Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question

1 Answer 1

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).

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.