Visits to the origin in a stochastic process

**FGT12** · August 11th 2011, 06:59 AM

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

**Moo** · August 11th 2011, 08:00 AM

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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