Weak convergence of a sequence of Markov chains

1975 ◽  
Vol 33 (1) ◽  
pp. 41-48 ◽  
Author(s):  
Alan F. Karr
Keyword(s):  
Weak Convergence ◽  
Markov Chains

scholarly journals Relative stability and weak convergence in non-decreasing stochastically monotone Markov chains

1991 ◽  
Vol 4 (4) ◽  
pp. 293-303
Author(s):  
P. Todorovic
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Markov Chains ◽  
Renewal Process ◽  
Relative Stability ◽  
Transition Probability ◽  
Markov Renewal Process

Let {ξn} be a non-decreasing stochastically monotone Markov chain whose transition probability Q(.,.) has Q(x,{x})=β(x)>0 for some function β(.) that is non-decreasing with β(x)↑1 as x→+∞, and each Q(x,.) is non-atomic otherwise. A typical realization of {ξn} is a Markov renewal process {(Xn,Tn)}, where ξj=Xn, for Tn consecutive values of j, Tn geometric on {1,2,…} with parameter β(Xn). Conditions are given for Xn, to be relatively stable and for Tn to be weakly convergent.

Extreme values, range and weak convergence of integrals of Markov chains

1982 ◽  
Vol 19 (02) ◽  
pp. 272-288 ◽  
Author(s):  
P. J. Brockwell ◽  
S. I. Resnick ◽  
N. Pacheco-Santiago
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
State Space ◽  
Extreme Values ◽  
Finite State ◽  
Integral Process ◽  
Maximum Minimum ◽  
Finite State Space

A study is made of the maximum, minimum and range on [0,t] of the integral processwhereSis a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.

Transient phenomena for Markov chains and applications

1992 ◽  
Vol 24 (02) ◽  
pp. 322-342
Author(s):  
A. A. Borovkov ◽  
G. Fayolle ◽  
D. A. Korshunov
Keyword(s):  
Steady State ◽  
Weak Convergence ◽  
Markov Chains ◽  
Critical Behaviour ◽  
Transient Phenomena ◽  
Limit Distributions ◽  
Uniform Distributions

We consider a family of irreducible, ergodic and aperiodic Markov chains X(ε) = {X(ε) n, n ≧0} depending on a parameter ε > 0, so that the local drifts have a critical behaviour (in terms of Pakes' lemma). The purpose is to analyse the steady-state distributions of these chains (in the sense of weak convergence), when ε↓ 0. Under assumptions involving at most the existence of moments of order 2 + γ for the jumps, we show that, whenever X (0) is not ergodic, it is possible to characterize accurately these limit distributions. Connections with the gamma and uniform distributions are revealed. An application to the well-known ALOHA network is given.

On -convergence of Markov chains

1981 ◽  
Vol 90 (2) ◽  
pp. 331-333
Author(s):  
D. J. Aldous
Keyword(s):  
Weak Convergence ◽  
Markov Chains ◽  
Stochastic Processes ◽  
Discrete Topology ◽  
Natural Topology ◽  
Stationary Stochastic Processes ◽  
Image Position

Let I be a countable set, with discrete topology, and let X = (Xn), Y = (Yn) be stationary stochastic processes taking values in I. To a probabilist, the natural topology on processes (strictly speaking, on distributions of processes) is weak convergence:

Extreme values, range and weak convergence of integrals of Markov chains

1982 ◽  
Vol 19 (2) ◽  
pp. 272-288 ◽  
Author(s):  
P. J. Brockwell ◽  
S. I. Resnick ◽  
N. Pacheco-Santiago
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
State Space ◽  
Extreme Values ◽  
Finite State ◽  
Integral Process ◽  
Maximum Minimum ◽  
Finite State Space

A study is made of the maximum, minimum and range on [0, t] of the integral process where S is a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.

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