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 Sibirsk. Mat. Zh., 2006, Volume 47, Number 6, Pages 1323–1341 (Mi smj937)

Large deviations of the first passage time for a random walk with semiexponentially distributed jumps

A. A. Mogul'skii

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: Suppose that $\xi,\xi(1),\xi(2),…$ are independent identically distributed random variables such that $-\xi$ is semiexponential; i.e., $\mathbf P(-\xi\geqslant t)=e^{-t^{\beta}L(t)}$, $\beta\in(0,1)$, $L(t)$ is a slowly varying function as $t\to\infty$ possessing some smoothness properties. Let $\mathbf E\xi=0$, $\mathbf D\xi=1$, and $S(k)=\xi(1)+…+\xi(k)$. Given $d>0$, define the first upcrossing time $\eta+(u)=\inf\{k\geqslant1:S(k)+kd>u\}$ at nonnegative level $u\geqslant0$ of the walk $S(k)+kd$ with positive drift $\d>0$. We prove that, under general conditions, the following relation is valid for $n\to\infty$ and for $u=u(n)\in[0,dn-N_n\sqrt{n}]$:
\begin{equation*} \mathbf P(\eta_+(u)>n)\thicksim\frac{\mathbf E_{\eta_+}(u)}{n}\mathbf P(S(n)\leqslant x), \tag{0.1} \end{equation*}
where $x=u-nd<0$ and an arbitrary fixed sequence $N_n$ not exceeding $d\sqrt{n}$ tends to $\infty$.
The conditions under which we prove (0.1) coincide exactly with the conditions under which the asymptotic behavior of the probability $\mathbf P(S(n)\leqslant x)$ for $x\leqslant-\sqrt{n}$ (for $x\in[-\sqrt{n},0]$ it follows from the central limit theorem).

Keywords: one-dimensional random walk, first passage time, large deviation, semiexponential distribution, integro-local theorem, integral theorem, deviation function, segment of the Cramér series

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English version:
Siberian Mathematical Journal, 2006, 47:6, 1084–1101

Bibliographic databases:

UDC: 519.21

Citation: A. A. Mogul'skii, “Large deviations of the first passage time for a random walk with semiexponentially distributed jumps”, Sibirsk. Mat. Zh., 47:6 (2006), 1323–1341; Siberian Math. J., 47:6 (2006), 1084–1101

Citation in format AMSBIB
\Bibitem{Mog06} \by A.~A.~Mogul'skii \paper Large deviations of the first passage time for a random walk with semiexponentially distributed jumps \jour Sibirsk. Mat. Zh. \yr 2006 \vol 47 \issue 6 \pages 1323--1341 \mathnet{http://mi.mathnet.ru/smj937} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2302848} \zmath{https://zbmath.org/?q=an:1150.60006} \transl \jour Siberian Math. J. \yr 2006 \vol 47 \issue 6 \pages 1084--1101 \crossref{https://doi.org/10.1007/s11202-006-0117-3} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000243454700010} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33845500940} 

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This publication is cited in the following articles:
1. A. A. Mogulskii, “Integralnye i integro-lokalnye teoremy dlya summ sluchainykh velichin s semieksponentsialnymi raspredeleniyami”, Sib. elektron. matem. izv., 6 (2009), 251–271
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