Let $\epsilon_1,\ldots,\epsilon_m$ be Bernoulli random variables, i.e. $P(\epsilon_i=1)=P(\epsilon_i=-1)=\frac 12$.
I wanted to calculate (or at least approximate) the following conditional expectation: $ E\left(\epsilon^{n_1}_1 \cdots \epsilon^{n_m}_m |\sum_{i=1}^m \epsilon_i=0 \right) $ Here $n_1+\cdots+n_m=n$, $n\in N$ and $n_j \in \{0,1,\ldots,n\}$.
Since we have condition that $\sum_{i=1}^m \epsilon_i=0$, then we can say that $m=2k$ is even.
Now, I rearrange random variables: $(\epsilon^{n_1}_1\cdots\epsilon^{n_k}_k) \left(\epsilon^{n_{k+1}}_{k+1}\cdots\epsilon^{n_2k}_{2k}\right)$ so that first $k$ of them are 1 and second $k$ of them are $-1$. Thus, $ E\left[\left(\epsilon^{n_1}_1\cdots\epsilon^{n_k}_k\right) \left(\epsilon^{n_{k+1}}_{k+1}\cdots \epsilon^{n_2k}_{2k}\right)\right]=E\left[(-1)^{\sum_{j=k+1}^{2k}n_j}\right]. $
And here I am stuck...