A Local Theorem for the First Hitting Time of a Fixed Level by a Random Walk

For the sums $S(n)=X(1)+\dots+X(n)$ of independent identically distributed random variables with zero mean, we determine the first passage time
$$ \eta_y=\inf\bigl\{n\ge 1:S(n)\ge y\bigr\} $$
across the level $y\ge 0$ from below to above by the random walk $\bigl\{S(n);\,n=1,2,\dots\bigr\}$. We obtain a local theorem for this random variable, i. e., we find asymptotics of $\mathbb P(\eta_y=n)$ for a fixed level $y\ge 0$ as $n\to\infty$.