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.

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.

scholarly journals A Diffusion Limit for Generalized Correlated Random Walks

2006 ◽  
Vol 43 (01) ◽  
pp. 60-73 ◽  
Author(s):  
Urs Gruber ◽  
Martin Schweizer
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Transition Probabilities ◽  
Partial Sums ◽  
Diffusion Limit ◽  
Correlated Random Walk ◽  
Binomial Trees ◽  
Correlated Random Walks ◽  
Limiting Behaviour ◽  
Image Position

A generalized correlated random walk is a process of partial sums such that (X, Y) forms a Markov chain. For a sequence (X n ) of such processes in which each takes only two values, we prove weak convergence to a diffusion process whose generator is explicitly described in terms of the limiting behaviour of the transition probabilities for the Y n . Applications include asymptotics of option replication under transaction costs and approximation of a given diffusion by regular recombining binomial trees.

scholarly journals SIMPLE PROCEDURES FOR FINDING MEAN FIRST PASSAGE TIMES IN MARKOV CHAINS

2007 ◽  
Vol 24 (06) ◽  
pp. 813-829 ◽  
Author(s):  
JEFFREY J. HUNTER
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Linear Equations ◽  
First Passage Times ◽  
First Passage ◽  
Mean First Passage Times ◽  
The Mean ◽  
Computational Procedures ◽  
Passage Times

The derivation of mean first passage times in Markov chains involves the solution of a family of linear equations. By exploring the solution of a related set of equations, using suitable generalized inverses of the Markovian kernel I - P, where P is the transition matrix of a finite irreducible Markov chain, we are able to derive elegant new results for finding the mean first passage times. As a by-product we derive the stationary distribution of the Markov chain without the necessity of any further computational procedures. Standard techniques in the literature, using for example Kemeny and Snell's fundamental matrix Z, require the initial derivation of the stationary distribution followed by the computation of Z, the inverse of I - P + eπT where eT = (1, 1, …, 1) and πT is the stationary probability vector. The procedures of this paper involve only the derivation of the inverse of a matrix of simple structure, based upon known characteristics of the Markov chain together with simple elementary vectors. No prior computations are required. Various possible families of matrices are explored leading to different related procedures.

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.

On ageing properties of first-passage times of increasing Markov processes

2002 ◽  
Vol 34 (01) ◽  
pp. 241-259
Author(s):  
Félix Belzunce ◽  
Eva-María Ortega ◽  
José M. Ruiz
Keyword(s):  
Markov Chain ◽  
Markov Processes ◽  
Continuous Time ◽  
Transition Matrix ◽  
Sufficient Conditions ◽  
First Passage Times ◽  
Generator Matrix ◽  
First Passage ◽  
Finite Case ◽  
Passage Times

The purpose of this paper is to study ageing properties of first-passage times of increasing Markov chains. We extend the literature to some new ageing classes, such as the IFR(2), NBU(2), DRLLt and NBULt classes. We also give sufficient conditions in the finite case, that are more efficient computationally, just in terms of the transition matrix K, in the discrete case, or the generator matrix Q, in the continuous case. For the uniformizable, continuous-time Markov processes, we derive conditions in terms of the discrete uniformized Markov chain for the NBU(2) and the NBULt classes. In the last section, a review of the main results in this direction in the literature is given, and we compare some of the conditions stated in this paper with others given in the literature about some other ageing classes. Some examples where these results are applied are given.

On ageing properties of first-passage times of increasing Markov processes

2002 ◽  
Vol 34 (1) ◽  
pp. 241-259 ◽  
Author(s):  
Félix Belzunce ◽  
Eva-María Ortega ◽  
José M. Ruiz
Keyword(s):  
Markov Chain ◽  
Markov Processes ◽  
Continuous Time ◽  
Transition Matrix ◽  
Sufficient Conditions ◽  
First Passage Times ◽  
Generator Matrix ◽  
First Passage ◽  
Finite Case ◽  
Passage Times

The purpose of this paper is to study ageing properties of first-passage times of increasing Markov chains. We extend the literature to some new ageing classes, such as the IFR(2), NBU(2), DRLLt and NBULt classes. We also give sufficient conditions in the finite case, that are more efficient computationally, just in terms of the transition matrix K, in the discrete case, or the generator matrix Q, in the continuous case. For the uniformizable, continuous-time Markov processes, we derive conditions in terms of the discrete uniformized Markov chain for the NBU(2) and the NBULt classes. In the last section, a review of the main results in this direction in the literature is given, and we compare some of the conditions stated in this paper with others given in the literature about some other ageing classes. Some examples where these results are applied are given.

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