Limit Theorems for Markov Renewal Processes

Limit Theorems for Renewal Processes

This article describes the Key renewal theorem and the Blackwell’s renewaltheorem. These two limit theorems for renewal processes are equivalent but of different forms.They are particularly useful for characterizing the asymptotic behavior of a probabilisticquantity of interest in a renewal process. We present two applications of these limit theorems:the limiting distributions o...

Limit Theorems for Multidimensional Markov Processes

An informal exposition of some recent results and conjectures. A multidimensional Markov process (mdmp) is a dynamical system (K, m, T) where: K space of the sequences of symbols from a finite alphabet / = (α, fo,... z) indexed by the elements η e Z = lattice formed by the d-ples of integers. K is regarded as K = TίηeZd I i.e. as a product space of copies of/; furthermore / is topologized by th...

Limit Theorems for Mixed Max–sum Processes with Renewal Stopping

This article is devoted to the investigation of limit theorems for mixed max–sum processes with renewal type stopping indexes. Limit theorems of weak convergence type are obtained as well as functional limit theorems. 1. Introduction. The main object of this article is the derivation of a number of limit theorems for mixed max–sum processes with renewal stopping. Such processes are constructed ...

Renewal Theorems for Markov Chains

Spectral Theory and Limit Theorems for Geometrically Ergodic Markov Processes

Consider the partial sums {St} of a real-valued functional F (Φ(t)) of a Markov chain {Φ(t)} with values in a general state space. Assuming only that the Markov chain is geometrically ergodic and that the functional F is bounded, the following conclusions are obtained: Spectral theory: Well-behaved solutions f̌ can be constructed for the “multiplicative Poisson equation” (eP )f̌ = λf̌ , where P is...