Counterexample to a theorem on quasi-birth-and-death processes

Spatial birth and death processes as solutions of stochastic equations

Spatial birth and death processes are obtained as solutions of a system of stochastic equations. The processes are required to be locally finite, but may involve an infinite population over the full (noncompact) type space. Conditions are given for existence and uniqueness of such solutions, and for temporal and spatial ergodicity. For birth and death processes with constant death rate, a sub-criticality condition on the birth rate implies that the process is ergodic and converges exponentially fast to the stationary distribution.

On the cycle maximum of birth-death processes and networks of queues

This paper considers the cycle maximum in birth-death processes as a stepping stone to characterisation of the cycle maximum in single queues and open Kelly-Whittle networks of queues. For positive recurrent birth-death processes we show that the sequence of sample maxima is stochastically compact. For transient birth-death processes we show that the sequence of sample maxima conditioned on the maximum being finite is stochastically compact. We show that the Markov chain recording the total number of customers in a Kelly-Whittle network is a birth-death process with birth and death rates deter…

On variation of eigenvalues of birth and death matrices and random walk matrices

The purpose of this note is twofold: firstly to improve the known results on variation of extreme eigenvalues of birth and death matrices and random walk matrices; and secondly to progress towards the solution of a thirty years old open problem concerning the variation of eigenvalues of these matrices. Keywords: Birth and death matrices, random walk matrices, eigenvalues, monotonicity

具有两类灾难的生灭过程

Birth-Death processes with two-type catastrophes

研究两类灾难作用下的一般生灭过程，利用无灾难过程的转移概率 Laplace 变换表达带灾难过程，并给出首次有效灾难发生时间的密度、期望与方差结果。

Non-Local Solvable Birth-Death Processes

In this paper we study strong solutions of some non-local difference-differential equations linked to a class of birth-death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth-death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth-death processes.

Bilateral Birth and death process in quantum calculus

In this paper I shall give the complete solution of the equations governing the bilateral birth and death process on path set $\mathbb{R}_q=\{q^n,\quad n\in\mathbb{Z}\}$ in which the birth and death rates $λ_n=q^{2ν-2n}$ and $μ_n=q^{-2n}$ where $0<q<1$ and $ν>-1$ . The mathematical methods employed here are based on $q$-Bessel Fourier analysis.

Applying Brownian motion to the study of birth-death chains

Basic properties of Brownian motion are used to derive two results concerning birth-death chains. First, the probability of extinction is calculated. Second, sufficient conditions on the transition probabilities of a birth-death chain are given to ensure that the expected value of the chain converges to a limit. The theory of Brownian motion local time figures prominently in the proof of the second result.

An Optional Stopping-like theorem for large stopping times on birth-death chains

An elementary proof is given for a theorem showing that certain birth-death chains show martingale-like behavior at large stopping times. This is a generalization of and new proof for a theorem from a earlier paper by the author.

有限生灭过程中稀疏计数的二阶性质

Second Order Properties of Thinned Counts in Finite Birth--Death Processes

本文研究了有限生灭过程中稀疏计数的二阶性质，推导了渐近方差率公式，并通过排队模型示例展示了其适用范围。

On the Strong Ratio Limit Property for Discrete-Time Birth-Death Processes

A sufficient condition is obtained for a discrete-time birth-death process to possess the strong ratio limit property, directly in terms of the one-step transition probabilities of the process. The condition encompasses all previously known sufficient conditions.

On some transformations of bilateral birth-and-death processes with applications

A method yielding simple relationships among bilateral birth-and-death processes is outlined. This allows one to relate birth and death rates of two processes in such a way that their transition probabilities, first-passage-time densities and ultimate crossing probabilities are mutually related by some product-form expressions.

Sensitivity analysis of Quasi-Birth-and-Death processes

We perform the sensitivity analysis of a level-dependent QBD with a particular focus on applications in modelling healthcare systems.

Hitting Time Distributions for Denumerable Birth and Death Processes

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.

