# Identity for simple 1D random walk

The question is to find a purely probabilistic proof of the following identity, valid for every integer $n\geqslant1$, where $(S_n)_{n\geqslant0}$ denotes a standard simple random walk:

$$E[(S_n)^2;S_{2n}=0]=\frac{n}2\,P[S_{2n-2}=0].$$

Standard simple random walk is defined as $S_0=0$ and $S_n=\sum\limits_{k=1}^nX_k$ for every $n\geqslant1$, where $(X_k)_{k\geqslant1}$ is an i.i.d. sequence such that $P[X_k=+1]=P[X_k=-1]=\frac12$.

Of course, the RHS of the identity is $$\frac{n}{2^{2n-1}}\,{2n-2\choose n-1}.$$ For a combinatorial proof, see this MSE question and its comments.

For an accessible introduction to the subject, see the corresponding chapter of the Chance project.

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Here, are you using $\mathbb{E}[X; A]$ to denote the integral taken only over the subset $A$? –  Nicholas R. Peterson Jul 14 at 13:38
@nrpeterson Yes, $E[X;A]=E[X\mathbf 1_A]$. –  Did Jul 14 at 21:28
This is an interesting question... I'll do some thinking on it. –  Nicholas R. Peterson Jul 14 at 21:36
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