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

## TL;DR

This paper extends the Azuma inequality to a broader class of martingales with initial conditions being subgaussian of rank p variables, generalizing the classic subgaussian case.

## Contribution

It introduces a version of the Azuma inequality for martingales starting with subgaussian of rank p initial variables, broadening its applicability.

## Key findings

- Provides a generalized Azuma inequality for subgaussian of rank p variables.
- Establishes bounds on deviations for martingales with non-zero initial conditions.
- Extends classical results to a wider class of 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.