(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1\Pi n1 < \infty .
If n /\in H, that is to say
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
EXACT RATES IN THE DAVIS – GUT LAW OF ITERATED LOGARITHM . . . 249
\BbbE | X| I
\biggl\{
| X| >
\surd
n
(\varphi (n))p
\biggr\}
>
1
8
\surd
n(\varphi (n))2
. (3.8)
By Lemma 2.3, we obtain
\sum
n/\in H
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\Pi n1 \leq
\sum
n/\in H
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\infty \int
0
\BbbP
\Bigl(
M (1)
n \geq \lambda
\sqrt{}
2n\varphi (n) + x
\Bigr)
dx \leq
\leq
\sum
n/\in H
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\infty \int
0
\Biggl(
\mathrm{e}\mathrm{x}\mathrm{p}
\Biggl\{
-
(\lambda
\sqrt{}
2n\varphi (n) + x)2
8Bn
\Biggr\}
+
+C
\biggl(
x+
\lambda
2
\sqrt{}
2n\varphi (n)
\biggr) - q n\sum
j=1
\BbbE | X(1)
nj |
q
\right) dx =:
=:
\sum
n/\in H
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
(Gn1 +Gn2).
While from (3.8), we have\sum
n/\in H
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
Gn1 \leq C
\sum
n/\in H
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\sqrt{}
Bn \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- \lambda 2n\varphi (n)
4Bn
\biggr\}
\leq
\leq C
\sum
n/\in H
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1 - \lambda 2/4
n
\surd
n(\varphi (n))2\BbbE | X| I
\biggl\{
| X| >
\surd
n
(\varphi (n))p
\biggr\}
\leq
\leq C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1 - \lambda 2/4
\surd
n(\varphi (n)) - 2
\infty \sum
j=n
\BbbE | X| I
\biggl\{ \surd
j
(\varphi (j))p
< | X| \leq
\surd
j + 1
(\varphi (j + 1))p
\biggr\}
=
= C
\infty \sum
j=1
\BbbE | X| I
\biggl\{ \surd
j
(\varphi (j))p
< | X| \leq
\surd
j + 1
(\varphi (j + 1))p
\biggr\} j\sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1 - \lambda 2/4
\surd
n(\varphi (n)) - 2
\leq
\leq C
\infty \sum
j=1
\BbbE | X| I
\biggl\{ \surd
j
(\varphi (j))p
< | X| \leq
\surd
j + 1
(\varphi (j + 1))p
\biggr\} \surd
j(\mathrm{l}\mathrm{o}\mathrm{g} j)r - 1 - \lambda 2/4
(\varphi (j)) - 2
\leq
\leq C\BbbE X2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )r - 1 - \lambda 2/4(\varphi (| X| ))2+p < \infty
and \sum
n/\in H
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
Gn2 \leq C
\sum
n/\in H
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n1/2
(n\varphi (n))(1 - q)/2\BbbE | X| qI
\biggl\{
| X| \leq
\surd
n
(\varphi (n))p
\biggr\}
\leq
\leq C
\infty \sum
n=1
(\varphi (n))(1 - q)/2
nq/2(\mathrm{l}\mathrm{o}\mathrm{g} n)1 - r
n\sum
j=1
\BbbE | X| qI
\biggl\{ \surd
j - 1
(\varphi (j - 1))p
< | X| \leq
\surd
j
(\varphi (j))p
\biggr\}
=
= C
\infty \sum
j=1
\BbbE | X| qI
\biggl\{ \surd
j - 1
(\varphi (j - 1))p
< | X| \leq
\surd
j
(\varphi (j))p
\biggr\} \infty \sum
n=j
(\varphi (n))(1 - q)/2
nq/2(\mathrm{l}\mathrm{o}\mathrm{g} n)1 - r
\leq
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
250 X.-Y. XIAO, H.-W. YIN
\leq C
\infty \sum
j=1
\BbbE | X| qI
\biggl\{ \surd
j - 1
(\varphi (j - 1))p
< | X| \leq
\surd
j
(\varphi (j))p
\biggr\}
j1 - q/2(\varphi (j))(1 - q)/2
(\mathrm{l}\mathrm{o}\mathrm{g} j)1 - r
\leq
\leq C\BbbE X2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )r - 1(\varphi (| X| )) - (q - 2)(p+1/2) - 1/2 < \infty .