Markov birth-and-death dynamics of populations

Spatial birth-and-death processes with a finite number of particles are obtained as unique solutions to certain stochastic equations. Conditions are given for existence and uniqueness of such solutions, as well as for continuous dependence on the initial conditions. The possibility of an explosion and connection with the heuristic generator of the process are discussed.

How do birth and death processes come down from infinity?

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…

Symmetry in the Green's function for birth-death chains

A symmetric relation in the probabilistic Green's function for birth-death chains is explored. Two proofs are given, each of which makes use of the known symmetry of the Green's functions in other contexts. The first uses as primary tool the local time of Brownian motion, while the second uses the reciprocity principle from electric network theory. We also show that the the second proof extends easily to cover birth-death chains (a.k.a. state-dependent random walks) on trees.

Truncated correlations in the stirring process with births and deaths

We consider the stirring process in the interval $\La_N:=[-N,N]$ of $\mathbb Z$ with births and deaths taking place in the intervals $I_+:=(N-K,N]$, $K>0$, and respectively $I_-:=[-N,-N+K)$. We prove bounds on the truncated moments uniform in $N$ which yield strong factorization properties.

A local large deviation principle for inhomogeneous birth-death processes

The paper considers a continuous-time birth-death process where the jump rate has an asymptotically polynomial dependence on the process position. We obtain a rough exponential asymptotics for the probability of excursions of a re-scaled process contained within a neighborhood of a given continuous non-negative function.

Alternating birth-death processes

We consider a continuous time Markov process on $\mathbb{N}_0$ which can be interpreted as generalized alternating birth-death process in a non-autonomous random environment. Depending on the status of the environment the process either increases until the environment changes and the process starts to decrease until the environment changes again, and the process restarts to increase, and so on, or its starts decreasing, reversing its direction due to environmental changes, et cetera. The birth and death rates depend on the state (height, population size) of the birth-death process and the envi…

Reproduction of initial distributions from the first hitting time distribution for birth-and-death processes

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.

Computing cutoff times of birth and death chains

Earlier work by Diaconis and Saloff-Coste gives a spectral criterion for a maximum separation cutoff to occur for birth and death chains. Ding, Lubetzky and Peres gave a related criterion for a maximum total variation cutoff to occur in the same setting. Here, we provide complementary results which allow us to compute the cutoff times and windows in a variety of examples.

Spectral computations for birth and death chains

We consider the spectrum of birth and death chains on a $n$-path. An iterative scheme is proposed to compute any eigenvalue with exponential convergence rate independent of $n$. This allows one to determine the whole spectrum in order $n^2$ elementary operations. Using the same idea, we also provide a lower bound on the spectral gap, which is of the correct order on some classes of examples.

Lineage-through-time plots of birth-death processes

We calculate the density and expectation for the number of lineages in a reconstructed tree with $n$ extant species. This is done with conditioning on the age of the tree as well as with assuming a uniform prior for the age of the tree.

A note on scaling limits for truncated birth-and-death processes with interaction

In this note we consider a Markov chain formed by a finite system of interacting birth-and-death processes on a finite state space. We study an asymptotic behaviour of the Markov chain as its state space becomes large. In particular, we show that the appropriately scaled Markov chain converges to a diffusion process, and derive conditions for existence of the stationary distribution of the limit diffusion process in special cases.

The Cutoff Phenomenon for Random Birth and Death Chains

For any distribution $π$ with support equal to $[n] = \{1, 2,..., n \}$, we study the set $\mathcal{A}_π$ of tridiagonal stochastic matrices $K$ satisfying $π(i) K[i,j] = π(j) K[j,i]$ for all $i, j \in [n]$. These matrices correspond to birth and death chains with stationary distribution $π$. We study matrices $K$ drawn uniformly from $\mathcal{A}_π$, following the work of Diaconis and Wood on the case $π(i) = \frac{1}{n}$. We analyze a `block sampler' version of their algorithm for drawing from $\mathcal{A}_π$ at random, and use results from this analysis to draw conclusions about typical mat…

