On the strong law of large numbers for ϕ-sub-Gaussian random variables
UDC 517.9 For $p\ge 1$ let $\varphi_p(x)=x^2/2$ if $|x|\le 1$ and $\varphi_p(x)=1/p|x|^p-1/p+1/2$ if $|x|>1.$  For a random variable $\xi$ let $\tau_{\varphi_p}(\xi)$ denote $\inf\{a\ge 0\colon \forall_{\lambda\in\mathbb{R}}$  $\ln\mathb...
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| author | Zajkowski, K. Zajkowski, K. |
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For $p\ge 1$ let $\varphi_p(x)=x^2/2$ if $|x|\le 1$ and $\varphi_p(x)=1/p|x|^p-1/p+1/2$ if $|x|>1.$  For a random variable $\xi$ let $\tau_{\varphi_p}(\xi)$ denote $\inf\{a\ge 0\colon \forall_{\lambda\in\mathbb{R}}$  $\ln\mathbb{E}\exp(\lambda\xi)\le\varphi_p(a\lambda)\};$ $\tau_{\varphi_p}$ is a norm in a space ${\rm Sub}_{\varphi_p}=\{\xi\colon \tau_{\varphi_p}(\xi)<\infty\}$ of $\varphi_p$-sub-Gaussian random variables. We prove that if for a sequence $(\xi_n)\subset {\rm Sub}_{\varphi_p},$  $p>1,$ there exist positive constants $c$ and $\alpha$ such that for every natural number $n$  the inequality $\tau_{\varphi_p} \Big(\sum_{i=1}^n\xi_i \Big)\le cn^{1-\alpha}$ holds, then  $n^{-1}\sum_{i=1}^n\xi_i$ converges almost surely to zero  as $n\to\infty.$ This result is a generalization of the strong law of large numbers for independent sub-Gaussian random variables [see   R. L. Taylor, T.-C. Hu, Sub-Gaussian techniques in proving strong laws of large numbers, Amer. Math. Monthly, 94, 295–299 (1987)] to the case of dependent $\varphi_p$-sub-Gaussian random variables. |
| doi_str_mv | 10.37863/umzh.v73i3.197 |
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DOI: 10.37863/umzh.v73i3.197
UDC 519.21
K. Zajkowski (Inst. Math., Univ. Bialystok, Poland)
ON THE STRONG LAW OF LARGE NUMBERS
FOR \bfitvarphi -SUB-GAUSSIAN RANDOM VARIABLES
ПРО ПОСИЛЕНИЙ ЗАКОН ВЕЛИКИХ ЧИСЕЛ
ДЛЯ \bfitvarphi -СУБГАУССОВИХ ВИПАДКОВИХ ВЕЛИЧИН
For p \geq 1 let \varphi p(x) = x2/2 if | x| \leq 1 and \varphi p(x) = 1/p| x| p - 1/p + 1/2 if | x| > 1. For a random variable \xi let
\tau \varphi p(\xi ) denote \mathrm{i}\mathrm{n}\mathrm{f}\{ a \geq 0 : \forall \lambda \in \BbbR \mathrm{l}\mathrm{n}\BbbE \mathrm{e}\mathrm{x}\mathrm{p}(\lambda \xi ) \leq \varphi p(a\lambda )\} ; \tau \varphi p is a norm in a space \mathrm{S}\mathrm{u}\mathrm{b}\varphi p = \{ \xi : \tau \varphi p(\xi ) < \infty \} of
\varphi p -sub-Gaussian random variables. We prove that if for a sequence (\xi n) \subset \mathrm{S}\mathrm{u}\mathrm{b}\varphi p , p > 1, there exist positive constants c
and \alpha such that for every natural number n the inequality \tau \varphi p
\Bigl( \sum n
i=1
\xi i
\Bigr)
\leq cn1 - \alpha holds, then n - 1
\sum n
i=1
\xi i converges
almost surely to zero as n \rightarrow \infty . This result is a generalization of the strong law of large numbers for independent sub-
Gaussian random variables [see R. L. Taylor, T.-C. Hu, Sub-Gaussian techniques in proving strong laws of large numbers,
Amer. Math. Monthly, 94, 295 – 299 (1987)] to the case of dependent \varphi p -sub-Gaussian random variables.