Proposition 3.3 is proved.
Proposition 3.4. For any \lambda > 0, we have
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\Pi n2 < \infty , (3.9)
where \Pi n2 =
\int \infty
0
\BbbP (Mn \geq \lambda
\sqrt{}
2n\varphi (n) + x, \Delta n >
\surd
n/(\varphi (n))p)dx, \Delta n is defined in (3.3).
Proof. It holds that
\BbbP
\biggl(
Mn \geq \lambda
\sqrt{}
2n\varphi (n) + x, \Delta n >
\surd
n
(\varphi (n))p
\biggr)
\leq
\leq \BbbP
\biggl(
M (1)
n \geq \lambda
3
\sqrt{}
2n\varphi (n) +
x
3
,\Delta n >
\surd
n
(\varphi (n))p
\biggr)
+
+\BbbP
\biggl(
M (2)
n \geq \lambda
3
\sqrt{}
2n\varphi (n) +
x
3
\biggr)
+ \BbbP
\biggl(
M (3)
n \geq \lambda
3
\sqrt{}
2n\varphi (n) +
x
3
\biggr)
.
From (1.3), we have
B(2)
n :=
n\sum
k=1
\mathrm{V}\mathrm{a}\mathrm{r}X
(2)
nk \leq n\BbbE X2I
\biggl\{
| X| >
\surd
n
(\varphi (n))p
\biggr\}
= o
\biggl(
n
\varphi (n)
\biggr)
.
Then applying Lemma 2.3, we obtain
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\infty \int
0
\BbbP
\biggl(
M (2)
n \geq \lambda
3
\sqrt{}
2n\varphi (n) +
x
3
\biggr)
dx \leq
\leq C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\infty \int
0
\Bigl(
x+ \lambda
\sqrt{}
2n\varphi (n)
\Bigr) - q
n\BbbE | X| qI\{ | X| \leq
\sqrt{}
n\varphi (n)\} dx+
+C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\infty \int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\Biggl\{
-
(\lambda
\sqrt{}
2n\varphi (n) + x)2
72B
(2)
n
\Biggr\}
dx \leq
\leq C
\infty \sum
n=1
n - q/2(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
(\varphi (n))(q - 1)/2
n\sum
j=1
\BbbE | X| qI\{
\sqrt{}
(j - 1)\varphi (j - 1) < | X| \leq
\sqrt{}
j\varphi (j)\} +
+C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n
\mathrm{e}\mathrm{x}\mathrm{p}
\Biggl\{
- \lambda 2n\varphi (n)
36B
(2)
n
\Biggr\}
(< \infty ) \leq
\leq C
\infty \sum
j=1
\BbbE | X| qI\{
\sqrt{}
(j - 1)\varphi (j - 1) < | X| \leq
\sqrt{}
j\varphi (j)\}
\infty \sum
n=j
n - q/2(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
(\varphi (n))(q - 1)/2
\leq
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
EXACT RATES IN THE DAVIS – GUT LAW OF ITERATED LOGARITHM . . . 251
\leq C
\infty \sum
j=1
\BbbE | X| qI
\Bigl\{ \sqrt{}
(j - 1)\varphi (j - 1) < | X| \leq
\sqrt{}
j\varphi (j)
\Bigr\} j1 - q/2(\mathrm{l}\mathrm{o}\mathrm{g} j)r - 1
(\varphi (j))(q - 1)/2
\leq
\leq C\BbbE X2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )r - 1(\varphi (| X| )) - 1/2 < \infty ,
where we have used the fact that
\mathrm{e}\mathrm{x}\mathrm{p}
\Biggl\{
- \lambda 2n\varphi (n)
36B
(2)
n
\Biggr\}
= (\mathrm{l}\mathrm{o}\mathrm{g} n) - \lambda 2n/(36B
(2)
n )
with n/B
(2)
n \rightarrow +\infty as n \rightarrow +\infty .
