ON THE CONVERGENCE OF STOCHASTICALLY MONOTONE SEQUENCES OF RANDOM-VARIABLES AND SOME APPLICATIONS

The relationship between various modes of convergence for stochastically monotone sequences of random variables, is considered. A necessary and sufficient condition, as well as a sufficient condition, for convergence in probability of a vaguely convergent sequence are given. If, in addition, the sequence is assumed Markovian the same conditions are shown to pertain to almost sure convergence. A counterexample in the case when stochastic monotonicity fails is presented and some applications to branching processes are discussed.