生灭过程的Ray-Knight紧化

Ray-Knight compactification of birth and death processes

该研究针对生灭过程非唯一情形的构造问题，提出用Ray-Knight紧化方法，兼具解析与概率方法优势，推导了生成元并给出新的概率构造。

关于灭绝时带移民的时间改变线性生灭过程

On Time-Changed Linear Birth-Death Process with Immigration at Extinction

本文研究了灭绝时带移民的时间改变线性生灭过程，提出TFLBDPwI模型，利用Adomian分解法求其暂态概率，推导前两阶矩，并详细分析特殊情形的分布性质。

Intertwining of birth-and-death processes

It has been known for a long time that for birth-and-death processes started in zero the first passage time of a given level is distributed as a sum of independent exponentially distributed random variables, the parameters of which are the negatives of the eigenvalues of the stopped process. Recently, Diaconis and Miclo have given a probabilistic proof of this fact by constructing a coupling between a general birth-and-death process and a process whose birth rates are the negatives of the eigenvalues, ordered from high to low, and whose death rates are zero, in such a way that the latter proce…

Intertwinings and Stein's magic factors for birth-death processes

This article investigates second order intertwinings between semigroups of birth-death processes and discrete gradients on the space of natural integers N. It goes one step beyond a recent work of Chafa{ï} and Joulin which establishes and applies to the analysis of birth-death semigroups a first order intertwining. Similarly to the first order relation, the second order intertwining involves birth-death and Feynman-Kac semigroups and weighted gradients on N, and can be seen as a second derivative relation. As our main application, we provide new quantitative bounds on the Stein factors of disc…

Functional Central Limit Theorems for Local Statistics of Spatial Birth-Death Processes in the Thermodynamic Regime

We present normal approximation results at the process level for local functionals defined on dynamic Poisson processes in $\mathbb{R}^d$. The dynamics we study here are those of a Markov birth-death process. We prove functional limit theorems in the so-called thermodynamic regime. Our results are applicable to several functionals of interest in the stochastic geometry literature, including subgraph and component counts in the random geometric graphs.

Long term behaviour of locally interacting birth-and-death processes

In this paper we study long-term evolution of a finite system of locally interacting birth-and-death processes labelled by vertices of a finite connected graph. A detailed description of the asymptotic behaviour is obtained in the case of both constant vertex degree graphs and star graphs. The model is motivated by modelling interactions between populations and is related to interacting particle systems, Gibbs models with unbounded spins, as well as urn models with interaction.

Stability on {0,1,2,...}^S: birth-death chains and particle systems

A strong negative dependence property for measures on {0,1}^n - stability - was recently developed in [5], by considering the zero set of the probability generating function. We extend this property to the more general setting of reaction-diffusion processes and collections of independent Markov chains. In one dimension the generalized stability property is now independently interesting, and we characterize the birth-death chains preserving it.

Minimal quasi-stationary distribution approximation for a birth and death process

In a first part, we prove a Lyapunov-type criterion for the $ξ\_1$-positive recurrence of absorbed birth and death processes and provide new results on the domain of attraction of the minimal quasi-stationary distribution. In a second part, we study the ergodicity and the convergence of a Fleming-Viot type particle system whose particles evolve independently as a birth and death process and jump on each others when they hit $0$. Our main result is that the sequence of empirical stationary distributions of the particle system converges to the minimal quasi-stationary distribution of the birth a…

零常返生灭动力环境中的随机游走

Random walk in a null recurrent birth-and-death dynamical environment

本文研究零常返生灭动力环境中的随机游走，证明其位置标度极限为布朗运动的次线性时间变换，确立次扩散性，与遍历版本的扩散性结果形成对比。

时间非齐次生灭过程的常返与暂态条件

Conditions for recurrence and transience for time-inhomogeneous birth-and-death processes

本文推导了时间非齐次生灭过程（视为正偏置漂移随机游走）的常返与暂态条件，建立的一般性结果涵盖了Menshikov和Volkov的已知特例。

On Spatial Point Processes with Uniform Births and Deaths by Random Connection

This paper is focused on a class of spatial birth and death process of the Euclidean space where the birth rate is constant and the death rate of a given point is the shot noise created at its location by the other points of the current configuration for some response function $f$. An equivalent view point is that each pair of points of the configuration establishes a random connection at an exponential time determined by $f$, which results in the death of one of the two points. We concentrate on space-motion invariant processes of this type. Under some natural conditions on $f$, we construct …

