On the strong law of large numbers for $\varphi$-subgaussian random variables

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 $\inf\{a\ge 0:\;\forall_{\lambda\in\mathbb{R}}\; \ln\mathbb{E}\exp(\lambda\xi)\le\varphi_p(a\lambda)\}$; $\tau_{\varphi_p}$ is a norm in a space $Sub_{\varphi_p}=\{\xi:\;\tau_{\varphi_p}(\xi)<\infty\}$ of $\varphi_p$-subgaussian random variables. We prove that if for a sequence $(\xi_n)\subset Sub_{\varphi_p}$ ($p>1$) there exist positive constants $c$ and $\alpha$ such that for every natural number $n$ the following inequality $\tau_{\varphi_p}(\sum_{i=1}^n\xi_i)\le cn^{1-\alpha}$ 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 SLLN for independent subgaussian random variables (Taylor and Hu \cite{TayHu}) to the case of dependent $\varphi_p$-subgaussian random variables.


Introduction
The classical Kolmogorov strong laws of large numbers are dealt with independent variables. Investigations of limit theorems for dependent r.v.s are extensive and episodic. The strong law of large numbers for various classes of many type associated random variables one can find for instance in Bulinski and Shashkin [2,Chap. 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 L p -spaces. In this paper we investigate almost sure convergence of the arithmetic mean (but not only) sequences of ϕ-subgaussian random variables.
The function ϕ p is an example of the quadratic N-function which is some standardization of the function |x| p (see [10,Lem. 2.5]). Let us emphasize that for p = 2 we have the case of subgaussian random variables.
Let us recall that the convex conjugate is order-reversing and possesses some scaling The convex conjugate of the cumulant generating function can be served to estimate of 'tails' distribution of a centered random variable. Let Eξ = 0 and ψ ξ denote the cumulant generating function of ξ, i.e. ψ ξ (λ) = ln E exp(λξ) then for ε > 0 ). Let us observe that for ξ ∈ Sub ϕ , by the definition of τ ϕ (ξ), we have the following inequality: ψ ξ (λ) ≤ ϕ(τ ϕ (ξ)λ) and by the order-reversing and the scaling property we get ψ * ξ (ε) ≥ ϕ * (ε/τ ϕ (ξ)). Now we can obtain some weaker form of the above estimation but with using the general function ϕ:

Results
First we show that if we have some upper estimate for τ ϕ then in (1) we can substitute this estimate instead of τ ϕ .
Proof. Since ϕ is even and increasing monotonic for x > 0, we get And again by the order-reversing and the scaling property we obtain which combined with (1) establishes the inequality.
With these preliminaries accounted for, we can prove the main result of the paper.
Theorem 2.2. Let (ξ n ) ⊂ Sub ϕp for some p > 1. If there exist positive constants c and α such that for every natural number n the following condition τ ϕp ( n i=1 ξ i ) ≤ cn 1−α holds then the term n −1 n i=1 ξ i converges almost surely to zero as n → ∞ .
Proof. Since ϕ * p = ϕ q , by Lemma 2.1 and the condition of the theorem we have For sufficiently large n (n > (c/ε) 1/α ) we have n α ε/c > 1 and, in consequence, Thus we get the following estimate for every ε and n > (c/ε) 1/α . Thus, by the integral test, we obtain convergence of the series ∞ n=1 P(| n i=1 ξ i | ≥ nε). It follows the completely and, in consequence, almost sure convergence of n −1 n i=1 ξ i to zero. For this reason we used a modified condition for a behavior of the norm τ p than Taylor and Hu, which I describe below.
Since τ ϕ is a norm, we obtain If for instance ξ i , i = 1, ..., n, are copies of the same variable ξ then in the above the equality holds and τ ϕ n i=1 ξ i = nτ ϕ (ξ). Let us observe that in this case the assumption of Theorem 2.2 is not satisfied. Additionally informations about form of dependence (or independence) sometime allow us to improve this estimate. And so, for an independence sequence (ξ n ) if there is some r ∈ (0, 2] such that ϕ(|x| 1/r ) is convex then (2) see [3, Sec.2, Th.5.2]. If r is bigger then the estimate is better. For the function ϕ p we can always take r = min{p, 2}. In Taylor's and Hu's SLLN variables ξ n were subgaussian and independent and it was taken p = 2. Let us emphasize that in this case if in addition ξ 1 , ..., ξ n have the same distribution as ξ then τ ϕ ( n i=1 ξ i ) ≤ √ nτ ϕ (ξ) and the condition of Theorem 2.2 is satisfied (c = τ ϕ (ξ) and α = 1/2). Let us emphasize that another assumptions on dependence of ξ 1 , ..., ξ n can give the same estimate of the norm of τ ϕ ( n i=1 ξ i ). In the paper Giuliano Antonini et al. [5,Lem.3] it was proved that for ϕ-subgaussian acceptable random variables the inequality (2) holds, if ϕ(|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 ϕ-subgaussian random variables as a corollary of much more general theorem. We give an independent proof of this corollary but under modified assumptions.
Proposition 2.4. For p > 1 let (ξ n ) be a bounded sequence of ϕ p -subgaussian random variables and let r = min{p, 2}. If in addition then n −1/s n i=1 ξ i → 0 almost surely for any 0 < s < r.
Remark 2.5. Since ϕ p (|x| 1/r ) is convex, the estimate (3) is satisfied by sequences of independent or acceptable random variables, for instance.
Remark 2.6. Because we apply the function ϕ 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,Cor. 7]).
Example 2.7. The proof of Hoeffding-Azuma's inequality for a sequence (ξ n ) of bounded random variables such that |ξ n | ≤ d n a.s. and Eξ n = 0 is based on an estimate of the moment generating function of the partial sum n i=1 ξ i . Under assumptions that ξ n are independent (Hoeffding) or ξ n are martingales increments (Azuma) the following inequality holds see Hoeffding [7] and Azuma [1]. Let us emphasize that in [1] Azuma has proved the above estimate under more general assumptions on (ξ n ) which satisfy centered bounded martingales increments. The inequality (4) means that If we take d n = 1 for n = 1, 2, ... then we get the following condition which follows that the sequence (ξ n ) satisfies the assumptions of Proposition 2.4 with p = r = 2 and the norm τ ϕ 2 (ξ n ) ≤ 1 and we get the almost sure convergence n −1/s n i=1 ξ i to 0 for any 0 < s < 2. Let us note that for s = 1 we obtain SLLN for this sequence.