Since \BbbE M (3)
n \leq 2n\BbbE | X| I\{ | X| >
\sqrt{}
n\varphi (n)\} , it follows that
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\infty \int
0
\BbbP
\biggl(
M (3)
n \geq \lambda
3
\sqrt{}
2n\varphi (n) +
x
3
\biggr)
dx =
=
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
3\BbbE
\biggl\{
M (3)
n - \lambda
3
\sqrt{}
2n\varphi (n)
\biggr\}
+
\leq
\leq C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
n\BbbE | X| I\{ | X| >
\sqrt{}
n\varphi (n)\} =
= C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
\surd
n
\infty \sum
j=n
\BbbE | X| I\{
\sqrt{}
j\varphi (j) < | X| \leq
\sqrt{}
(j + 1)\varphi (j + 1)\} =
= C
\infty \sum
j=1
\BbbE | X| I\{
\sqrt{}
j\varphi (j) < | X| \leq
\sqrt{}
(j + 1)\varphi (j + 1)\}
j\sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
\surd
n
\leq
\leq C
\infty \sum
j=1
\BbbE | X| I\{
\sqrt{}
j\varphi (j) < | X| \leq
\sqrt{}
(j + 1)\varphi (j + 1)\}
\sqrt{}
j(\mathrm{l}\mathrm{o}\mathrm{g} j)r - 1 \leq
\leq C\BbbE X2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )r - 1(\varphi (| X| )) - 1/2 < \infty .
Then from Proposition, 3.3, we get the desired result.
Proof of the direct part of Theorem 1.1. From Propositions 3.2 – 3.4, we have for large n,
\BbbE
\Bigl\{
Mn - (\varepsilon + an(\varepsilon ))
\sqrt{}
2Bn\varphi (n)
\Bigr\}
+
=
=
\infty \int
0
\BbbP
\biggl(
Mn \geq (\varepsilon + an(\varepsilon ))
\sqrt{}
2Bn\varphi (n) + x,\Delta n \leq
\surd
n
(\varphi (n))p
\biggr)
dx+
+
\infty \int
0
\BbbP
\biggl(
Mn \geq (\varepsilon + an(\varepsilon ))
\sqrt{}
2Bn\varphi (n) + x,\Delta n >
\surd
n
(\varphi (n))p
\biggr)
dx \leq
\leq
\infty \int
0
\BbbP
\biggl(
M (1)
n \geq (\varepsilon + an(\varepsilon ))
\sqrt{}
2Bn\varphi (n) -
\surd
n
(\varphi (n))p
+ x
\biggr)
dx+
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
252 X.-Y. XIAO, H.-W. YIN
+
\infty \int
0
\BbbP
\biggl(
Mn \geq \varepsilon
2
\sqrt{}
2n\varphi (n) + x,\Delta n >
\surd
n
(\varphi (n))p
\biggr)
dx \leq
\leq \BbbE
\biggl\{
M (1)
n -
\sqrt{}
Bn
\biggl(
(\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n) - 2
(\varphi (n))p
\biggr) \biggr\}
+
+\Pi n2 \leq
\leq
\sqrt{}
Bn\BbbE
\biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
0\leq s\leq 1
| W (s)| - (\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n) +
2
(\varphi (n))p
+
1
(\varphi (n))p\prime
\biggr\}
+
+ pn +\Pi n2
and
\BbbE
\Bigl\{
Mn - (\varepsilon + an(\varepsilon ))
\sqrt{}
2Bn\varphi (n)
\Bigr\}
+
\geq
\geq
\infty \int
0
\BbbP
\biggl(
Mn \geq (\varepsilon + an(\varepsilon ))
\sqrt{}
2Bn\varphi (n) + x,\Delta n \leq
\surd
n
(\varphi (n))p
\biggr)
dx \geq
\geq
\infty \int
0
\BbbP
\biggl(
M (1)
n \geq (\varepsilon + an(\varepsilon ))
\sqrt{}
2Bn\varphi (n) +
\surd
n
(\varphi (n))p
+ x,\Delta n \leq
\surd
n
(\varphi (n))p
\biggr)
dx \geq
\geq
\infty \int
0
\BbbP
\biggl(
M (1)
n \geq (\varepsilon + an(\varepsilon ))
\sqrt{}
2Bn\varphi (n) +
\surd
n
(\varphi (n))p
+ x
\biggr)
dx -
-
\infty \int
0
\BbbP
\biggl(
M (1)
n \geq \varepsilon
2
\sqrt{}
2n\varphi (n) + x,\Delta n >
\surd
n
(\varphi (n))p
\biggr)
dx \geq
\geq \BbbE
\biggl\{
M (1)
n -
\sqrt{}
Bn
\biggl(
(\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n) +
2
(\varphi (n))p
\biggr) \biggr\}
+
- \Pi n1 \geq
\geq
\sqrt{}
Bn\BbbE
\biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
0\leq s\leq 1
| W (s)| - (\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n) - 2
(\varphi (n))p
- 1
(\varphi (n))p\prime
\biggr\}
+
-
- pn - \Pi n1,
where \Pi n1 and \Pi n2 are defined in Propositions 3.3 and 3.4 respectively with \lambda = \varepsilon /2.
