On a property related to convergence in probability and some applications to branching processes

Harry Cohn

Stochastic Processes and their Applications, 1981, vol. 12, issue 1, 59-72

Abstract: It is shown that any real-valued sequence of random variables {Xn} converging in probability to a non-degenerate, not necessarily a.s. finite limit X possesses the following property: for any c with P(X [epsilon] (c - [delta], c + [delta])) > 0 for all [delta] > 0, there exists a sequence {cn} with limn-->[infinity] cn = c such that for any [var epsilon] > 0, limn-->[infinity] P(X [delta] (c - [var epsilon], c + [var epsilon]) Xn = cn) = 1. This property is applied to various types of branching processes where Xn = Zn/Cn or Xn = U(Zn)/Cn{Cn} being a sequence of constants or random variables and U a slowly varying function. If {Zn} is a supercritical branching process in varying or random environment, X is shown to have a continuous and strictly increasing distribution function on (0, [infinity]). Characterizations of the tail of the liniting distribution of the finite mean and the infinite mean supercritical Galton-Watson processes are also obtained.

Keywords: Convergence; in; probability; Galton-Watson; process; random; environment; law; of; large; numbers; limiting; distribution; varying; environment; regular; infinite; mean; process (search for similar items in EconPapers)
Date: 1981
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