Estimation in Stationary Markov Renewal Processes,with Application to Earthquake Forecasting in Turkey

Consider a process in which different events occur, with random inter-occurrence times. In Markov renewal processes as well as in semi-Markov processes, the sequence of events is a Markov chain and the waiting distributions depend only on the types of the last and the next event. Suppose that the state-space is finite and that the process started far in the past, achieving stationary. Weibull distributions are proposed for the waiting times and their parameters are estimated jointly with the transition probabilities through maximum likelihood, when one or several realizations of the process are observed over finite windows. The model is illustrated with data of earthquakes of three types of severity that occurred in Turkey during the 20th century.AMS 2000 Subject Classification: 60K20     

Bayesian Nonparametric Estimation for Reinforced Markov Renewal Processes

Starting from the definitions and the properties of reinforced renewal processes and reinforced Markov renewal processes, we characterize, via exchangeability and de Finetti’s representation theorem, a prior that consists of a family of Dirichlet distributions on the space of Markov transition matrices and beta-Stacy processes on distribution functions. Then, we show that this family is conjugate and give some estimate results.

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Limit Theorems for Stopped Functionals of Markov Renewal Processes

Some results for stopped random walks are extended to the Markov renewal setup where the random walk is driven by a Harris recurrent Markov chain. Some interesting applications are given; for example, a generalization of the alternating renewal process.     

The Geometric Markov Renewal Processes with Application to Finance

We introduce the geometric Markov renewal processes as a model for a security market and study this processes in a series scheme. We consider its approximations in the form of averaged, merged and double averaged geometric Markov renewal processes. Weak convergence analysis and rates of convergence of ergodic geometric Markov renewal processes are presented. Martingale properties, infinitesimal operators of geometric Markov renewal processes are presented and a Markov renewal equation for expectation is derived. As an application, we consider the case of two ergodic classes. Moreover, we consider a generalized binomial model for a security market induced by a position dependent random map as a special case of a geometric Markov renewal process.     

Moment Solutions for the State Exiting Counting Processes of a Markov Renewal Process

Important performance measures for many Markov renewal processes are the counts of the exits from each state. We present solutions for the conditional first, second, and covariance moments of the state exiting counting processes for a Markov renewal process, and solutions for the unconditional equilibrium versions of the moments. We demonstrate the relationship between the conditional first moments for the state exiting and the state entering counting processes. For analytical and illustrative purposes, we concentrate on the two state case. Two asymptotic expansions for the moment functions are proposed and evaluated both analytically and empirically. The two approximations are shown to be competitive in terms of absolute relative error, but the second approximation has a simpler analytical form which is useful in analyzing more complex stochastic processes having an underlying MRP structure.     

Self-Intersections of Markov Processes

We describe explicit conditions on the transition density functionsof a time-homogeneous continuous. Markov process so that almostevery path has multiple points. The application to two- andthree-dimensional diffusions is exhibited.     

Extreme Value Asymptotics for Multivariate Renewal Processes

For a sequence of partial sums ofd-dimensional independent identically distributed random vectors a corresponding multivariate renewal process is defined componentwise. Via strong invariance together with an extreme value limit theorem for Rayleigh processes, a number of weak asymptotic results are established for thed-dimensional renewal process. Similar theorems for the estimated version of this process are also derived. These results are suggested to serve as simultaneous asymptotic testing devices for detecting changes in the multivariate setting.     

Functional Limit Theorems for Compound Renewal Processes

Siberian Mathematical Journal - We generalize Anscombe’s Theorem to the case of stochastic processes converging to a continuous random process. As applications, we find a simple proof of an...     

Weighted Approximation of the Renewal Spacing Processes

In this paper, we provide a weighted approximation for the renewal spacing empirical and quantile processes. Some linear bounds for the empirical distribution and quantile functions are also given.     

