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 Mat. Tr., 2009, Volume 12, Number 2, Pages 126–138 (Mi mt184)

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 $X,X(1),X(2),…$ be independent identically distributed random variables with mean zero and a finite variance. Put $S(n)=X(1)+…+X(n)$, $n=1,2,…$, 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 $S(n)$, $n=1,2,…$. 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 limità $\lim_{n\to\infty}n^{3/2}\mathbb P(\eta_y=n)$ 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 òthe 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.

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English version:
Siberian Advances in Mathematics, 2010, 20:3, 191–200

Bibliographic databases:

UDC: 519.21

Citation: A. A. Mogul'skii, “Local limit theorem for the first crossing time of a fixed level by a random walk”, Mat. Tr., 12:2 (2009), 126–138; Siberian Adv. Math., 20:3 (2010), 191–200

Citation in format AMSBIB
\Bibitem{Mog09} \by A.~A.~Mogul'skii \paper Local limit theorem for the first crossing time of a~fixed level by a~random walk \jour Mat. Tr. \yr 2009 \vol 12 \issue 2 \pages 126--138 \mathnet{http://mi.mathnet.ru/mt184} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2599428} \elib{http://elibrary.ru/item.asp?id=13021590} \transl \jour Siberian Adv. Math. \yr 2010 \vol 20 \issue 3 \pages 191--200 \crossref{https://doi.org/10.3103/S1055134410030041} \elib{http://elibrary.ru/item.asp?id=15325618}