Total Positivity of Markov Chains and the Failure Rate Character of Some First Passage Times

1997 ◽  
Vol 29 (3) ◽  
pp. 713-732 ◽  
Author(s):  
Shiowjen Lee ◽  
J. Lynch
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Failure Rate ◽  
Total Positivity ◽  
First Passage Times ◽  
First Passage ◽  
Positive Order ◽  
Totally Positive ◽  
Ordered State ◽  
Passage Times

It is shown that totally positive order 2 (TP2) properties of the infinitesimal generator of a continuous-time Markov chain with totally ordered state space carry over to the chain's transition distribution function. For chains with such properties, failure rate characteristics of the first passage times are established. For Markov chains with partially ordered state space, it is shown that the first passage times have an IFR distribution under a multivariate total positivity condition on the transition function.

Total Positivity of Markov Chains and the Failure Rate Character of Some First Passage Times

1997 ◽  
Vol 29 (03) ◽  
pp. 713-732 ◽  
Author(s):  
Shiowjen Lee ◽  
J. Lynch
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Failure Rate ◽  
Total Positivity ◽  
First Passage Times ◽  
First Passage ◽  
Positive Order ◽  
Totally Positive ◽  
Ordered State ◽  
Passage Times

It is shown that totally positive order 2 (TP2) properties of the infinitesimal generator of a continuous-time Markov chain with totally ordered state space carry over to the chain's transition distribution function. For chains with such properties, failure rate characteristics of the first passage times are established. For Markov chains with partially ordered state space, it is shown that the first passage times have an IFR distribution under a multivariate total positivity condition on the transition function.

Spectral Theory for Skip-Free Markov Chains

1989 ◽  
Vol 3 (1) ◽  
pp. 77-88 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Markov Chains ◽  
Spectral Theory ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage Times ◽  
Simple Relationship ◽  
First Passage ◽  
Birth And Death Processes ◽  
The Laplace Transform ◽  
Passage Times

The distribution of upward first passage times in skip-free Markov chains can be expressed solely in terms of the eigenvalues in the spectral representation, without performing a separate calculation to determine the eigenvectors. We provide insight into this result and skip-free Markov chains more generally by showing that part of the spectral theory developed for birth-and-death processes extends to skip-free chains. We show that the eigenvalues and eigenvectors of skip-free chains can be characterized in terms of recursively defined polynomials. Moreover, the Laplace transform of the upward first passage time from 0 to n is the reciprocal of the nth polynomial. This simple relationship holds because the Laplace transforms of the first passage times satisfy the same recursion as the polynomials except for a normalization.

Geometric Markov chains

1995 ◽  
Vol 32 (02) ◽  
pp. 349-374
Author(s):  
William Rising
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Nearest Neighbor ◽  
First Passage Times ◽  
First Passage ◽  
Death Rates ◽  
Passage Times ◽  
Birth Death

A generalization of the familiar birth–death chain, called the geometric chain, is introduced and explored. By the introduction of two families of parameters in addition to the infinitesimal birth and death rates, the geometric chain allows transitions beyond the nearest neighbor, but is shown to retain the simple computational formulas of the birth–death chain for the stationary distribution and the expected first-passage times between states. It is also demonstrated that even when not reversible, a reversed geometric chain is again a geometric chain.

scholarly journals First Passage and Collective Marks

2019 ◽  
Vol 8 (6) ◽  
pp. 47
Author(s):  
Yiping Zhang ◽  
Myron Hlynka ◽  
Percy H. Brill
Keyword(s):  
Markov Chains ◽  
Generating Functions ◽  
First Passage Times ◽  
System Of Equations ◽  
First Passage ◽  
Probability Generating Functions ◽  
Collective Marks ◽  
Passage Times

Probability generating functions for first passage times of Markov chains are found using the method of collective marks. A system of equations is found which can be used to obtain moments of the first passage times. Second passage probabilities are discussed.

First-passage times with PFr densities

1985 ◽  
Vol 22 (1) ◽  
pp. 185-196 ◽  
Author(s):  
David Assaf ◽  
Moshe Shared ◽  
J. George shanthikumar
Keyword(s):  
First Passage Time ◽  
Compound Poisson Process ◽  
Passage Time ◽  
First Passage Times ◽  
Embedded Markov Chain ◽  
First Passage ◽  
Completely Monotone ◽  
Totally Positive ◽  
Finite State ◽  
Passage Times

It is shown that if a finite-state continuous-time Markov process can be uniformized such that the embedded Markov chain has a TPr (totally positive of order r) transition matrix, then the first-passage time from state 0 to any other state has a PFr (Polya frequency of order r) density. As a consequence, results of Keilson (1971), Esary, Marshall and Proschan (1973), Ghosh and Ebrahimi (1982) and Derman, Ross and Schechner (1983) are strengthened. It is also shown that some cumulative damage shock models, with an underlying compound Poisson process and ‘damages' which are not necessarily non-negative, are associated with wear processes having PFr first-passage times to any threshold. First-passage times with completely monotone densities are also discussed.

Weak moment conditions for time coordinates in first-passage percolation models

1980 ◽  
Vol 17 (4) ◽  
pp. 968-978 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Time Constant ◽  
First Passage Times ◽  
First Passage Percolation ◽  
First Passage ◽  
Moment Conditions ◽  
Positive Order ◽  
Finite Moment ◽  
Time Coordinate ◽  
Point To Point ◽  
Passage Times

A generalization of first-passage percolation theory proves that the fundamental convergence theorems hold provided only that the time coordinate distribution has a finite moment of a positive order. The existence of a time constant is proved by considering first-passage times between intervals of sites, rather than the usual point-to-point and point-to-line first-passage times. The basic limit theorems for the related stochastic processes follow easily by previous techniques. The time constant is evaluated as 0 when the atom at 0 of the time-coordinate distribution exceeds½.

Weak moment conditions for time coordinates in first-passage percolation models

1980 ◽  
Vol 17 (04) ◽  
pp. 968-978 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Time Constant ◽  
First Passage Times ◽  
First Passage Percolation ◽  
First Passage ◽  
Moment Conditions ◽  
Positive Order ◽  
Finite Moment ◽  
Time Coordinate ◽  
Point To Point ◽  
Passage Times

A generalization of first-passage percolation theory proves that the fundamental convergence theorems hold provided only that the time coordinate distribution has a finite moment of a positive order. The existence of a time constant is proved by considering first-passage times between intervals of sites, rather than the usual point-to-point and point-to-line first-passage times. The basic limit theorems for the related stochastic processes follow easily by previous techniques. The time constant is evaluated as 0 when the atom at 0 of the time-coordinate distribution exceeds½.

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