The Simple Random Walk II: First Passage Times

2021 ◽  
pp. 27-39
Author(s):  
Rabi Bhattacharya ◽  
Edward C. Waymire
Keyword(s):  
Random Walk ◽  
Simple Random Walk ◽  
First Passage Times ◽  
First Passage ◽  
Passage Times

Random walk type models for indicator-dilution studies: comparison of a local density random walk and a first passage times distribution

1986 ◽  
Vol 20 (11) ◽  
pp. 789-796 ◽  
Author(s):  
J M BOGAARD ◽  
J R C JANSEN ◽  
E A VON RETH ◽  
A VERSPRILLE ◽  
M E WISE
Keyword(s):  
Random Walk ◽  
Local Density ◽  
Indicator Dilution ◽  
First Passage Times ◽  
First Passage ◽  
Passage Times

Some asymptotic results for transient random walks

1996 ◽  
Vol 28 (1) ◽  
pp. 207-226 ◽  
Author(s):  
J. Bertoin ◽  
R. A. Doney
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Limit Theorems ◽  
First Passage Times ◽  
Asymptotic Results ◽  
First Passage ◽  
Cramér’S Condition ◽  
Tail Behaviour ◽  
Passage Times ◽  
Asymptotic Tail

We consider a real-valued random walk S which drifts to –∞ and is such that E(exp θS1) < ∞ for some θ > 0, but for which Cramér's condition fails. We investigate the asymptotic tail behaviour of the distributions of the all time maximum, the upwards and downwards first passage times and the last passage times. As an application, we obtain new limit theorems for certain conditional laws.

scholarly journals Asymptotics of first passage times for random walk in an orthant

1999 ◽  
Vol 9 (1) ◽  
pp. 110-145 ◽  
Author(s):  
D. R. McDonald
Keyword(s):  
Random Walk ◽  
First Passage Times ◽  
First Passage ◽  
Passage Times

scholarly journals The joint distribution of occupation totals for a simple random walk

1964 ◽  
Vol 4 (4) ◽  
pp. 518-528 ◽  
Author(s):  
V. D. Barnett
Keyword(s):  
Random Walk ◽  
Lattice Point ◽  
Joint Distribution ◽  
Average Duration ◽  
Simple Random Walk ◽  
First Passage ◽  
Absorption Probabilities ◽  
Passage Times ◽  
Quantities Of Interest ◽  
Explicit Expressions

SummaryA great deal of attention has been given in the literature to the various properties of the simple binomial random walk. Explicit expressions are available for first passage times, absorption probabilities, average duration of the walk up to absorption and other quantities of interest. One aspect of the behaviour of this work which has, however, attracted little attention is the form of the distribution of occupation totals. This paper is devoted to the derivation of an explict expression for the joint probility generating function of the occupation totals up to absorption, for the binomial random walk in the presence of two absorbing points. The appropriate marginal form of this p.g.f. yields the distribution of the occupation total, and expected occupation total, at any particular lattice point. The limiting forms of these results provide explicit expressions for the corresponding quatities in the case of a binomial random walk having a single absorbing point and, where relevent, in the case of the unrestricted binomial random walk.

First-passage times for the partial sums of a sequence of geometric distributions

1982 ◽  
Vol 19 (02) ◽  
pp. 430-432
Author(s):  
A. J. Woods
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Transition Probabilities ◽  
Probability Distributions ◽  
First Passage Times ◽  
Partial Sums ◽  
Homogeneous Markov Chain ◽  
First Passage ◽  
Epidemic Process ◽  
Passage Times

It is shown here that questions about the probability distributions of the partial sums of a sequence of geometric distributions, all with different parameters, can be answered by considering the transition probabilities of a homogeneous Markov chain. The result is applied to the embedded random walk of an epidemic process.

First-passage times for the partial sums of a sequence of geometric distributions

1982 ◽  
Vol 19 (2) ◽  
pp. 430-432 ◽  
Author(s):  
A. J. Woods
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Transition Probabilities ◽  
Probability Distributions ◽  
First Passage Times ◽  
Partial Sums ◽  
Homogeneous Markov Chain ◽  
First Passage ◽  
Epidemic Process ◽  
Passage Times

It is shown here that questions about the probability distributions of the partial sums of a sequence of geometric distributions, all with different parameters, can be answered by considering the transition probabilities of a homogeneous Markov chain. The result is applied to the embedded random walk of an epidemic process.

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