Time-fractional Birth and Death Processes

In this article, we provide different representations for a time-fractional birth and death process $N_α(t)$, whose transition probabilities are governed by a time-fractional system of differential equations. More specifically, we present two equivalent characterizations for its trajectories: the first one as a time-changed classic birth and death process, whereas the second one is a Markov renewal process. Also, we provide results for the asymptotic behavior of the process conditioned not to be killed. The most important is that the concept of quasi-limiting distribution and quasi-stationary …

生灭过程的逼近

Approximation of birth-death processes

本文研究生灭过程在 $|ν|=\infty$ 等情形下的轨迹描述难题，提出用两类更简单的生灭过程逼近原过程，并证明相应概率测度在 càdlàg 函数空间上的弱收敛。

生灭过程是经过时间变换的 Feller 布朗运动

Birth-death processes are time-changed Feller's Brownian motions

本文通过特定时间变换建立 Feller 布朗运动与生灭过程的双向等价关系，并给出参数精确对应与构造性路径表示框架。

Lattice birth-and-death processes

Lattice birth-and-death Markov dynamics of particle systems with spins from the set of non-negative integers are constructed as unique solutions to certain stochastic equations. Pathwise uniqueness, strong existence, Markov property and joint uniqueness in law are proven, and a martingale characterization of the process is given. Sufficient conditions for the existence of an invariant distribution are formulated in terms of Lyapunov functions. We apply obtained results to discrete analogs of the Bolker--Pacala--Dieckmann--Law model and an aggregation model.

On a bilateral birth-death process with alternating rates

We consider a bilateral birth-death process characterized by a constant transition rate $λ$ from even states and a possibly different transition rate $μ$ from odd states. We determine the probability generating functions of the even and odd states, the transition probabilities, mean and variance of the process for arbitrary initial state. Some features of the birth-death process confined to the non-negative integers by a reflecting boundary in the zero-state are also analyzed. In particular, making use of a Laplace transform approach we obtain a series form of the transition probability from s…

Fecundity regulation in a spatial birth-and-death process

We study a Markov birth-and-death process on a space of locally finite configurations, which describes an ecological model with a density dependent fecundity regulation mechanism. We establish existence and uniqueness of this process and analyze its properties. In particular, we show global time-space boundedness of the population density and, using a constructed Foster-Lyapunov-type function, we study return times to certain level sets of tempered configurations. We find also sufficient conditions that the degenerate invariant distribution is unique for the considered process.

A Hard-core Stochastic Process with Simultaneous Births and Deaths

We consider a stochastic spatial point process with births and deaths on $\mathbb{R}^d$, with the hard-core property that at any time the balls of radius half of any two points do not overlap. We give explicit construction of the process. Under some more conditions, we show the stochastic stability of this Markov process by constructing a stationary regime for the dynamics. The main tool used to construct the stationary regime is the coupling form the past technique. Further, we study the exponential convergence of the probability distribution to the stationary distribution.

Quasi-stationary distribution for multi-dimensional birth and death processes conditioned to survival of all coordinates

This article studies the quasi-stationary behaviour of multidimensional birth and death processes, modeling the interaction between several species, absorbed when one of the coordinates hits 0. We study models where the absorption rate is not uniformly bounded, contrary to most of the previous works. To handle this natural situation, we develop original Lyapunov function arguments that might apply in other situations with unbounded killing rates. We obtain the exponential convergence in total variation of the conditional distributions to a unique stationary distribution, uniformly with respect…

A note on the invariant distribution of a quasi-birth-and-death process

The aim of this paper is to give an explicit formula of the invariant distribution of a quasi-birth-and-death process in terms of the block entries of the transition probability matrix using a matrix-valued orthogonal polynomials approach. We will show that the invariant distribution can be computed using the squared norms of the corresponding matrix-valued orthogonal polynomials, no matter if they are or not diagonal matrices. We will give an example where the squared norms are not diagonal matrices, but nevertheless we can compute its invariant distribution.

