Theorems For Random Walks Research Articles

Conditioned limit theorems for random walks with negative drift

In this paper we will solve a problem posed by Iglehart. In (1975) he conjectured that if S n is a random walk with negative mean and finite variance then there is a constant α so that (S [n.]/αn 1/2¦N>n) converges weakly to a process which he called the Brownian excursion. It will be shown that his conjecture is false or, more precisely, that if ES 1=−a<0, ES 1 2 <∞, and there is a slowly varying function L so that P(S 1>x)∼x −q L(x) as x→∞ then (S [n.]/n¦S n >0) and (S [n.]/n¦N>n) converge weakly to nondegenerate limits. The limit processes have sample paths which have a single jump (with d.f. (1−(x/a)−q )+) and are otherwise linear with slope −a. The jump occurs at a uniformly distributed time in the first case and at t=0 in the second.

Some Limit Theorems of Renewal Theory

Previous article Next article Some Limit Theorems of Renewal TheoryA. V. NagaevA. V. Nagaevhttps://doi.org/10.1137/1120036PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] E. B. Dynkin, Some limit theorems for sums of independent random quantities with infinite mathematical expectations, Izv. Akad. Nauk SSSR. Ser. Mat., 19 (1955), 247–266, (In Russian.) MR0076214 Google Scholar[2] William Feller, An introduction to probability theory and its applications. Vol. II, Second edition, John Wiley & Sons Inc., New York, 1971xxiv+669 MR0270403 0219.60003 Google Scholar[3] A. V. Nagaev, A local limit theorem for the number of renewals, Litovsk. Mat. Sb., 10 (1970), 109–119, (In Russian.) MR0260051 0213.19602 Google Scholar[4] I. A. Ibragimov and , Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971, 443– MR0322926 0219.60027 Google Scholar[5] Charles Stone, A local limit theorem for nonlattice multi-dimensional distribution functions, Ann. Math. Statist., 36 (1965), 546–551 MR0175166 0135.19204 CrossrefGoogle Scholar[6] Jean Bretagnolle and , Didier Dacunha-Castelle, Théorèmes limites à distance finie pour les marches aléatoires, Ann. Inst. H. Poincaré Sect. B (N.S.), 4 (1968), 25–73 MR0235598 0164.47402 Google Scholar[7] Bengt Rosén, On the asymptotic distribution of sums of independent indentically distributed random variables, Ark. Mat., 4 (1962), 323–332 (1962) MR0142135 0103.11902 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Limit theorems for random walks that avoid bounded sets, with applications to the largest gap problemStochastic Processes and their Applications, Vol. 125, No. 5 Cross Ref Large Deviation Theorems for the First Time of Crossing an Increasing Level in a Transient Random WalkA. V. Nagaev17 July 2006 | Theory of Probability & Its Applications, Vol. 38, No. 1AbstractPDF (581 KB)On a Boundary Problem for a Nonrecurrent Random WalkA. V. Nagaev3 August 2006 | Theory of Probability & Its Applications, Vol. 31, No. 2AbstractPDF (493 KB)Summary of Reports Presented at Sessions of the Probability and Mathematical Statistics Seminar at the Mathematics Institute of the Siberian Section of the USSR Academy of Sciences, February–May 198317 July 2006 | Theory of Probability & Its Applications, Vol. 29, No. 2AbstractPDF (525 KB) Volume 20, Issue 2| 1976Theory of Probability & Its Applications History Submitted:13 December 1973Published online:17 July 2006 InformationCopyright © 1976 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1120036Article page range:pp. 323-336ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Conditioned limit theorems for random walks

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Conditioned limit theorems for random walks

An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Marches al�atoires sur un groupe nilpotent

Under the assumption that there exist moments of order four we give a proof of the renewal theorem for random walks on connected compactly generated nilpotent groups. In three dimension, we get a bound for the potential kernel.

A limit theorem for random walks with drift

Let Xi, i = 1, 2, 3, … be a sequence of independent and identically distributed random variables. Write and for x ≧ 0 define M(x) + 1 is then the first passage time out of the interval (– ∞, x] for the random walk process Sn.

A limit theorem for random walks with drift

Let Xi , i = 1, 2, 3, … be a sequence of independent and identically distributed random variables. Write and for x ≧ 0 define M(x) + 1 is then the first passage time out of the interval (– ∞, x] for the random walk process Sn.

Ratio Limit Theorems for Random Walks on Groups

Ratio Limit Theorems for Random Walks on Groups

Ratio limit theorems for random walks on groups

Ratio limit theorems for random walks on groups

Limit Theorems for Random Walks. II

Limit Theorems for Random Walks. II

Some Limit Theorems for Large Deviations

Some Limit Theorems for Large Deviations

Limit Theorems for Random Walks, I

Limit Theorems for Random Walks, I

Ratio theorems for random walks II

Ratio theorems for random walks II

Ratio theorems for random walks I

Ratio theorems for random walks I

Limit theorems for random walks, birth and death processes, and diffusion processes

Limit theorems for random walks, birth and death processes, and diffusion processes