Vector valued unified martingale and ergodic theorems with continuous parameter

TL;DR

This paper establishes new vector-valued martingale and ergodic theorems with continuous parameters, proving convergence, inequalities, and conditions under which these theorems coincide, advancing the theoretical understanding of stochastic processes.

Contribution

It introduces unified vector-valued martingale and ergodic theorems with continuous parameters, including convergence results and inequalities, which were not previously established.

Findings

Almost everywhere convergence of vector-valued martingales with continuous parameter.

Norm and almost everywhere convergence of averages.

Dominant and maximal inequalities for these processes.

Abstract

We prove martingale-ergodic and ergodic-martingale theorems with continuous parameter for vector valued Bochner integrable functions. We first prove almost everywhere convergence of vector valued martingales with continuous parameter. The norm as well as almost everywhere convergence of martingale-ergodic and ergodic-martingale averages are given. We also obtain the dominant and maximal inequalities. Finally, we show that a.e. martingale-ergodic and ergodic-martingale theorems will coincide under certain assumptions.