Therefore, for sufficiently large n, we get\sqrt{}
Bn\BbbE
\biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
0\leq s\leq 1
| W (s)| - (\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n) - 3
(\varphi (n))p\prime
\biggr\}
+
- pn - \Pi n1 \leq
\leq \BbbE
\Bigl\{
Mn - (\varepsilon + an(\varepsilon ))
\sqrt{}
2Bn\varphi (n)
\Bigr\}
+
\leq
\leq
\sqrt{}
Bn\BbbE
\biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
0\leq s\leq 1
| W (s)| - (\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n) +
3
(\varphi (n))p\prime
\biggr\}
+
+ pn +\Pi n2.
Similarly, for n large enough, it holds that\sqrt{}
Bn\BbbE
\biggl\{
| N | - (\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n) - 3
(\varphi (n))p\prime
\biggr\}
+
- pn - \Pi n1 \leq
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EXACT RATES IN THE DAVIS – GUT LAW OF ITERATED LOGARITHM . . . 253
\leq \BbbE
\Bigl\{
| Sn| - (\varepsilon + an(\varepsilon ))
\sqrt{}
2Bn\varphi (n)
\Bigr\}
+
\leq
\leq
\sqrt{}
Bn\BbbE
\biggl\{
| N | - (\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n) +
3
(\varphi (n))p\prime
\biggr\}
+
+ pn +\Pi n2.
While by Propositions 3.2 – 3.4, for 1/2 < p\prime < p \leq 2 and q \geq 2 + \mathrm{m}\mathrm{a}\mathrm{x}\{ (p\prime + 1/2)/(p - p\prime ),
4p/(2p+ 1)\} , we have
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
(pn +\Pi n1 +\Pi n2) < \infty .
Next, let a\prime n(\varepsilon ) = an(\varepsilon )\pm 3/(
\surd
2(\varphi (n))p
\prime +1/2), then we get
a\prime n(\varepsilon )\varphi (n) \rightarrow \tau , as n \rightarrow \infty and \varepsilon \searrow
\surd
r.
Since Bn/n \rightarrow 1 as n \rightarrow \infty , then by Proposition 3.1, we obtain
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\times
\times
\sqrt{}
Bn\BbbE
\biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
0\leq s\leq 1
| W (s)| - (\varepsilon + an(\varepsilon ))
\sqrt{}
2\varphi (n)\pm 3
(\varphi (n))p\prime
\biggr\}
+
=
= \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n
\BbbE
\biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
0\leq s\leq 1
| W (s)| - (\varepsilon + a\prime n(\varepsilon ))
\sqrt{}
2\varphi (n)
\biggr\}
+
=
=
2e - 2\tau
\surd
r
\surd
2\pi r
.
