Generalization of tail inequalities for random variables, using in the martingale theory

TL;DR

This paper extends Doob's tail inequality for non-negative random variables within martingale theory, applying to broader data sources and Grand Lebesgue Spaces, with examples demonstrating estimate accuracy.

Contribution

It introduces generalized tail inequalities for non-negative random variables in martingale theory, expanding applicability to more general data and function spaces.

Findings

Extended Doob's inequality to broader data sources.

Applied inequalities to Grand Lebesgue Spaces.

Provided examples confirming estimate precision.

Abstract

We generalize a famous tail Doob's inequality, relative two non-negative random variables, arising in the martingale theory, in two directions: on the more general source data and on the random variables belonging to the so-called Grand Lebesgue Spaces. We bring also several examples in each sections in order to show the exactness of our estimates.