Limit Theorem For Markov Chains Research Articles

A central limit theorem for Markov chains, and its application to the calculation of transition probabilities in a mine field

Abstract In this paper we consider a sequence of real valued Markov chains xn (t), = 1, 2, ....

Ratio limit theorems for Markov chains

Ratio limit theorems for Markov chains

Ratio Limit Theorems for Markov Chains

Ratio Limit Theorems for Markov Chains

Ratio limit theorems for Markov chains

Introduction. We consider Markov chains with stationary transition probabilities and state space S=0, 1, 2, * , in both discrete and continuous time. As in [1 ] the transition probabilities are denoted by p(.) (in discrete time) and pij(t) (in continuous time). We assume that the chains considered are irreducible and recurrent, and in addition that discrete time chains are aperiodic and that continuous time chains are standard, i.e., p,j(t)->8ij as t->O+. As usual pij is written for p(). The Doblin ratio limit theorem tells us that there exist positive numbers 7ro , 7r2, such that

Information theory and the limit-theorem for Markov chains and processes with a countable infinity of states

Information theory and the limit-theorem for Markov chains and processes with a countable infinity of states

A limit theorem for Markov chains with continuous state space

Let R denote the set of real numbers, B the σ-field of all Borel subsets of R. A homogeneous Markov Chain with state space a Borel subset Ω of R is a sequence {an}, n≧ 0, of random variables, taking values in Ω, with one-step transition probabilities P(1) (ξ, A) defined by for each choice of ξ, ξ0, …, ξn−1 in ω and all Borel subsets A of ω The fact that the right-hand side of (1.1) does not depend on the ξi, 0 ≧ i > n, is of course the Markovian property, the non-dependence on n is the homogeneity of the chain.

Limit Theorems for Markov Chains with a Finite Number of States

Consider the scheme of trial sequences \[ \begin{gathered} \nu _{11} \hfill \nu _{21} ,\nu _{22} \hfill \cdots \hfill \nu _{n1} ,\nu _{n2} , \cdots ,\nu _{nn} \hfill \cdots \cdots \cdots \hfill \end{gathered} \] The sequence $\nu _{nk} $, $k = 1, \cdots ,n$, is a uniform Markov chain with a finite number of states $E_1 , \cdots ,E_s $ and a given matrix of transition probabilities \[ P = P(n) = \left\| {p_{uv} (n)} \right\|_{u,v = 1}^s . \]Let $\mu = \mu (n)$ denote the number of passages up in the n-th sequence of trials of the system through $E_1 $ on condition that the system is in state $E_1 $ at the initial (or zero-th) time. We consider the limit distribution for a sequence of random variables \[ \alpha (\mu - n\theta ),\quad \alpha = \alpha (n),\quad \theta = \theta (n). \]Theorems 1–5 give characteristic functions for some possible limit distributions.The main result of this paper is Theorem 6:If the limit distribution for$\alpha (\mu - n\theta )$exists, then it does not differ from...