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Convergence in probability and almost sure with applications

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  • Cohn, Harry

Abstract

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.

Suggested Citation

  • Cohn, Harry, 2001. "Convergence in probability and almost sure with applications," Stochastic Processes and their Applications, Elsevier, vol. 94(1), pages 135-154, July.
    • Handle: RePEc:eee:spapps:v:94:y:2001:i:1:p:135-154
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