Нехай для p \geq 1 \varphi p(x) = x2/2, якщо | x| \leq 1, i \varphi p(x) = 1/p| x| p - 1/p + 1/2, якщо | x| > 1. Для випадкової
величини \xi нехай \tau \varphi p(\xi ) позначає \mathrm{i}\mathrm{n}\mathrm{f}\{ a \geq 0 : \forall \lambda \in \BbbR \mathrm{l}\mathrm{n}\BbbE \mathrm{e}\mathrm{x}\mathrm{p}(\lambda \xi ) \leq \varphi p(a\lambda )\} ; \tau \varphi p — норма у просторi \mathrm{S}\mathrm{u}\mathrm{b}\varphi p = \{ \xi :
\tau \varphi p(\xi ) < \infty \} \varphi p -субгауссових випадкових величин. У цiй роботi доведено наступне: якщо для послiдовностi
(\xi n) \subset \mathrm{S}\mathrm{u}\mathrm{b}\varphi p , p > 1, iснують додатнi сталi c i \alpha такi, що для будь-якого натурального числа n виконується
нерiвнiсть \tau \varphi p
\Bigl( \sum n
i=1
\xi i
\Bigr)
\leq cn1 - \alpha , то n - 1
\sum n
i=1
\xi i збiгається майже напевно до нуля при n \rightarrow \infty . Цей результат
узагальнює посилений закон великих чисел для незалежних субгауссових випадкових величин [див. R. L. Taylor,
T.-C. Hu, Sub-Gaussian techniques in proving strong laws of large numbers, Amer. Math. Monthly, 94, 295 – 299 (1987)]
у випадку, коли розглядаються залежнi \varphi p -субгауссовi випадковi величини.
1. Introduction. The classical Kolmogorov strong laws of large numbers are dealt with independent
variables. Investigations of limit theorems for dependent random variables are extensive and episodic.
The strong law of large numbers (SLLN) for various classes of many type associated random variables
one can find for instance in Bulinski and Shashkin [2] (Ch. 4). Most of them are considered in the
spaces of integrable functions. It is also interested to describe general conditions under which the
SLLN holds in other spaces of random variables than Lp-spaces. In this paper we investigate
almost sure convergence of the arithmetic mean (but not only) sequences of \varphi -sub-Gaussian random
variables.
The notion of sub-Gaussian random variables was introduced by Kahane in [8]. A random
variable \xi is sub-Gaussian if its moment generating function is majorized by the moment gene-
rating function of some centered Gaussian random variable with variance \sigma 2 that is \BbbE \mathrm{e}\mathrm{x}\mathrm{p}(\lambda \xi ) \leq
\leq \BbbE \mathrm{e}\mathrm{x}\mathrm{p}(\lambda g) = \mathrm{e}\mathrm{x}\mathrm{p}(\sigma 2\lambda 2/2), where g \sim \scrN (0, \sigma 2) (see [4] or [3], Ch. 1). In terms of the cumulant
generating functions this condition takes a form \mathrm{l}\mathrm{n}\BbbE \mathrm{e}\mathrm{x}\mathrm{p}(\lambda \xi ) \leq \sigma 2\lambda 2/2.
One can generalize the notion of sub-Gaussian random variables to classes of \varphi -sub-Gaussian
random variables (see [3], Ch. 2). A continuous even convex function \varphi (x) (x \in \BbbR ) is called a
N -function, if the following condition holds:
(a) \varphi (0) = 0 and \varphi (x) is monotone increasing for x > 0,
(b) \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow 0 \varphi (x)/x = 0 and \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow \infty \varphi (x)/x = \infty .
c\bigcirc K. ZAJKOWSKI, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3 431
432 K. ZAJKOWSKI
It is called a quadratic N -function, if in addition \varphi (x) = cx2 for all | x| \leq x0 with c > 0 and
x0 > 0. The quadratic condition is needed to ensure nontriviality for classes of \varphi -sub-Gaussian
random variables (see [3, p. 67]).
Example 1.1. Let for p \geq 1
\varphi p(x) =
\left\{
x2
2
, \mathrm{i}\mathrm{f} | x| \leq 1,
1
p
| x| p - 1
p
+
1
2
, \mathrm{i}\mathrm{f} | x| > 1.
The function \varphi p is an example of the quadratic N -function which is some standardization of the
function | x| p (see [10], Lemma 2.5). Let us emphasize that for p = 2 we have the case of sub-
Gaussian random variables.
