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Place exactly one random walker at each integer in $\Bbb Z$ and define $Y_n$ as the number of these who are at the origin at time n. Show that $0<\displaystyle\lim_{n\to\infty}P\{Y_n=0\}<1$ and find $\displaystyle\lim_{n\to\infty}P\{Y_n=k\}$ for $k\in\Bbb Z^+$.

I'd like to get a hint to start, I don't know how to tackle this problem.

1 Answers 1

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To start you off:

If somebody starts at $c$ then the probability they are at the origin at time $n$ if $n \lt |c|$ or $n-c$ is odd is zero, and otherwise is ${n \choose \frac{n-c}{2}}\dfrac{1}{2^n}$ and so the probability of not being at the origin is $1-{n \choose \frac{n-c}{2}}\frac{1}{2^n}.$ So for the first part of the question, you need to show that the limit of the product of that expression over $c$ is between $0$ and $1$. By letting $d=\frac{n-c}{2}$ you can change the product into $\prod_{d=0}^n \left(1- {n \choose d}\frac{1}{2^n} \right)$ and this does indeed have a simply expressed limit between $0$ and $1$.