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Math Help - Visits to the origin in a stochastic process

  1. #1
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    Visits to the origin in a stochastic process

    Let X_1,X_2,... be independent identically distributed random variables with \mathbb{P}(X_1=-1)=\frac{1}{3}, \ \mathbb{P}(X_1=1)=\frac{2}{3}

    Let Y_0=0 and set Y_n=X_1+X_2+...X_n

    By the S.L.L.N's \frac{Y_n}{n}\rightarrow \frac{1}{3} \; as  \ n \rightarrow \infty

    Deduce that with probability 1 the stochastic process {Y_n: n=0,1,2,...} visits the origin at most a finite number of times.

    I have no clue where to start with this question could someone please give me a hint

    thanks
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    Re: Visits to the origin in a stochastic process

    Hello,

    You have to think of Borel-Cantelli's lemma !

    Let A_n=\{Y_{2n}=0\} (we know it's impossible for Y_{2n+1} to equal 0).

    We want to prove that almost surely, An happens a finite number of times, that is to say not an infinite number of times. So that is P(A_n \quad i.o.)=P(\limsup A_n)=0

    So we just have to prove that \sum P(A_n)<\infty, which is easy since Y_{2n}=0 \Leftrightarrow there are n Xi's equal to -1 and n equal to 1.
    So P(A_n)={2n \choose n} \frac{1}{3^n}\cdot\frac{2^n}{3^n}.

    So \sum_n P(A_n)=\sum_{n=0}^{\infty} \tfrac{n!}{(2n)!n!}\cdot \left(\tfrac{\sqrt{2}}{3}\right)^{2n}<\sum_{k=0}^{  \infty} \tfrac{1}{k!} \cdot \left(\tfrac{\sqrt{2}}{3}\right)^k = \exp (\sqrt{2}/3) <\infty.

    Hence the result.
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