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Let $X_n$ takes its values in [0,1] and $p$ is a fixed number in $[0,1]$ Now if $ X_{n+1} = 1-p+pX_n $ with probability $X_n$ and $X_{n+1} = pX_n$ with probability $1-X_n$ . I know that $X_n$ is a martingale but why it converges almost surely? with which one of the theorems?

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up vote 1 down vote accepted

Since $X_n$ takes its values in $[0,1]$, we have in particular

$$\sup_{n \in \mathbb{N}} \|X_n\|_1=1<\infty$$

Hence $X_n \to X_{\infty}$ almost surely for some random variable $X_\infty$ ($L^1$-bounded martingales are a.s. convergent).

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Thanks and what would be the distribution of the limit? – peanut Nov 28 '12 at 22:44
Concerning the distribution you can take a look at this question (choose $\alpha:=1-p$, $\beta:=p$). – saz Nov 29 '12 at 6:46

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