Let \varphi be a quadratic N -function. A random variable \xi is said to be \varphi -sub-Gaussian if there is
a constant a > 0 such that \mathrm{l}\mathrm{n}\BbbE \mathrm{e}\mathrm{x}\mathrm{p}(\lambda \xi ) \leq \varphi (a\lambda ). The \varphi -sub-Gaussian standard (norm) \tau \varphi (\xi ) is
defined as
\tau \varphi (\xi ) = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
a \geq 0 : \forall \lambda \in \BbbR \mathrm{l}\mathrm{n}\BbbE \mathrm{e}\mathrm{x}\mathrm{p}(\lambda \xi ) \leq \varphi (a\lambda )
\bigr\}
;
a space \mathrm{S}\mathrm{u}\mathrm{b}\varphi = \{ \xi : \tau \varphi (\xi ) <\infty \} with the norm \tau \varphi is a Banach space (see [3], Ch. 2, Theorem 4.1)
Let \varphi (x), x \in \BbbR , be a real-valued function. The function \varphi \ast (y) (y \in \BbbR ) defined by \varphi \ast (y) =
= \mathrm{s}\mathrm{u}\mathrm{p}x\in \BbbR \{ xy - \varphi (x)\} is called the Young – Fenchel transform or the convex conjugate of \varphi (in
general, \varphi \ast may take value \infty ). It is known that if \varphi is a quadratic N -function, then \varphi \ast is quadratic
N -function too. For instance, since our \varphi p is a differentiable (even at \pm 1) function one can easy
check (see [10], Lemma 2.6) that \varphi \ast
p = \varphi q for p, q > 1, if 1/p+ 1/q = 1.
Remark 1.1. One can define the space \mathrm{S}\mathrm{u}\mathrm{b}\varphi p by using the Luxemburg norm of the form
\| \xi \| \psi q = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
K > 0 : \BbbE \mathrm{e}\mathrm{x}\mathrm{p} | \xi /K| q \leq 2
\bigr\}
, q = p/(p - 1),
and then \mathrm{S}\mathrm{u}\mathrm{b}\varphi p = \{ \xi : \| \xi \| \psi q < \infty \mathrm{a}\mathrm{n}\mathrm{d} \BbbE \xi = 0\} (compare [10], Theorem 2.7), the space L0
\psi q
=
= \mathrm{S}\mathrm{u}\mathrm{b}\varphi p ). Note that \| \BbbE \xi \| \psi q = \| \xi \| \psi q , and we get that if \| \xi \| \psi q <\infty , then \xi - \BbbE \xi \in \mathrm{S}\mathrm{u}\mathrm{b}\varphi p .
Example 1.2. The standard normal random variable g belongs to \mathrm{S}\mathrm{u}\mathrm{b}\varphi 2 and \tau \varphi 2(g) = 1, since
\BbbE \mathrm{e}\mathrm{x}\mathrm{p}(tg) = \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
t2/2
\bigr)
= \mathrm{e}\mathrm{x}\mathrm{p}(\varphi 2(t)).
Because g2 has \chi 2
1-distribution with one degree of freedom whose moment generating function is
\BbbE \mathrm{e}\mathrm{x}\mathrm{p}(tg) = (1 - 2t) - 1/2 for t < 1/2, then
\BbbE \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
g2/K2
\bigr)
=
\bigl(
1 - 2/K2
\bigr) - 1/2
,
which is less or equal 2 if K \geq
\sqrt{}
8/3. It gives that \| \xi \| \psi 2 =
\sqrt{}
8/3. Let us observe that \psi 2-norm
of | g| is equal to \psi 2-norm of g. It implies that | g| - \BbbE | g| \in \mathrm{S}\mathrm{u}\mathrm{b}\varphi 2 . Similarly as above one can show
that
\| | g| 2/q\| \psi q = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
K > 0; \BbbE \mathrm{e}\mathrm{x}\mathrm{p}(g2/Kq) \leq 2
\bigr\}
= (8/3)1/q <\infty .
Thus we get that | g| 2/q - \BbbE | g| 2/q \in \mathrm{S}\mathrm{u}\mathrm{b}\varphi p , where 1/p+ 1/q = 1.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ON THE STRONG LAW OF LARGE NUMBERS . . . 433
Let us recall that the convex conjugate is order-reversing and possesses some scaling property. If
\varphi 1 \geq \varphi 2, then \varphi \ast
1 \leq \varphi \ast
2. Let for a > 0 and b \not = 0 \psi (x) = a\varphi (bx), then \psi \ast (y) = a\varphi \ast (y/(ab)) (see,
e.g., [6], Ch. X, Proposition 1.3.1).