It follows that
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\BbbE
\Bigl\{
Mn - (\varepsilon + an(\varepsilon ))
\sqrt{}
2Bn\varphi (n)
\Bigr\}
+
=
2e - 2\tau
\surd
r
\surd
2\pi r
. (3.10)
Similarly, it holds that
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\BbbE
\Bigl\{
| Sn| - (\varepsilon + an(\varepsilon ))
\sqrt{}
2Bn\varphi (n)
\Bigr\}
+
=
e - 2\tau
\surd
r
\surd
2\pi r
. (3.11)
Finally, since \BbbE X2I\{ | X| \geq t\} = o((\varphi (t)) - 1) as t \rightarrow \infty , it follows that
0 \leq n - Bn \leq 2n\BbbE X2I
\biggl\{
| X| >
\surd
n
(\varphi (n))p
\biggr\}
= o
\biggl(
n
\varphi (n)
\biggr)
.
Let
a\ast n(\varepsilon ) =
\sqrt{}
Bn
n
(\varepsilon + an(\varepsilon )) - \varepsilon =
\sqrt{}
Bn
n
an(\varepsilon ) -
\varepsilon (n - Bn)\surd
n(
\surd
n+
\surd
Bn)
.
Then
a\ast n(\varepsilon ) = an(\varepsilon ) + o
\biggl(
1
\varphi (n)
\biggr)
, as n \rightarrow \infty , \varepsilon \searrow
\surd
r,
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
254 X.-Y. XIAO, H.-W. YIN
and
\BbbE
\Bigl\{
Mn - (\varepsilon + an(\varepsilon ))
\sqrt{}
2Bn\varphi (n)
\Bigr\}
+
= \BbbE
\Bigl\{
Mn - (\varepsilon + a\ast n(\varepsilon ))
\sqrt{}
2n\varphi (n)
\Bigr\}
+
,
\BbbE
\Bigl\{
| Sn| - (\varepsilon + an(\varepsilon ))
\sqrt{}
2Bn\varphi (n)
\Bigr\}
+
= \BbbE
\Bigl\{
| Sn| - (\varepsilon + a\ast n(\varepsilon ))
\sqrt{}
2n\varphi (n)
\Bigr\}
+
.
Therefore, (1.5) and (1.4) follow from (3.10) and (3.11), respectively.
Proof of the converse part of Theorem 1.1. First we prove \BbbE X = 0. From (1.2), we get, for
all \varepsilon >
\surd
r, that
\infty >
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\BbbE
\Bigl\{
| Sn| - (\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n)
\Bigr\}
+
=
=
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\infty \int
0
\BbbP
\Bigl\{
| Sn| \geq (\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n) + x
\Bigr\}
dx \geq
\geq
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\tau (n)\int
0
\BbbP
\Bigl\{
| Sn| \geq (\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n) + x
\Bigr\}
dx \geq
\geq
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\tau (n)\int
0
\BbbP
\Bigl\{
| Sn| \geq (\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n) + \tau (n)
\Bigr\}
dx \geq
\geq C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r
n
\BbbP
\Bigl\{
| Sn| \geq \varepsilon \sigma
\surd
2n \mathrm{l}\mathrm{o}\mathrm{g} n
\Bigr\}
\geq
\geq C
\infty \sum
n=1
1
n
\BbbP \{ | Sn| \geq n\} , (3.12)
where \tau (n) = \sigma \varepsilon
\surd
n \mathrm{l}\mathrm{o}\mathrm{g} n, then we have \BbbE X = 0.