Refined Asymptotic Costs for Renewal Reward Processes

A common criteria function for the selection of decision variables in renewal reward processes is the leading term in the asymptotic expansion of the expected cost per unit time, namely the ratio of expected cost per cycle to expected cycle length. The general implications of refining this criteria function by adopting the first two terms in the asymptotic expansion as an objective function are discussed. An analysis for both an age based and a cumulative damage based renewal reward process is presented.     

关于马尔可夫更新定理的若干局部渐近结果

在文[3]的基础上,考虑了逗留时间服从Δ-次指数分布的马尔可夫更新测度的局部渐近表达式,同时推广了关键更新定理.     

On weakly mixing Markov Processes

A result of England and Martin on weak mixing (see [6]) is extended to Markov Processes in a strengthened form, and also to continuous time Markov Processes.     

Markov Processes with Equal Capacities

Let X and be transient standard Markov processes in weak duality with respect to a -finite measure m. Let (Y, , ) be a second dual pair with the same state space E as (X, , m). Let Cap ^X and Cap ^Y be the 0-order capacities associated with (X, , m) and (Y, , ), and let V and denote the potential kernels for Y and . Assume that singletons are polar with respect to both X and Y, and that semipolar sets are of capacity zero for both dual pairs. We show that if Cap ^X (B)=Cap ^Y (B) for every Borel subset of E then there is a strictly increasing continuous additive functional D=(D _t) _t0 of (X, , m) such that

A Markov Renewal Based Model for Wireless Networks

We propose a queueing network model which can be used for the integration of the mobility and teletraffic aspects that are characteristic of wireless networks. In the general case, the model is an open network of infinite server queues where customers arrive according to a non-homogeneous Poisson process. The movement of a customer in the network is described by a Markov renewal process. Moreover, customers have attributes, such as a teletraffic state, that are driven by continuous time Markov chains and, therefore, change as they move through the network. We investigate the transient and limit number of customers in disjoint sets of nodes and attributes. These turn out to be independent Poisson random variables. We also calculate the covariances of the number of customers in two sets of nodes and attributes at different time epochs. Moreover, we conclude that the arrival process per attribute to a node is the sum of independent Poisson cluster processes and derive its univariate probability generating function. In addition, the arrival process to an outside node of the network is a non-homogeneous Poisson process. We illustrate the applications of the queueing network model and the results derived in a particular wireless network.     

马尔可夫过程的与时间有关的函数

设（Ｘｔ）是有转移函数的马尔可夫过程,其中Ｘｔ取值于状态空间（Ｅｔ,ξ,ｔ≥０。设ｆｔ是（Ｅｔ,ξ）到状态空间（Ｅｔ,ξ是的可测变换。本文给出了使（ｆｔ（Ｘｔ）仍是有转移函数的马尔可夫过程的充分条件,对于有函数的马尔可夫过程族,也讨论了类似的问题。     

Strong Limit Theorems for Increments of Renewal Processes

We study the almost sure behavior of increments of renewal processes. We derive a universal form of norming functions in the strong limit theorems for increments of such processes. This result unifies the following well-known theorems for increments of renewal processes: the strong law of large numbers, Erdos-Renyi law, Csorgo-Revesz law, and law of the iterated logarithm. New results are obtained for processes with distributions of renewal times from domains of attraction of the normal law and completely asymmetric stable laws with index (1, 2). Bibliography: 15 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 298, 2003, pp. 208–225.     

On Large Deviation Principles for Compound Renewal Processes

Mathematical Notes - The notion of a Hadamard decomposition of a semisimple associative finite-dimensional complex algebra generalizes the notion of classical Hadamard matrices, which correspond to...     

Markov Jump Processes in Estimating Sharing of Identity by Descent

Identity by descent(IBD) sharing is a very important genomic feature in population genetics which can be used to reconstruct recent demographic history.In this paper we provide a framework to estimate IBD sharing for a demographic model called two-population model with migration.We adopt the structured coalescent theory and use a continuous-time Markov jump process {X(t),t≥0} to describe the genealogical process in such model.Then we apply Kolmogorov backward equation to calculate the distributi...