Local limit theorem for the first crossing time of a fixed level by a random walk

Let $X,X(1),X(2),\dots$ be independent identically distributed random variables with mean zero and a finite variance. Put $S(n)=X(1)+\dots+X(n)$, $n=1,2,\dots$, and define the Markov stopping time $\eta_y=\inf\{n\ge1\colon S(n)\ge y\}$ of the first crossing a level $y\ge0$ by the random walk $S(n)$, $n=1,2,\dots$. In the case $\mathbb E|X|^3<\infty$ the following relation was obtained in [5]: $\mathbb P(\eta_0=n)=\frac1{n\sqrt n}(R+\nu_n+o(1))$, $n\to\infty$, where the constant $R$ and the bounded sequence $\nu_n$ were calculated in an explicit form. Moreover, there were obtained necessary and sufficient conditions for the limit existence $H(y):=\lim_{n\to\infty}n^{3/2}\mathbb P(\eta_y=n)$ for every fixed $y\ge0$, and there was found a representation for $H(y)$. The present paper was motivated by the following reason. In [5], the authors unfortunately did not cite papers [8,9] where the above-mentioned relations were obtained under weaker restrictions. Namely, it was proved in [8] the existence of the limitа $\lim_{n\to\infty}n^{3/2}\mathbb P(\eta_y=n)$ for every fixed $y\ge0$ under the condition $\mathbb EX^2<\infty$ only. In [9], an explicit form of the limit $\lim_{n\to\infty}n^{3/2}\mathbb E(\eta_0=n)$ was found under тthe same condition $\mathbb EX^2<\infty$ in the case when the summand $X$ has an arithmetic distribution. In the present paper, we prove that the main assertion in [8] fails and we correct the original proof. It worth noting that this corrected version was formulated in [5] as a conjecture.