Now we show \BbbE X2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )r - 1(\varphi (| X| )) - 1/2 < \infty . From (3.12) and Lemma 2.4, we take
\tau (n) = a\sigma
\sqrt{}
2n\varphi (n) with any a > 0 and \varepsilon \prime = \varepsilon + a >
\surd
r, then
\infty > C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
\sqrt{}
\varphi (n)
n
\BbbP
\Bigl\{
| Sn| \geq (\varepsilon \prime + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n)
\Bigr\}
\geq
\geq C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
\sqrt{}
\varphi (n)
n
\BbbP
\Bigl\{
| Sn| \geq 2\varepsilon \prime \sigma
\sqrt{}
2n\varphi (n)
\Bigr\}
=
= C1 + C
\infty \sum
n=N0
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
\sqrt{}
\varphi (n)
n
\BbbP
\Bigl\{
| Sn| \geq 2\varepsilon \prime \sigma
\sqrt{}
2n\varphi (n)
\Bigr\}
\geq
\geq C
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
\sqrt{}
\varphi (n)\BbbP
\Bigl\{
| X| \geq T
\sqrt{}
n\varphi (n)
\Bigr\}
(T \geq 8
\surd
2\varepsilon \prime \sigma ) =
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
EXACT RATES IN THE DAVIS – GUT LAW OF ITERATED LOGARITHM . . . 255
= C
\infty \sum
n=1
\sqrt{}
\varphi (n)
(\mathrm{l}\mathrm{o}\mathrm{g} n)1 - r
\infty \sum
j=n
\BbbE
\biggl(
I
\biggl\{ \sqrt{}
j\varphi (j) \leq | X|
T
<
\sqrt{}
(j + 1)\varphi (j + 1)
\biggr\} \biggr)
\geq
\geq C
\infty \sum
j=1
j
\sqrt{}
\varphi (j)
(\mathrm{l}\mathrm{o}\mathrm{g} j)1 - r
\BbbE
\biggl(
I
\biggl\{ \sqrt{}
j\varphi (j) \leq | X|
T
<
\sqrt{}
(j + 1)\varphi (j + 1)
\biggr\} \biggr)
\geq
\geq C\BbbE X2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )r - 1(\varphi (| X| )) - 1/2.
Next we prove \BbbE X2 < \infty . Let \{ \~X, \~Xn, n \geq 1\} be the symmetrization of \{ X,Xn, n \geq 1\} . Set
\~Sn =
\sum n
i=1
\~Xi, then, for large n and \varepsilon >
\surd
r, it holds that
\BbbE
\Bigl\{
| \~Sn| - 2(\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n)
\Bigr\}
+
\leq 2\BbbE
\Bigl\{
| Sn| - (\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n)
\Bigr\}
+
.
By (1.2), we have
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\BbbE
\Bigl\{
| \~Sn| - 2(\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n)
\Bigr\}
+
\leq 2\sigma e - 2\tau
\surd
r
\surd
2\pi r
.
We define Y = \~XI\{ | \~X| < K\} and Yn = \~XnI\{ | \~Xn| < K\} for K > 0. Notice that \~XI\{ | \~X| <
< K\} - \~XI\{ | \~X| \geq K\} has the same distribution as \~X, and 2Y = \~XI\{ | \~X| < K\} - \~XI\{ | \~X| \geq
\geq K\} + \~X, then by (1.2), we obtain
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\BbbE
\Biggl\{ \bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
i=1
2Yi
\bigm| \bigm| \bigm| \bigm| \bigm| - 4(\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n)
\Biggr\}
+
\leq
\leq 2 \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\BbbE
\Bigl\{
| \~Sn| - 2(\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n)
\Bigr\}
+
\leq
\leq 4\sigma e - 2\tau
\surd
r
\surd
2\pi r
.
Since | Y | < K, following the proof of the direct part, we have
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \searrow
\surd
r
1
- \mathrm{l}\mathrm{o}\mathrm{g}(\varepsilon 2 - r)
\infty \sum
n=1
(\mathrm{l}\mathrm{o}\mathrm{g} n)r - 1
n3/2
\BbbE
\Biggl\{ \bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
i=1
2Yi
\bigm| \bigm| \bigm| \bigm| \bigm| - 2(\varepsilon + an(\varepsilon ))
\sqrt{}
(\BbbE Y 2)2n\varphi (n)
\Biggr\}
+
=
=
2
\surd
\BbbE Y 2e - 2\tau
\surd
r
\surd
2\pi r
.
Therefore, \BbbE ( \~X2I\{ | \~X| < K\} ) = \BbbE Y 2 \leq 4\sigma 2. Then let K \rightarrow \infty , we get \BbbE X2 < \infty . At last,
following the proof of the direct part, we have \BbbE X2 = \sigma 2.