The convex conjugate of the cumulant generating function can be served to estimate of ‘tails’
distribution of a centered random variable. Let \BbbE \xi = 0 and \psi \xi denote the cumulant generating
function of \xi , i.e., \psi \xi (\lambda ) = \mathrm{l}\mathrm{n}\BbbE \mathrm{e}\mathrm{x}\mathrm{p}(\lambda \xi ), then for \varepsilon > 0
\BbbP (\xi \geq \varepsilon ) \leq \mathrm{e}\mathrm{x}\mathrm{p}( - \psi \ast
\xi (\varepsilon )).
Let us observe that for \xi \in \mathrm{S}\mathrm{u}\mathrm{b}\varphi , by the definition of \tau \varphi (\xi ), we have the inequality \psi \xi (\lambda ) \leq
\leq \varphi (\tau \varphi (\xi )\lambda ) and, by the order-reversing and the scaling property, we get \psi \ast
\xi (\varepsilon ) \geq \varphi \ast (\varepsilon /\tau \varphi (\xi )).
Now we can obtain some weaker form of the above estimation but with using the general func-
tion \varphi :
\BbbP (| \xi | \geq \varepsilon ) \leq 2 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
- \varphi \ast
\biggl(
\varepsilon
\tau \varphi (\xi )
\biggr) \biggr)
(1)
(see [3], Ch. 2, Lemma 4.3).
2. Results. First we show that if we have some upper estimate for \tau \varphi , then in (1) we can
substitute this estimate instead of \tau \varphi .
Lemma 2.1. If \tau \varphi (\xi ) \leq C(\xi ) for every \xi \in \mathrm{S}\mathrm{u}\mathrm{b}\varphi , then
\BbbP (| \xi | \geq \varepsilon ) \leq 2 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
- \varphi \ast
\biggl(
\varepsilon
C(\xi )
\biggr) \biggr)
.
Proof. Since \varphi is even and increasing monotonic for x > 0, we get
\varphi (\tau \varphi (\xi )x) = \varphi (\tau \varphi (\xi )| x| ) \leq \varphi (C(\xi )| x| ) = \varphi (C(\xi )x).
And again by the order-reversing and the scaling property we obtain
\varphi \ast
\biggl(
y
\tau \varphi (\xi )
\biggr)
\geq \varphi \ast
\biggl(
y
C(\xi )
\biggr)
,
which combined with (1) establishes the inequality.
With these preliminaries accounted for, we can prove the main result of the paper.
Theorem 2.1. Let (\xi n) \subset \mathrm{S}\mathrm{u}\mathrm{b}\varphi p for some p > 1. If there exist positive constants c and \alpha
such that for every natural number n the condition \tau \varphi p
\Bigl( \sum n
i=1
\xi i
\Bigr)
\leq cn1 - \alpha holds, then the term
n - 1
\sum n
i=1
\xi i converges almost surely to zero as n\rightarrow \infty .
Proof. Since \varphi \ast
p = \varphi q, by Lemma 2.1 and the condition of the theorem we have
\BbbP
\Biggl( \bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
i=1
\xi i
\bigm| \bigm| \bigm| \bigm| \bigm| \geq n\varepsilon
\Biggr)
\leq 2 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
- \varphi q
\biggl(
n\alpha \varepsilon
c
\biggr) \biggr)
.
For sufficiently large n (n > (c/\varepsilon )1/\alpha ) we have n\alpha \varepsilon /c > 1 and, in consequence,
\varphi q
\biggl(
n\alpha \varepsilon
c
\biggr)
= nq\alpha
1
q
\Bigl( \varepsilon
c
\Bigr) q
- 1
q
+
1
2
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
434 K. ZAJKOWSKI
Thus, we get the estimate
\BbbP
\Biggl( \bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
i=1
\xi i
\bigm| \bigm| \bigm| \bigm| \bigm| \geq n\varepsilon
\Biggr)
\leq 2 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
1
q
- 1
2
\biggr)
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
- nq\alpha 1
q
\Bigl( \varepsilon
c
\Bigr) q\biggr)
for every \varepsilon and n > (c/\varepsilon )1/\alpha . Thus, by the integral test, we obtain convergence of the series\sum \infty
n=1
\BbbP
\Bigl( \bigm| \bigm| \bigm| \sum n
i=1
\xi i
\bigm| \bigm| \bigm| \geq n\varepsilon
\Bigr)
. It follows the completely and, in consequence, almost sure conver-
gence of n - 1
\sum n
i=1
\xi i to zero.
