scholarly journals Asymptotic Expansions for the Distributions of Stopped Random Walks and First Passage Times

1994 ◽  
Vol 22 (4) ◽  
pp. 1957-1992 ◽  
Author(s):  
Tze Leung Lai ◽  
Julia Qizhi Wang
Keyword(s):  
Random Walks ◽  
Asymptotic Expansions ◽  
First Passage Times ◽  
First Passage ◽  
Stopped Random Walks ◽  
Passage Times

Wald's equations, first passage times and moments of ladder variables in Markov random walks

1998 ◽  
Vol 35 (3) ◽  
pp. 566-580 ◽  
Author(s):  
Cheng Der Fuh ◽  
Tze Leung Lai
Keyword(s):  
Random Walks ◽  
Asymptotic Expansions ◽  
Generating Functions ◽  
First Passage Times ◽  
First Passage ◽  
New Approach ◽  
Markov Random ◽  
Moment Generating Functions ◽  
Passage Times

Previous work in extending Wald's equations to Markov random walks involves finiteness of moment generating functions and uniform recurrence assumptions. By using a new approach, we can remove these assumptions. The results are applied to establish finiteness of moments of ladder variables and to derive asymptotic expansions for expected first passage times of Markov random walks. Wiener–Hopf factorizations for Markov random walks are also applied to analyse ladder variables and related first passage problems.

Wald's equations, first passage times and moments of ladder variables in Markov random walks

1998 ◽  
Vol 35 (03) ◽  
pp. 566-580 ◽  
Author(s):  
Cheng Der Fuh ◽  
Tze Leung Lai
Keyword(s):  
Random Walks ◽  
Asymptotic Expansions ◽  
Generating Functions ◽  
First Passage Times ◽  
First Passage ◽  
New Approach ◽  
Markov Random ◽  
Moment Generating Functions ◽  
Passage Times

Previous work in extending Wald's equations to Markov random walks involves finiteness of moment generating functions and uniform recurrence assumptions. By using a new approach, we can remove these assumptions. The results are applied to establish finiteness of moments of ladder variables and to derive asymptotic expansions for expected first passage times of Markov random walks. Wiener–Hopf factorizations for Markov random walks are also applied to analyse ladder variables and related first passage problems.

scholarly journals Asymptotics for the First Passage Times of Lévy Processes and Random Walks

2013 ◽  
Vol 50 (1) ◽  
pp. 64-84 ◽  
Author(s):  
Denis Denisov ◽  
Vsevolod Shneer
Keyword(s):  
Large Deviations ◽  
Random Walks ◽  
Levy Processes ◽  
Levy Process ◽  
First Passage Times ◽  
Exact Asymptotics ◽  
First Passage ◽  
Heavy Tailed ◽  
First Time ◽  
Passage Times

We study the exact asymptotics for the distribution of the first time, τx, a Lévy process Xt crosses a fixed negative level -x. We prove that ℙ{τx >t} ~V(x) ℙ{Xt≥0}/t as t→∞ for a certain function V(x). Using known results for the large deviations of random walks, we obtain asymptotics for ℙ{τx>t} explicitly in both light- and heavy-tailed cases.

scholarly journals Random walks on random trees

1973 ◽  
Vol 15 (1) ◽  
pp. 42-53 ◽  
Author(s):  
J. W. Moon
Keyword(s):  
Random Walks ◽  
Random Trees ◽  
First Passage Times ◽  
General Reference ◽  
First Passage ◽  
Labelled Trees ◽  
Passage Times

Let T denote one of the nn−2 trees with n labelled nodes that is rooted at a given node x (see [6] or [8] as a general reference on trees). If i and j are any two nodes of T, we write i ∼ j if they are joined by an edge in T. We want to consider random walks on T; we assume that when we are at a node i of degree d the probability that we proceed to node j at the next step is di–1 if i ∼ j and zero otherwise. Our object here is to determine the first two moments of the first return and first passage times for random walks on T when T is a specific tree and when T is chosen at random from the set of all labelled trees with certain properties.

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