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The Passage Time Distribution for a Birth-and-Death Chain: Strong Stationary Duality Gives a First Stochastic Proof

On Times to Quasi-stationarity for Birth and Death Processes

On Hitting Times and Fastest Strong Stationary Times for Skip-Free and More General Chains

Reflected BSDE Driven by a Lévy Process

Uniform Shrinking and Expansion under Isotropic Brownian Flows

Exact L $_{2}$-Small Ball Asymptotics of Gaussian Processes and the Spectrum of Boundary-Value Problems

On Distributional Properties of Perpetuities

Support Theorems for the Radon Transform and Cramér-Wold Theorems

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On Hitting Times and Fastest Strong Stationary Times for Skip-Free and More General Chains

Content Provider	Springer Nature Link
Author	Fill, James Allen
Copyright Year	2009
Abstract	An (upward) skip-free Markov chain with the set of nonnegative integers as state space is a chain for which upward jumps may be only of unit size; there is no restriction on downward jumps. In a 1987 paper, Brown and Shao determined, for an irreducible continuous-time skip-free chain and any d, the passage time distribution from state 0 to state d. When the nonzero eigenvalues ν $_{ j }$ of the generator on {0,…,d}, with d made absorbing, are all real, their result states that the passage time is distributed as the sum of d independent exponential random variables with rates ν $_{ j }$. We give another proof of their theorem. In the case of birth-and-death chains, our proof leads to an explicit representation of the passage time as a sum of independent exponential random variables. Diaconis and Miclo recently obtained the first such representation, but our construction is much simpler.We obtain similar (and new) results for a fastest strong stationary time T of an ergodic continuous-time skip-free chain with stochastically monotone time-reversal started in state 0, and we also obtain discrete-time analogs of all our results.In the paper’s final section we present extensions of our results to more general chains.
Ending Page	600
Page Count	14
Starting Page	587
File Format	PDF
ISSN	08949840
e-ISSN	15729230
Journal	Journal of Theoretical Probability
Issue Number	3
Volume Number	22
Language	English
Publisher	Springer US
Publisher Date	2009-06-02
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Birth-and-death chains Absorption time Markov chains Probability Theory and Stochastic Processes Transition functions, generators and resolvents Statistics Markov chains (discrete-time Markov processes on discrete state spaces) Passage time Strong stationary duality Skip-free chains Stochastic monotonicity Eigenvalues Stopping times; optimal stopping problems; gambling theory Continuous-time Markov processes on general state spaces Fastest strong stationary times
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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On hitting times and fastest strong stationary times for skip-free and more general chains

The Passage Time Distribution for a Birth-and-Death Chain: Strong Stationary Duality Gives a First Stochastic Proof

On hitting times and fastest strong stationary times for skip-free chains (2007).

Mixing Times are Hitting Times of Large Sets

On hitting times and fastest strong stationary times for skip-free and more general chains (2009).

On hitting times and fastest strong stationary times for skip-free and more general chains (2009)

Explicit criteria on separation cutoff for birth and death chains

Strong stationary duality for Möbius monotone Markov chains

Hitting Times and Interlacing Eigenvalues: A Stochastic Approach Using Intertwinings

On Hitting Times and Fastest Strong Stationary Times for Skip-Free and More General Chains

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