
Local limit theorem for the first crossing time of a fixed level by a random walk
A. A. Mogul'skii^{ab} ^{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 $\mathbb EX^3<\infty$ 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 abovementioned 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.
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English version:
Siberian Advances in Mathematics, 2010, 20:3, 191–200
Bibliographic databases:
UDC:
519.21 Received: 24.01.2007
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
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\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 126138
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\mathscinet{http://www.ams.org/mathscinetgetitem?mr=2599428}
\elib{http://elibrary.ru/item.asp?id=13021590}
\transl
\jour Siberian Adv. Math.
\yr 2010
\vol 20
\issue 3
\pages 191200
\crossref{https://doi.org/10.3103/S1055134410030041}
\elib{http://elibrary.ru/item.asp?id=15325618}
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