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Miklós Schweitzer 1985, Problem 12

Source: Miklós Schweitzer 1985

September 5, 2016
Miklos Schweitzercollege contestsprobabilitysigma-algebrasrandom variables

Problem Statement

Let (Ω,A,P)(\Omega, \mathcal A, P) be a probability space, and let (Xn,Fn)(X_n, \mathcal F_n) be an adapted sequence in (Ω,A,P)(\Omega, \mathcal A, P) (that is, for the σ\sigma-algebras Fn\mathcal F_n, we have F1⊆F2⊆⋯⊆A\mathcal F_1\subseteq \mathcal F_2\subseteq \dots \subseteq \mathcal A, and for all nn, XnX_n is an Fn\mathcal F_n-measurable and integrable random variable). Assume that E(Xn+1∣Fn)=12Xn+12Xn−1     (n=2,3,…)\mathrm E (X_{n+1} \mid \mathcal F_n )=\frac12 X_n+\frac12 X_{n-1}\,\,\,\,\, (n=2, 3, \ldots ) Prove that supnE∣Xn∣<∞\mathrm{sup}_n \mathrm{E}|X_n|<\infty implies that XnX_n converges with probability one as n→∞n\to\infty. [I. Fazekas]
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