On the Azuma inequality in spaces of subgaussian of rank $p$ random variables

Abstract

For $p > 1$ let a function $φ_p(x) = x^2/2$ if $|x|\le 1$ and $φ_p(x) = 1/p|x|^p -1/p + 1/2$ if $|x| > 1$. For a random variable $ξ$ let $τ_{φ_p}(ξ)$ denote $\inf\{c\ge 0 :\; \forall_{λ\in\mathbb{R}}\; \ln\mathbb{E}\exp(λξ)\leφ_p(cλ)\}$; $τ_{φ_p}$ is a norm in a space $Sub_{φ_p}(Ω) =\{ξ: \; τ_{φ_p}(ξ) <\infty\}$ of $φ_p$-subgaussian random variables which we call {\it subgaussian of rank $p$ random variables}. For $p = 2$ we have the classic subgaussian random variables. The Azuma inequality gives an estimate on the probability of the deviations of a zero-mean martingale $(ξ_n)_{n\ge 0}$ with bounded increments from zero. In its classic form is assumed that $ξ_0 = 0$. In this paper it is shown a version of the Azuma inequality under assumption that $ξ_0$ is any subgaussian of rank $p$ random variable.