Limit theorem for first passage times in the random walk described by the generalization of the autoregressive process

2020 ◽  
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Autoregressive Process ◽  
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.

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.

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

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.

Some asymptotic results for transient random walks

1996 ◽  
Vol 28 (01) ◽  
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 θS 1) &lt; ∞ for some θ &gt; 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.

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