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Local limit theorem for the first crossing time of a fixed level by a random walk
A. A. Mogul'skiiab a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
b Novosibirsk State University, Novosibirsk, Russia
Abstract:
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.
Key words:
random walk, first crossing time of a fixed level, arithmetic distribution, nonarithmetic distribution, local limit theorem.
UDC:
519.21 Received: 24.01.2007
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