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

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.