Remark 2.1. Let us emphasize that the above theorem is a generalization of the theorem (SLLN)
(see [9, p. 297]) to the case of \varphi p-sub-Gaussian random variables, not only sub-Gaussian ones.
Moreover, we do not assume their independence. For this reason we used a modified condition for a
behavior of the norm \tau p than Taylor and Hu, which describes below.
Since \tau \varphi is a norm, we obtain
\tau \varphi
\Biggl(
n\sum
i=1
\xi i
\Biggr)
\leq
n\sum
i=1
\tau \varphi (\xi i).
If for instance \xi i, i = 1, . . . , n, are copies of the same variable \xi , then in the above the equality
holds and \tau \varphi
\Bigl( \sum n
i=1
\xi i
\Bigr)
= n\tau \varphi (\xi ). Let us observe that in this case the assumption of Theorem 2.1
is not satisfied. Additionally informations about form of dependence (or independence) sometime
allow us to improve this estimate. So, for an independence sequence (\xi n), if there is some r \in (0, 2]
such that \varphi (| x| 1/r) is convex, then
\tau \varphi
\Biggl(
n\sum
i=1
\xi i
\Biggr) r
\leq
n\sum
i=1
\tau \varphi (\xi i)
r (2)
(see [3], Sec. 2, Theorem 5.2). If r is bigger then the estimate is better. For the function \varphi p we
can always take r = \mathrm{m}\mathrm{i}\mathrm{n}\{ p, 2\} . In Taylor’s and Hu’s SLLN variables \xi n were sub-Gaussian and
independent and it was taken p = 2. Let us emphasize that in this case if, in addition, \xi 1, . . . , \xi n,
have the same distribution as \xi then \tau \varphi
\Bigl( \sum n
i=1
\xi i
\Bigr)
\leq
\surd
n\tau \varphi (\xi ) and the condition of Theorem 2.1 is
satisfied (c = \tau \varphi (\xi ) and \alpha = 1/2).
Let us emphasize that another assumptions on dependence of \xi 1, . . . , \xi n can give the same es-
timate of the norm of \tau \varphi
\Bigl( \sum n
i=1
\xi i
\Bigr)
. In the paper Giuliano Antonini et al. [5] (Lemma 3) it was
proved that for \varphi -sub-Gaussian acceptable random variables the inequality (2) holds, if \varphi (| x| 1/r) is
convex. The definition of acceptability of sequence of random variable one can find therein. For us
it is the most important that these estimates are the same. In this article there is some version of the
Marcinkiewicz – Zygmund law of large numbers for \varphi -sub-Gaussian random variables as a corollary
of much more general theorem. We give an independent proof of this corollary but under modified
assumptions.
Proposition 2.1. Let (\xi n), p > 1, be a bounded sequence of \varphi p-sub-Gaussian random variables
and let r = \mathrm{m}\mathrm{i}\mathrm{n}\{ p, 2\} . If, in addition,
\tau \varphi p
\Biggl(
n\sum
i=1
\xi i
\Biggr) r
\leq
n\sum
i=1
\tau \varphi p(\xi i)
r, (3)
then n - 1/s
\sum n
i=1
\xi i \rightarrow 0 almost surely for any 0 < s < r.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ON THE STRONG LAW OF LARGE NUMBERS . . . 435
Remark 2.2. Since \varphi p(| x| 1/r) is convex, the estimate (3) is satisfied by sequences of independent
or acceptable random variables, for instance.
