Central limit theorems for ergodic continuous-time Markov chains with applications to single birth processes

We obtain sufficient criteria for central limit theorems (CLTs) for ergodic continuous-time Markov chains (CTMCs). We apply the results to establish CLTs for continuous-time single birth processes. Moreover, we present an explicit expression of the time average variance constant for a single birth process whenever a CLT exists. Several examples are given to illustrate these results.     

Classical limit theorems for measure-valued Markov processes

Laws of large numbers, central limit theorems, and laws of the iterated logarithm are obtained for discrete and continuous time Markov processes whose state space is a set of measures. These results apply to each measure-valued stochastic process itself and not simply to its real-valued functionals.     

Switching diffusion logistic models involving singularly perturbed Markov chains: Weak convergence and stochastic permanence

Focusing on stochastic dynamics involve continuous states as well as discrete events, this article investigates stochastic logistic model with regime switching modulated by a singular Markov chain involving a small parameter. This Markov chain undergoes weak and strong interactions, where the small parameter is used to reflect rapid rate of regime switching among each state class. Two-time-scale formulation is used to reduce the complexity. We obtain weak convergence of the underlying system so that the limit has much simpler structure. Then we utilize the structure of limit system as a bridge, to invest stochastic permanence of original system driving by a singular Markov chain with a large number of states. Sufficient conditions for stochastic permanence are obtained. A couple of examples and numerical simulations are given to illustrate our results.     

On an extremal property of Markov chains and sufficiency of Markov strategies in Markov decision processes with the Dubins-Savage criterion

An inequality regarding the minimum ofP(lim inf(X _n D _n )) is proved for a class of random sequences. This result is related to the problem of sufficiency of Markov strategies for Markov decision processes with the Dubins-Savage criterion, the asymptotical behaviour of nonhomogeneous Markov chains, and some other problems.     

Waiting time distributions of runs in higher order Markov chains

We consider a {0,1}-valuedm-th order stationary Markov chain. We study the occurrences of runs where two 1’s are separated byat most/exactly/at least k 0’s under the overlapping enumeration scheme wherek≥0 and occurrences of scans (at leastk ₁ successes in a window of length at mostk, 1≤k ₁≤k) under both non-overlapping and overlapping enumeration schemes. We derive the generating function of first two types of runs. Under the conditions, (1) strong tendency towards success and (2) strong tendency towards reversing the state, we establish the convergence of waiting times of ther-th occurrence of runs and scans to Poisson type distributions. We establish the central limit theorem and law of the iterated logarithm for the number of runs and scans up to timen.     

Clustering in coagulation-fragmentation processes, random combinatorial structures and additive number systems: Asymptotic formulae and limiting laws

We develop a unified approach to the problem of clustering in the three different fields of applications indicated in the title of the paper, in the case when the parametric function of the models is regularly varying with positive exponent. The approach is based on Khintchine's probabilistic method that grew out of the Darwin-Fowler method in statistical physics. Our main result is the derivation of asymptotic formulae for the distribution of the largest and the smallest clusters (= components), as the total size of a structure (= number of particles) goes to infinity. We discover that is the threshold for the limiting distribution of the largest cluster. As a by-product of our study, we prove the independence of the numbers of groups of fixed sizes, as This is in accordance with the general principle of asymptotic independence of sites in mean-field models. The latter principle is commonly accepted in statistical physics, but not rigorously proved.

     

Sooner and later waiting time problems in a two-state Markov chain

LetX ₁,X ₂,... be a time-homogeneous {0, 1}-valued Markov chain. LetF ₀ be the event thatl runs of 0 of lengthr occur and letF ₁ be the event thatm runs of 1 of lengthk occur in the sequenceX ₁,X ₂, ... We obtained the recurrence relations of the probability generating functions of the distributions of the waiting time for the sooner and later occurring events betweenF ₀ andF ₁ by the non-overlapping way of counting and overlapping way of counting. We also obtained the recurrence relations of the probability generating functions of the distributions of the sooner and later waiting time by the non-overlapping way of counting of 0-runs of lengthr or more and 1-runs of lengthk or more.