# Thread: Visits to the origin in a stochastic process

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

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

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

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

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

2. ## Re: Visits to the origin in a stochastic process

Hello,

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

Let $\displaystyle A_n=\{Y_{2n}=0\}$ (we know it's impossible for $\displaystyle Y_{2n+1}$ 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 $\displaystyle P(A_n \quad i.o.)=P(\limsup A_n)=0$

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

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

Hence the result.