Stationary distribution for birth and death process with one-side bounded jumps

In this paper, we study a birth and death process $\{N_t\}_{t\ge0}$ on positive half lattice, which at each discontinuity jumps at most a distance $R\ge 1$ to the right or exactly a distance $1$ to the left. The transitional probabilities at each site are nonhomogeneous. Firstly, sufficient conditions for the recurrence and positive recurrence are presented. Then by the branching structure within random walk with one-side bounded jumps set up in Hong and Wang (2013), the explicit form of the stationary distribution of the process $\{N_t\}_{t\ge0}$ is formulated.

Separation cut-offs for birth and death chains

This paper gives a necessary and sufficient condition for a sequence of birth and death chains to converge abruptly to stationarity, that is, to present a cut-off. The condition involves the notions of spectral gap and mixing time. Y. Peres has observed that for many families of Markov chains, there is a cut-off if and only if the product of spectral gap and mixing time tends to infinity. We establish this for arbitrary birth and death chains in continuous time when the convergence is measured in separation and the chains all start at 0.

灭绝与灾难在出生-死亡过程中的区别及其在种群遗传学中的应用

Killing versus catastrophes in birth-death processes and an application to population genetics

本文研究了出生-死亡过程中的灭绝与灾难现象，通过数学工具建立两者联系，并应用于种群遗传学中的祖先过程分析。

生灭连续时间随机游走的爆炸与内爆

Explosion and implosion of birth-and-death continuous-time random walks

本文针对非马尔可夫生灭连续时间随机游走，推导其爆炸、内爆的充要正则性判据，并给出两类半马尔可夫情形下的显式判定条件。

关于线性生灭过程的分数阶变体

On a Fractional Variant of Linear Birth-Death Process

该研究引入广义分数阶线性生灭过程（GFLBDP），通过正则化Hilfer-Prabhakar导数定义，推导其时间变换表示、状态概率及矩的显式表达式，揭示灭绝概率与广义分数阶泊松过程的关系并探讨积分过程的渐近特性与应用。

实值结构出生-死亡过程的长时间行为与Yaglom极限

Long time behavior and Yaglom limit for real trait-structured Birth and Death Processes

本文研究连续时间测度值出生-死亡过程的长时间行为，在临界、次临界和超临界三种状态下，证明了矩和灭绝概率的新递推及时间渐近性，并建立了条件于非灭绝的Q-过程和Yaglom极限的存在性，且给出了三类自然例子。

Multivariate Krawtchouk polynomials and composition birth and death processes

This paper defines the multivariate Krawtchouk polynomials, orthogonal on the multinomial distribution, and summarizes their properties as a review. The multivariate Krawtchouk polynomials are symmetric functions of orthogonal sets of functions defined on each of N multinomial trials. The dual multivariate Krawtchouk polynomials, which also have a polynomial structure, are seen to occur naturally as spectral orthogonal polynomials in a Karlin and McGregor spectral representation of transition functions in a composition birth and death process. In this Markov composition process in continuous t…

On the first-visit-time problem for birth and death processes with catastrophes

For a birth-death process subject to catastrophes, defined on the state-space $S=\{r,r+1,r+2,...\}$, with $r$ a positive integer or zero, the first-visit time to a state $k\in S$ is considered and the Laplace transform of its probability density function is determined, use of which is then made to obtain mean and variance. The Laplace transform of the probability density function of the first effective catastrophe occurrence time and its expected value are also obtained. Some extensions to time-non-homogeneous processes are then provided. Finally, certain additional results concerning the dete…

分布依赖的出生-死亡过程：$\mathbb{W}_p$-估计、遍历性与混沌传播

Distribution Dependent Birth-Death Processes: $\mathbb{W}_p$-Estimate, Ergodicity and Propagation of Chaos

本文研究了时间非齐次分布依赖出生-死亡过程的well-posedness、$\mathbb{W}_p$-估计、指数遍历性及时间一致的混沌传播，扩展了分布依赖SDE和均场粒子系统的相关结果。