Title: On the Azuma inequality in spaces of subgaussian of rank $p$ random variables,
Abstract: For $p > 1$ let a function $\varphi_p(x) = x^2/2$ 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 $\inf\{c\ge 0 :\;
\forall_{\lambda\in\mathbb{R}}\;
\ln\mathbb{E}\exp(\lambda\xi)\le\varphi_p(c\lambda)\}$; $\tau_{\varphi_p}$ is a
norm in a space $Sub_{\varphi_p}(\Omega) =\{\xi:
\; \tau_{\varphi_p}(\xi) <\infty\}$ of $\varphi_p$-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.