Convergence in probability and almost sure with applications

Abstract

A necessary and sufficient condition is given for the convergence in probability of a stochastic process { X t }. Moreover, as a byproduct, an almost sure convergent stochastic process { Y t } with the same limit as { X t } is identified. In a number of cases { Y t } reduces to { X t } thereby proving a.s. convergence. In other cases it leads to a different sequence but, under further assumptions, it may be shown that { X t } and { Y t } are a.s. equivalent, implying that { X t } is a.s. convergent. The method applies to a number of old and new cases of branching processes providing an unified approach. New results are derived for supercritical branching random walks and multitype branching processes in varying environment.