Convergence in probability and almost sure with applications

A necessary and sufficient condition is given for the convergence in probability of a stochastic process {Xt}. Moreover, as a byproduct, an almost sure convergent stochastic process {Yt} with the same limit as {Xt} is identified. In a number of cases {Yt} reduces to {Xt} thereby proving a.s. convergence. In other cases it leads to a different sequence but, under further assumptions, it may be shown that {Xt} and {Yt} are a.s. equivalent, implying that {Xt} 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.