Reward Algorithms for Semi-Markov Processes

New algorithms for computing power moments of hitting times and accumulated rewards of hitting type for semi-Markov processes are developed. The algorithms are based on special techniques of sequential phase space reduction and recurrence relations connecting moments of rewards. Applications are discussed as well as possible generalizations of presented results and examples.     

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...     

Central Limit Theorem for Nonlinear Semi-Markov Reward Processes

Abstract

In this work, we obtain a central limit theorem for reward processes defined on a finite state space semi-Markov process, when reward functions assumed to have general forms and are not of constant rates. Martingale theory is the main tool which have been used for establishing the convergence of scaled and shifted reward process to a zero mean Brownian motion. The striking point in this article is considering general forms for the reward functions which are realistic in applications. The conditions needed for these results are existence of variances for sojourn times in each state and second order integrability of reward functions with respect to sojourn times distributions.     

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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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...     

Maximum-Likelihood Asymptotic Inference for Autoregressive Hilbertian Processes

The autoregressive Hilbertian process framework has been introduced in Bosq (2000). This book provides the nonparametric estimation of the autocorrelation and covariance operators of the autoregressive Hilbertian processes. The asymptotic properties of these estimators are also provided. The maximum likelihood approach still remains unexplored. This paper obtains the asymptotic distribution of the maximum likelihood (ML) estimators of the auto-covariance operator of the Hilbert-valued innovation process, and of the autocorrelation operator of a Gaussian autoregressive Hilbertian process of order one. A real data example is analyzed in the financial context for illustration of the performance of the projection maximum likelihood estimation methodology in the context of missing data.     

Limit Theorems for Maxima of Sums and Renewal Processes

Let {X_i}_i=1,2,... be a sequence of i.i.d. random variables, let S_n = X₁ + ... + X_n, and let S_n a.s. We discuss necessary and sufficient conditions for the Kolmogorov and Marcinkiewicz–Zygmund type strong laws of large numbers and for the law of the iterated logarithm for renewal processes defined in two different ways. Bibliography: 16 titles.     

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.     

Estimation for Discrete-time Semi-Markov Reward Processes: Analysis and Inference

In this paper, the moments of a class of reward processes defined on a discrete-time semi-Markov process and the asymptotic behaviors of the corresponding empirical estimators have been investigated. Some known results concerning the asymptotic distribution and properties of semi-Markov kernel have been obtained by a different approach. By using the empirical estimator of the semi-Markov kernel and the mentioned approach, the estimators for the moments of the reward process have been introduced and their asymptotic properties have been established. As a consequence of the strong consistency and asymptotic normality, the confidence intervals have also been obtained. A numerical example illustrates the results.     

Asymptotic Behaviors for Critical Branching Processes with Immigration

In this paper, we investigate the asymptotic behaviors of the critical branching process with immigration {Z_n, n ≥ 0}. First we get some estimation for the probability generating function of Zn. Based on it, we get a large deviation for Z_(n+1)/Z_n. Lower and upper deviations for Zn are also studied. As a by-product, an upper deviation for max_(1≤i≤n) Z_i is obtained.     

Asymptotic Recurrence and Waiting Times for Stationary Processes

Let be a discrete-valued stationary ergodic process distributed according to P and let x=(..., x _–1, x ₀, x ₁,...) denote a realization from X. We investigate the asymptotic behavior of the recurrence time R _n defined as the first time that the initial n-block reappears in the past of x. We identify an associated random walk, on the same probability space as X, and we prove a strong approximation theorem between log R _n and . From this we deduce an almost sure invariance principle for log R _n. As a byproduct of our analysis we get unified proofs for several recent results that were previously established using methods from ergodic theory, the theory of Poisson approximation and the analysis of random trees. Similar results are proved for the waiting time W _n defined as the first time until the initial n-block from one realization first appears in an independent realization generated by the same (or by a different) process.     

加权和线性过程的渐近正态性

本文建立了α-混合序列情形的加权和平稳线性过程的渐近正态性.获得的结论基于最少的权条件.所得结论将Abadir等[Econometric Theory,2014,30(1):252-284]中的结论推广至α-混合序列情形.     

Asymptotic Normality of Autoregressive Processes

In the present paper, the form of iterated limits of the moderate deviation principle for dependent variables is considered and as an application, the moderate deviation principle of m-dependent random variables is obtained.     

Asymptotic Results for Ruin Probability of a Two-Dimensional Renewal Risk Model

In this article, some asymptotic formulas of the finite-time ruin probability for a two-dimensional renewal risk model are obtained. In the model, the distributions of two claim amounts belong to the intersection of the long-tailed distributions class and the dominated varying distributions class and the claim arrival-times are extended negatively dependence structures. Assumption that the claim arrivals of two classes are governed by a common renewal counting process. The asymptotic formulas hold uniformly for t ∈ [f(x), ∞), where f(x) is an infinitely increasing function.     

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.     

Integro-Local Theorems in Boundary Crossing Problems for Compound Renewal Processes

We find sharp asymptotics for the probability that the moment when the trajectory of a compound renewal process crosses an arbitrary remote boundary lies in a prescribed small time interval. As a key step in our proof, we obtain limit theorems for the conditional distribution of jumps of the process when the endpoint of the trajectory of a compound renewal process is fixed.

     

About Earthquake Forecasting by Markov Renewal Processes

We propose and validate a new method for the evaluation of seismic hazard. In particular, our aim is to model large earthquakes consistently with the underlying geophysics. Therefore we propose a non-Poisson model, which takes into account occurrence history, improved with some physical constraints. Among the prevalent non-Poisson models, we chose the Markov renewal process, which is expected to be sufficient to capture the main characteristics, maintaining simplicity in analysis. However, due to the introduction of some physical constraint, our process differs significantly from others already presented in literature. A mixture of exponential + Weibull distributions is proposed for the waiting times and their parameters are estimated following the likelihood method. We validated our model, using data of earthquakes of high severity occurred in Turkey during the 20th century. Our results exhibit a good accordance with the real events.     

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.