We proved the explicit formulas in Laplace transform of the hitting times for the birth and death processes on a denumerable state space with $\ift$ the exit or entrance boundary. This extends the well known Keilson's theorem from finite state space to infinite state space.
We also apply these formulas to the fastest strong stationary time for strongly ergodic birth and death processes, and obtain the explicit convergence rate in separation.
For birth-and-death processes, we show that every initial distribution is reproduced from the first hitting time distribution. The reproduction is done by applying to the distribution function a differential operator defined through the eigenfunction of the generator. Using the spectral theory for generalized second-order differential operators, we study asymmetric random walks.
In a recent paper in the Journal of Theoretical Probability Gong, Mao and Zhang, using the theory of Dirichlet forms, extended Karlin and McGregor's classical results on first-hitting times of a birth-death process on the nonnegative integers by establishing a representation for the Laplace transform $\mathbb{E}[e^{sT_{ij}}]$ of the first-hitting time $T_{ij}$ for $any$ pair of states $i$ and $j$, as well as asymptotics for $\mathbb{E}[e^{sT_{ij}}]$ when either $i$ or $j$ tends to infinity. It will be shown here that these results may also be obtained by employing tools from the orthogonal-pol…
In this paper we study the iterated birth process of which we examine the first-passage time distributions and the hitting probabilities. Furthermore, linear birth processes, linear and sublinear death processes at Poisson times are investigated. In particular, we study the hitting times in all cases and examine their long-range behavior. The time-changed population models considered here display upward (Birth process) and downward jumps (death processes) of arbitrary size and, for this reason, can be adopted as adequate models in ecology, epidemics and finance situations, under stress conditi…
We finely describe the speed of "coming down from infinity" for birth and death processes which eventually become extinct. Under general assumptions on the birth and death rates, we firstly determine the behavior of the successive hitting times of large integers. We put in light two different regimes depending on whether the mean time for the process to go from $n+1$ to $n$ is negligible or not compared to the mean time to reach $n$ from infinity. In the first regime, the coming down from infinity is very fast and the convergence is weak. In the second regime, the coming down from infinity is …
We finely describe the "coming down from infinity" for birth and death processes which eventually become extinct. Our biological motivation is to study the decrease of regulated populations which are initially large. Under general assumptions on the birth and death rates, we describe the behavior of the hitting time of large integers. We let two regimes appear and derive an expression of the speed of coming down from infinity. In the case of death rates with regular variations, we also get a central limit theorem and the asymptotic probability of extinction in small times. Finally, we apply ou…
We consider a model of open quantum random walk and together with a quantum trajectory approach we are able to examine a notion of hitting time. We see that many constructions, such as minimal solutions to hitting time problems, are variations of well-known classical probability results, but the density matrix degree of freedom on each site gives rise to systems which are seen to be nonclassical. As a more specific application we study the collection of walks induced by normal commuting contractions, for which the corresponding probability expressions are obtained. We examine quantum versions …
An aperiodic and irreducible Markov chain on a finite state space converges to its stationary distribution. When convergence to equilibrium is measured by total variation distance, there exists an optimal coupling and a maximal coupling time. In this article, the maximal coupling time is compared to the hitting time of a specific state or set. Such sets, named halting sets, are studied in the case of symmetric birth-and-death chains and in some other examples. Some applications to the cutoff phenomenon are given. These results yield new methods to calculate cutoff times for some monotone birth…
