1. At what speed does $\mathrm{E} \min \left(\sigma_{2 n}, 2 n\right)$ tend to $\infty$ as $n \rightarrow \infty$? (Here $\sigma_{2 n}=\min \left\{1 \leqslant k \leqslant 2 n: S_{k}=0\right\}$ and we assume $\sigma_{2 n}=\infty$ (or $\left.\sigma_{2 n}=2 n\right)$, if $S_{k} \neq 0$ for all $1 \leqslant k \leqslant 2 n$.)
Solution. According to item 1 of V1.I. 10, we have

$\mathrm{E} \min \left(\sigma_{2 n}, 2 n\right)=\sum_{k=1}^{n} u_{2(k-1)}+2 n u_{2 n}=$

$$
=\sum_{k=0}^{n-1} 2^{-2 k} C_{2 k}^{k}+2 n \cdot 2^{-2 n} C_{2 n}^{n}
$$

By Stirling's formula, $2 n \cdot 2^{-2 n} C_{2 n}^{n} \sim 2 \sqrt{n \pi^{-1}}$, and by problem I.9.3, the first term in formula (*) is equivalent to $2 \sqrt{n \pi^{-1}}$. Therefore, $\mathrm{E} \min \left(\sigma_{2 n}, 2 n\right)$ is equivalent to $4 \sqrt{n \pi^{-1}}$.