Finally, we show (1.6). Suppose that (1.6) is not true. Without loss of generality, we can assume
that \sigma - 2\BbbE X2I\{ | X| \geq
\surd
n/(\varphi (n))2\} \geq \tau 0/\varphi (n), for some \tau 0 > 0 and all n \geq 1. Then
n\sigma 2 - Bn \geq n\BbbE X2I
\biggl\{
| X| \geq
\surd
n
(\varphi (n))2
\biggr\}
\geq n\sigma 2\tau 0/\varphi (n).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
256 X.-Y. XIAO, H.-W. YIN
Let
a\ast n(\varepsilon ) =
\sqrt{}
1 +
\tau 0
\varphi (n)
(\varepsilon + an(\varepsilon )) - \varepsilon .
Then
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty ,\varepsilon \searrow
\surd
r
a\ast n(\varepsilon )\varphi (n) = \tau + \tau 0
\surd
r/2
and
\BbbE
\Bigl\{
Mn - (\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n)
\Bigr\}
+
\leq \BbbE
\Bigl\{
Mn - (\varepsilon + a\ast n(\varepsilon ))
\sqrt{}
2Bn\varphi (n)
\Bigr\}
+
,
\BbbE
\Bigl\{
| Sn| - (\varepsilon + an(\varepsilon ))\sigma
\sqrt{}
2n\varphi (n)
\Bigr\}
+
\leq \BbbE
\Bigl\{
| Sn| - (\varepsilon + a\ast n(\varepsilon ))
\sqrt{}
2Bn\varphi (n)
\Bigr\}
+
.
It follows that (1.4) and (1.5) are contradictory to (3.11) and (3.10), respectively.
By the way, the result of Corollary 1.1 is obvious from the proof of Theorem 1.1, and we omit
its proof.
References
1. Billingsley P. Convergence of probability measures. – New York: Wiley, 1968.
2. Csörgö M., Révész P. Strong approximations in probability and statistics. – New York: Acad. Press, 1981.
3. Davis J. A. Convergence rates for the law of the iterated logarithm // Ann. Math. Statist. – 1968. – 39. – P. 1479 – 1485.
4. Erdös P. On a theorem of hsu and robbins // Ann. Math. Statist. – 1949. – 20. – P. 261 – 291.
5. Erdös P. Remark on my paper “on a theorem of hsu and robbins” // Ann. Math. Statist. – 1950. – 21. – P. 138.
6. Gut A. Convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional
indices // Ann. Probab. – 1980. – 8. – P. 298 – 313.
7. Gut A., Spǎtaru A. Precise asymptotics in the law of the iterated logarithm // Ann. Probab. – 2000. – 28. – P. 1870 –
1883.
8. Hsu P. L., Robbins H. Complete convergence and the strong law of large numbers // Proc. Nat. Acad. Sci. USA. –
1947. – 33. – P. 25 – 31.
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Anal. and Appl. – 2007. – 327. – P. 695 – 714.
10. Jiang Y., Zhang L.-X., Pang T.-X. Precise rates in the law of iterated logarithm for the moment of i.i.d. random
variables // Acta Math. Sinica. English Ser. – 2006. – 22. – P. 781 – 792.
11. Li D., Wang X.-C., Rao M.-B. Some results on convergence rates for probabilities of moderate deviations for sums
of random variables // Int. J. Math. Sci. – 1992. – 15. – P. 481 – 498.
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13. Petrov V. V. Limit theorem of probability theory. – Oxford: Oxford Univ. Press, 1995.
