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Problem:
Let (Xn) be adapted to (Fn) with 0 ≤ Xn ≤ 1. Let α, β > 0 with α + β = 1. Suppose X0 = x0 and P(Xn+1 = α + βXn|Fn) = Xn, P(Xn+1 = βXn|Fn) = 1 - Xn
Show that Xn → X∞ a.s., where P(X∞ = 1) = x0 and P(X∞ = 0) = 1- x0
Additional Information:
This problem is basically from Mathematics as well as it is about martingale calculation
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