Doob's type optional sampling theorems and a central limit theorem for demimartingales with applications to associated sequences

TL;DR

This paper generalizes classical probabilistic theorems to demimartingales, providing new optional sampling, maximal inequalities, and limit theorems, with applications to associated sequences under weak conditions.

Contribution

It introduces variants of Doob's optional sampling theorem and limit results for demimartingales, extending their applicability to dependent sequences like associated variables.

Findings

Established optional sampling theorems for demimartingales.

Derived maximal and concentration inequalities of Azuma-Hoeffding-Bernstein type.

Proved a strong law of large numbers for associated sequences under mild conditions.

Abstract

This paper extends classical probabilistic results to the broader class of demimartingales and demisubmartingales. We establish variants of Doob's-type optional sampling theorem under minimal structural conditions on stopping times, relying on monotonicity properties of indicator functions. Building on these foundations, we derive maximal inequalities and a concentration inequality of Azuma-Hoeffding-Bernstein type for demimartingales. A central limit theorem and a strong law of large numbers are also obtained demonstrating convergence under conditions considerably weaker than those required for martingales or independent sequences. These results are then applied to partial sums of positively associated random variables, yielding concentration inequalities and exponential bounds without requiring covariance decay or truncation arguments. The optional sampling theorems are used to establish Wald inequalities for positively associated random variables while the paper concludes with a strong law of large numbers for associated random variables, established under very mild conditions, highlighting the power of the demimartingale framework in handling dependence.