Given a target distribution $μ$ and a proposal chain with generator $Q$ on a finite state space, in this paper we study two types of Metropolis-Hastings (MH) generator $M_1(Q,μ)$ and $M_2(Q,μ)$ in a continuous-time setting. While $M_1$ is the classical MH generator, we define a new generator $M_2$ that captures the opposite movement of $M_1$ and provide a comprehensive suite of comparison results ranging from hitting time and mixing time to asymptotic variance, large deviations and capacity, which demonstrate that $M_2$ enjoys superior mixing properties than $M_1$. To see that $M_1$ and $M_2$ …
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 nu_j of the generator are all real, their result states that the passage time is distributed as the sum of d independent exponential random variables with rates nu_j. We give another proof of their theorem. In the case o…
We consider a neighbourhood random walk on a quadrant, $\{(X_1(t),X_2(t),φ(t)):t\geq 0\}$, with state space \begin{eqnarray*} \mathcal{S}&=&\{(n,m,i):n,m=0,1,2,\ldots;i=1,2,\ldots,k(n,m)\}. \end{eqnarray*} Assuming start in state $(n,m,i)$, the process spends exponentially distributed amount of time in $(n,m,i)$ according to some parameter $λ_i^{(n,m)}$. Upon leaving state $(n,m,i)$ the process moves to some state $(n^{'},m^{'},j)$ with $j\in\{1,\ldots,k(n^{'},m^{'})\}$ and $n^{'}\in\{n-1,n,n+1\}$, $m^{'}\in\{m-1,m,m+1\}$, according to some probabilities $(p_{n;a}^{m;b})_{i,j}$ with $a,b\in\{+…
The cutoff phenomenon describes a case where a Markov chain exhibits a sharp transition in its convergence to stationarity. In 1996, Diaconis surveyed this phenomenon, and asked how one could recognize its occurrence in families of finite ergodic Markov chains. In 2004, the third author noted that a necessary condition for cutoff in a family of reversible chains is that the product of the mixing-time and spectral-gap tends to infinity, and conjectured that in many settings, this condition should also be sufficient. Diaconis and Saloff-Coste (2006) verified this conjecture for continuous-time b…
We show that a sequence of birth-and-death chains, given by lazy random walks in a (transient) environment (RWRE) on [0; n], exhibits a cutoff in the ballistic regime but does not exhibit a cutoff in the (interior of) the subballistic regime. We investigate the growth of the mixing times for this model. As an important step in the proof, we derive bounds for the quenched expectation and the quenched variance of the hitting times of the RWRE, which are of independent interest.
This paper provides full classification of dynamics for continuous time Markov chains (CTMCs) on the non-negative integers with polynomial transition rate functions. Such stochastic processes are abundant in applications, in particular in biology. More precisely, for CTMCs of bounded jumps, we provide necessary and sufficient conditions in terms of calculable parameters for explosivity, recurrence vs transience, certain absorption, positive recurrence vs null recurrence, and implosivity. Simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary di…
We are interested in the long-time behavior of a diploid population with sexual reproduction, characterized by its genotype composition at one bi-allelic locus. The population is modeled by a 3-dimensional birth-and-death process with competition, cooperation and Mendelian reproduction. This stochastic process is indexed by a scaling parameter $K$ that goes to infinity, following a large population assumption. When the birth and natural death parameters are of order $K$, the sequence of stochastic processes indexed by $K$ converges toward a slow-fast dynamics. We indeed prove the convergence t…
This survey concerns the study of quasi-stationary distributions with a specific focus on models derived from ecology and population dynamics. We are concerned with the long time behavior of different stochastic population size processes when 0 is an absorbing point almost surely attained by the process. The hitting time of this point, namely the extinction time, can be large compared to the physical time and the population size can fluctuate for large amount of time before extinction actually occurs. This phenomenon can be understood by the study of quasi-limiting distributions. In this paper…
This paper contributes an in-depth study of properties of continuous time Markov chains (CTMCs) on non-negative integer lattices $\N_0^d$, with particular interest in one-dimensional CTMCs with polynomial transitions rates. Such stochastic processes are abundant in applications, in particular in biology. We characterize the structure of the state space of general CTMCs on $\N_0^d$ in terms of the set of jump vectors and their corresponding transition rate functions. For CTMCs on $\N_0$ with polynomial transition rate functions, we provide threshold criteria in terms of easily computable parame…