Received 04.06.13,
after revision — 09.10.16
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
|
| id | umjimathkievua-article-1689 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:10:41Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/55/fa15de4c00b03e24fc440c0baedc9455.pdf |
| spelling | umjimathkievua-article-16892019-12-05T09:23:56Z Еxact rates in the Davis – Gut law of iterated logarithm for the first moment convergence of independent identically distributed random variables Точнi швидкостi в законi повторного логарифма Девiса–Гута для збiжностi першого моменту незалежних однаково розподiлених випадкових величин Xiao, X.-Y. Yin, H.-W. Сяо, Х.-І. Інь, Х.-В. Let $\{X, X_n, n \geq 1\}$ be a sequence of independent identically distributed random variables and let $S_n = \sum^n_{i=1} X_i$, $M_n = \max_{1\leq k\leq n} |S_k|$. For $r > 0$, let $a_n(\varepsilon)$ be a function of $\varepsilon$ such that $a_n(\varepsilon ) \mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} n \rightarrow \tau$ as $n \rightarrow \infty$ and $\varepsilon \searrow \surd r$. If $EX^2I\{|X| \geq t\} = o(\text{log}\text{log}t)^{-1})$ as $t \rightarrow \infty$ , then, by using the strong approximation, we show that $$\lim_{\varepsilon \searrow \surd r} \frac 1{-\text{log}(\varepsilon^2 - r)} \sum ^{\infty}_{n=1}\frac{(\text{log} n)^{r-1}}{n^{3/2}}E \Bigl\{ M_n - (\varepsilon + a_n(\varepsilon ))\sigma \sqrt{2n \text{log log} n} \Bigr\}_{+} = \frac{2\sigma \varepsilon^{-2\tau \sqrt{r}}}{\sqrt{2\pi}r}$$ holds if and only if $EX = 0, EX^2 = \sigma^2$, and $EX = 0, EX^2 = \sigma^2$ та $EX^2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )^{r-1}(\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} | X| )^{-\frac 12} < \infty$. Нехай $\{X, X_n, n \geq 1\}$ — множина незалежних однаково розподiлених випадкових величин та $S_n = \sum^n_{i=1} X_i$, $M_n = \max_{1\leq k\leq n} |S_k|$. Крiм того, для $r > 0$ нехай $a_n(\varepsilon)$ — функцiя $\varepsilon$ така, що $a_n(\varepsilon ) \mathrm{l}\mathrm{o}\mathrm{g}\; \mathrm{l}\mathrm{o}\mathrm{g}\; n \rightarrow \tau$ при $n \rightarrow \infty $та $\varepsilon \searrow \surd r$. У випадку $EX^2I\{|X| \geq t\} = o(\text{log}\text{log}t)^{-1})$ при $t \rightarrow \infty$ за допомогою сильної апроксимацiї доведено, що спiввiдношення $$\lim_{\varepsilon \searrow \surd r} \frac 1{-\text{log}(\varepsilon^2 - r)} \sum ^{\infty}_{n=1}\frac{(\text{log} n)^{r-1}}{n^{3/2}}E \Bigl\{ M_n - (\varepsilon + a_n(\varepsilon ))\sigma \sqrt{2n \text{log log} n} \Bigr\}_{+} = \frac{2\sigma \varepsilon^{-2\tau \sqrt{r}}}{\sqrt{2\pi}r}$$ виконуються тодi i тiльки тодi, коли $EX = 0, EX^2 = \sigma^2$ та $EX^2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )^{r-1}(\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} | X| )^{-\frac 12} < \infty$. Institute of Mathematics, NAS of Ukraine 2017-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1689 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 2 (2017); 240-256 Український математичний журнал; Том 69 № 2 (2017); 240-256 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1689/671 Copyright (c) 2017 Xiao X.-Y.; Yin H.-W. |
| spellingShingle | Xiao, X.-Y. Yin, H.-W. Сяо, Х.-І. Інь, Х.-В. Еxact rates in the Davis – Gut law of iterated logarithm for the first moment convergence of independent identically distributed random variables |
| title | Еxact rates in the Davis – Gut law of iterated logarithm for the first
moment convergence of independent identically distributed random variables |
| title_alt | Точнi швидкостi в законi повторного логарифма
Девiса–Гута для збiжностi першого моменту незалежних однаково розподiлених випадкових величин |
| title_full | Еxact rates in the Davis – Gut law of iterated logarithm for the first
moment convergence of independent identically distributed random variables |
| title_fullStr | Еxact rates in the Davis – Gut law of iterated logarithm for the first
moment convergence of independent identically distributed random variables |
| title_full_unstemmed | Еxact rates in the Davis – Gut law of iterated logarithm for the first
moment convergence of independent identically distributed random variables |
| title_short | Еxact rates in the Davis – Gut law of iterated logarithm for the first
moment convergence of independent identically distributed random variables |
| title_sort | еxact rates in the davis – gut law of iterated logarithm for the first
moment convergence of independent identically distributed random variables |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1689 |
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