Let (Ω,A,P) be a probability space, and let (Xn,Fn) be an adapted sequence in (Ω,A,P) (that is, for the σ-algebras Fn, we have F1⊆F2⊆⋯⊆A, and for all n, Xn is an Fn-measurable and integrable random variable). Assume that
E(Xn+1∣Fn)=21Xn+21Xn−1(n=2,3,…)
Prove that supnE∣Xn∣<∞ implies that Xn converges with probability one as n→∞. [I. Fazekas] Miklos Schweitzercollege contestsprobabilitysigma-algebrasrandom variables