Proof. Let b = \mathrm{s}\mathrm{u}\mathrm{p}n\geq 1 \tau \varphi p(\xi n), then
\sum n
i=1
\tau \varphi p(\xi i)
r \leq nbr and, in consequence,
\tau \varphi p
\Bigl( \sum n
i=1
\xi i
\Bigr)
\leq n1/rb. For positive number s less than r, by Lemma 2.1, we obtain
\BbbP
\Biggl( \bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
i=1
\xi i
\bigm| \bigm| \bigm| \bigm| \bigm| \geq n
1
s \varepsilon
\Biggr)
\leq 2 \mathrm{e}\mathrm{x}\mathrm{p}
\Biggl(
- \varphi q
\Biggl(
n1/s\varepsilon
n1/rb
\Biggr) \Biggr)
= 2 \mathrm{e}\mathrm{x}\mathrm{p}
\Bigl(
- \varphi q
\Bigl(
n(1/s - 1/r) \varepsilon
b
\Bigr) \Bigr)
.
For n > (b/\varepsilon )(1/s - 1/r) - 1
, we have
\varphi q
\Bigl(
n(1/s - 1/r) \varepsilon
b
\Bigr)
= nq(1/s - 1/r) 1
q
\Bigl( \varepsilon
b
\Bigr) q
- 1
q
+
1
2
and, in consequence,
\infty \sum
n=1
\mathrm{e}\mathrm{x}\mathrm{p}
\Bigl(
- \varphi q
\Bigl(
n(1/s - 1/r) \varepsilon
b
\Bigr) \Bigr)
<\infty ,
which, in view of Borel – Cantelli lemma, completes the proof.
Remark 2.3. Because we apply the function \varphi p(x) instead of | x| p, then we must not restrict p to
be less or equal 2 to ensure the fulfillment of the quadratic condition for the function | x| p. Moreover,
we use the metric property (3) instead of assumptions on some form of dependence random variables
(compare [5], Corollary 7).
Example 2.1. The proof of Hoeffding – Azuma’s inequality for a sequence (\xi n) of bounded ran-
dom variables such that | \xi n| \leq dn a.s. and \BbbE \xi n = 0 is based on an estimate of the moment generating
function of the partial sum
\sum n
i=1
\xi i. Under assumptions that \xi n are independent (Hoeffding) or \xi n
are martingales increments (Azuma) the following inequality holds:
\BbbE \mathrm{e}\mathrm{x}\mathrm{p}
\Biggl(
\lambda
n\sum
i=1
\xi i
\Biggr)
\leq \mathrm{e}\mathrm{x}\mathrm{p}
\left( \lambda 2\sum n
i=1
d2i
2
\right) (4)
(see [7, 1]). Let us emphasize that in [1] Azuma has proved the above estimate under more general
assumptions on (\xi n) which satisfy centered bounded martingales increments. The inequality (4)
means that
\tau \varphi 2
\Biggl(
n\sum
i=1
\xi i
\Biggr)
\leq
\Biggl(
n\sum
i=1
d2i
\Biggr) 1/2
.
If we take dn = 1 for n = 1, 2, . . . , then we get the condition
\tau \varphi 2
\Biggl(
n\sum
i=1
\xi i
\Biggr)
\leq
\surd
n,
which follows that the sequence (\xi n) satisfies the assumptions of Proposition 2.1 with p = r = 2
and the norm \tau \varphi 2(\xi n) \leq 1 and we get the almost sure convergence n - 1/s
\sum n
i=1
\xi i to 0 for any
0 < s < 2. Let us note that for s = 1 we obtain SLLN for this sequence.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
436 K. ZAJKOWSKI
References
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Providence, RI (2000).
