On the Strong Law of Large Numbers for φ-Sub-Gaussian Random Variables

For p ≥ 1, let φ_p(x) = x²/2 if |x| ≤ 1 and φ_p(x) = 1/p|x|^p − 1/p + 1/2 if |x| > 1. For a random variable ξ, let ${\tau}_{\varphi_p}$ (ξ) denote inf {a ≥ 0 : ∀_λ∈ℝ ln 𝔼 exp(λξ) ≤ φ_p(aλ)}; ${\tau}_{\varphi_p}$ is a norm in is a norm in a space ${\mathrm{Sub}}_{\varphi_p}$ = {ξ : ${\tau}_{\varphi_p}$ (ξ) < ∞} of φ_p-sub-Gaussian random variables. We prove that if, for a sequence (ξ_n) ⊂ ${\mathrm{Sub}}_{\varphi_p}$, p > 1, there exist positive constants c and α such that, for every natural number n, the inequality ${\tau}_{\varphi_p}\left({\sum}_{i=1}^n{\xi}_i\right)\le {cn}^{1-\alpha }$ holds, then ${n}^{-1}{\sum}_{i=1}^n{\xi}_i$ converges almost surely to zero as n → ∞. This result is a generalization of the strong law of large numbers for independent sub-Gaussian random variables [see R. L. Taylor and T.-C. Hu, Amer. Math. Monthly, 94, 295 (1987)] to the case of dependent φ_p-sub-Gaussian random variables.