Doob-type optional sampling theorems for demimartingales with applications to associated sequences

Abstract

We establish optional sampling inequalities for demimartingales and demisubmartingales under suitable monotonicity assumptions on the stopping rule. First, we establish comparison inequalities for processes stopped at bounded stopping times and identify broad classes of threshold-type stopping times for which the required assumptions are naturally satisfied. We then prove Doob-type optional sampling inequalities for possibly unbounded stopping times under standard integrability conditions. As applications, we derive maximal inequalities for demi(sub)martingales, obtain an almost sure bound for the maximal negative excursion of partial sums of positively associated random variables, and establish generalized Wald-type inequalities, including nonlinear convex-transform inequalities for associated sequences.