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

Abstract

For $p>1$ let a function ${\phi }_p(x)=x^2/2$ if $|x|\le 1$ and ${\phi }_p(x)=1/p|x{|}^p-1/p+1/2$ if $|x|>1$. For a random variable $\xi$ let ${\tau }_{{\phi }_p}(\xi )$ denote $\inf \{c\ge 0:{\forall }_{\lambda \in \mathbb{R}}\ln \mathbb{E}\exp (\lambda \xi )\le {\phi }_p(c\lambda )\}$; ${\tau }_{{\phi }_p}$ is a norm in a space $Su{b}_{{\phi }_p}(\Omega )=\{\xi :{\tau }_{{\phi }_p}(\xi )<\infty \}$ of ${\phi }_p$-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 $({\xi }_n{)}_{n\ge 0}$ with bounded increments from zero. In its classic form is assumed that ${\xi }_0=0$. In this paper it is shown a version of the Azuma inequality under assumption that ${\xi }_0$ is any subgaussian of rank $p$ random variable.

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