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Received 11.07.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
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| id | umjimathkievua-article-197 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:04Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/94/aa1bf770296947f24206d1da3af9db94.pdf |
| spelling | umjimathkievua-article-1972025-03-31T08:48:21Z On the strong law of large numbers for ϕ-sub-Gaussian random variables On the strong law of large numbers for ϕ-sub-Gaussian random variables Zajkowski, K. Zajkowski, K. ϕ-subgaussian random variables strong law of large numbers ϕ-subgaussian random variables strong law of large numbers UDC 517.9 For $p\ge 1$ let $\varphi_p(x)=x^2/2$ if $|x|\le 1$ and $\varphi_p(x)=1/p|x|^p-1/p+1/2$ if $|x|&gt;1.$&nbsp;&nbsp;For a random variable $\xi$ let $\tau_{\varphi_p}(\xi)$ denote&nbsp;$\inf\{a\ge 0\colon \forall_{\lambda\in\mathbb{R}}$&nbsp; $\ln\mathbb{E}\exp(\lambda\xi)\le\varphi_p(a\lambda)\};$ $\tau_{\varphi_p}$ is a norm in a space ${\rm Sub}_{\varphi_p}=\{\xi\colon \tau_{\varphi_p}(\xi)&lt;\infty\}$ of $\varphi_p$-sub-Gaussian random variables. We prove that if for a sequence $(\xi_n)\subset {\rm Sub}_{\varphi_p},$&nbsp; $p&gt;1,$ there exist positive constants $c$ and $\alpha$ such that for every natural number $n$&nbsp;&nbsp;the inequality $\tau_{\varphi_p} \Big(\sum_{i=1}^n\xi_i \Big)\le cn^{1-\alpha}$ holds, then&nbsp; $n^{-1}\sum_{i=1}^n\xi_i$ converges almost surely to zero&nbsp; as $n\to\infty.$ This result is a generalization of the strong law of large numbers for independent sub-Gaussian random variables [see &nbsp; R. L. Taylor, T.-C. Hu, Sub-Gaussian techniques in proving strong laws of large numbers, Amer. Math. Monthly, 94, 295–299 (1987)] to the case of dependent $\varphi_p$-sub-Gaussian random variables. УДК 517.9Про посилений закон великих чисел для ϕ-субгаусових випадкових величин Нехай для $p\ge 1$&nbsp; $\varphi_p(x)=x^2/2$, якщо $|x|\le 1$, і $\varphi_p(x)=1/p|x|^p-1/p+1/2$, якщо $|x|&gt;1.$&nbsp;&nbsp;Для випадкової величини $\xi$ нехай $\tau_{\varphi_p}(\xi)$ позначає $\inf\{a\ge 0\colon \forall_{\lambda\in\mathbb{R}}$ $\ln\mathbb{E}\exp(\lambda\xi)\le\varphi_p(a\lambda)\};$ $\tau_{\varphi_p}$ --- норма у просторі ${\rm Sub}_{\varphi_p}=\{\xi\colon \tau_{\varphi_p}(\xi)&lt;\infty\}$ $\varphi_p$-субгауссових випадкових величин.&nbsp;&nbsp;У цій роботі доведено наступне: якщо для послідовності $(\xi_n)\subset {\rm Sub}_{\varphi_p},$ $p&gt;1,$ існують додатні сталі $c$ і $\alpha$ такі, що для будь-якого натурального числа $n$ виконується нерівність $\tau_{\varphi_p} \Big(\sum_{i=1}^n\xi_i \Big)\le cn^{1-\alpha}$, то&nbsp; $n^{-1}\sum_{i=1}^n\xi_i$ збігається майже напевно до нуля при $n\to\infty.$&nbsp;&nbsp;Цей результат узагальнює посилений закон великих чисел для незалежних субгауссових випадкових величин [див. R. L. Taylor, T.-C. Hu, Sub-Gaussian techniques in proving strong laws of large numbers, Amer. &nbsp;Math. Monthly, 94, 295–299 (1987)] у випадку, коли розглядаються залежні $\varphi_p$-субгауссові випадкові величини. Institute of Mathematics, NAS of Ukraine 2021-03-19 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/197 10.37863/umzh.v73i3.197 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 3 (2021); 431 - 436 Український математичний журнал; Том 73 № 3 (2021); 431 - 436 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/197/8986 Copyright (c) 2021 Krzysztof Zajkowski |
| spellingShingle | Zajkowski, K. Zajkowski, K. On the strong law of large numbers for ϕ-sub-Gaussian random variables |
| title | On the strong law of large numbers for ϕ-sub-Gaussian random variables |
| title_alt | On the strong law of large numbers for ϕ-sub-Gaussian random variables |
| title_full | On the strong law of large numbers for ϕ-sub-Gaussian random variables |
| title_fullStr | On the strong law of large numbers for ϕ-sub-Gaussian random variables |
| title_full_unstemmed | On the strong law of large numbers for ϕ-sub-Gaussian random variables |
| title_short | On the strong law of large numbers for ϕ-sub-Gaussian random variables |
| title_sort | on the strong law of large numbers for ϕ-sub-gaussian random variables |
| topic_facet | ϕ-subgaussian random variables strong law of large numbers ϕ-subgaussian random variables strong law of large numbers |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/197 |
| work_keys_str_mv | AT zajkowskik onthestronglawoflargenumbersforphsubgaussianrandomvariables AT zajkowskik onthestronglawoflargenumbersforphsubgaussianrandomvariables |