Subgeometric Rates of Convergence of f-Ergodic Markov Chains

1994 ◽  
Vol 26 (3) ◽  
pp. 775-798 ◽  
Author(s):  
Pekka Tuominen ◽  
Richard L. Tweedie
Keyword(s):  
State Space ◽  
Invariant Probability Measure ◽  
Sufficient Conditions ◽  
Rates Of Convergence ◽  
General State ◽  
General State Space ◽  
Drift Conditions ◽  
Image Position ◽  
Regular Sets ◽  
Positive Recurrent

Let Φ = {Φ n} be an aperiodic, positive recurrent Markov chain on a general state space, π its invariant probability measure and f ≧ 1. We consider the rate of (uniform) convergence of Ex[g(Φ n)] to the stationary limit π (g) for |g| ≦ f: specifically, we find conditions under which as n →∞, for suitable subgeometric rate functions r. We give sufficient conditions for this convergence to hold in terms of(i) the existence of suitably regular sets, i.e. sets on which (f, r)-modulated hitting time moments are bounded, and(ii) the existence of (f, r)-modulated drift conditions (Foster–Lyapunov conditions).The results are illustrated for random walks and for more general state space models.

Subgeometric Rates of Convergence of f-Ergodic Markov Chains

1994 ◽  
Vol 26 (03) ◽  
pp. 775-798 ◽  
Author(s):  
Pekka Tuominen ◽  
Richard L. Tweedie
Keyword(s):  
State Space ◽  
Invariant Probability Measure ◽  
Sufficient Conditions ◽  
Rates Of Convergence ◽  
General State ◽  
General State Space ◽  
Drift Conditions ◽  
Image Position ◽  
Regular Sets ◽  
Positive Recurrent

Let Φ = {Φ n } be an aperiodic, positive recurrent Markov chain on a general state space, π its invariant probability measure and f ≧ 1. We consider the rate of (uniform) convergence of E x [g(Φ n )] to the stationary limit π (g) for |g| ≦ f: specifically, we find conditions under which as n →∞, for suitable subgeometric rate functions r. We give sufficient conditions for this convergence to hold in terms of (i) the existence of suitably regular sets, i.e. sets on which (f, r)-modulated hitting time moments are bounded, and (ii) the existence of (f, r)-modulated drift conditions (Foster–Lyapunov conditions). The results are illustrated for random walks and for more general state space models.

Reversibility of Markov chains with applications to storage models

1985 ◽  
Vol 22 (01) ◽  
pp. 123-137 ◽  
Author(s):  
Hideo Ōsawa
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Discrete Time ◽  
Risk Model ◽  
Sufficient Conditions ◽  
Single Server ◽  
General State ◽  
Insurance Risk ◽  
General State Space ◽  
Necessary And Sufficient

This paper studies the reversibility conditions of stationary Markov chains (discrete-time Markov processes) with general state space. In particular, we investigate the Markov chains having atomic points in the state space. Such processes are often seen in storage models, for example waiting time in a queue, insurance risk reserve, dam content and so on. The necessary and sufficient conditions for reversibility of these processes are obtained. Further, we apply these conditions to some storage models and present some interesting results for single-server queues and a finite insurance risk model.

Reversibility of Markov chains with applications to storage models

1985 ◽  
Vol 22 (1) ◽  
pp. 123-137 ◽  
Author(s):  
Hideo Ōsawa
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Discrete Time ◽  
Risk Model ◽  
Sufficient Conditions ◽  
Single Server ◽  
General State ◽  
Insurance Risk ◽  
General State Space ◽  
Necessary And Sufficient

This paper studies the reversibility conditions of stationary Markov chains (discrete-time Markov processes) with general state space. In particular, we investigate the Markov chains having atomic points in the state space. Such processes are often seen in storage models, for example waiting time in a queue, insurance risk reserve, dam content and so on. The necessary and sufficient conditions for reversibility of these processes are obtained. Further, we apply these conditions to some storage models and present some interesting results for single-server queues and a finite insurance risk model.

scholarly journals Subgeometric rates of convergence for a class of continuous-time Markov process

2005 ◽  
Vol 42 (03) ◽  
pp. 698-712
Author(s):  
Zhenting Hou ◽  
Yuanyuan Liu ◽  
Hanjun Zhang
Keyword(s):  
Markov Process ◽  
Continuous Time ◽  
Invariant Probability Measure ◽  
Sufficient Conditions ◽  
Rates Of Convergence ◽  
Transition Function ◽  
Total Variation Norm ◽  
Rate Functions ◽  
General State Space ◽  
Explicit Bounds

Let (Φ t ) t∈ℝ+ be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure π. We investigate the rates of convergence of the transition function P t (x, ·) to π; specifically, we find conditions under which r(t)||P t (x, ·) − π|| → 0 as t → ∞, for suitable subgeometric rate functions r(t), where ||·|| denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.

scholarly journals Subgeometric rates of convergence for a class of continuous-time Markov process

2005 ◽  
Vol 42 (3) ◽  
pp. 698-712 ◽  
Author(s):  
Zhenting Hou ◽  
Yuanyuan Liu ◽  
Hanjun Zhang
Keyword(s):  
Markov Process ◽  
Continuous Time ◽  
Invariant Probability Measure ◽  
Sufficient Conditions ◽  
Rates Of Convergence ◽  
Transition Function ◽  
Total Variation Norm ◽  
Rate Functions ◽  
General State Space ◽  
Explicit Bounds

Let (Φt)t∈ℝ+ be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure π. We investigate the rates of convergence of the transition function Pt(x, ·) to π; specifically, we find conditions under which r(t)||Pt(x, ·) − π|| → 0 as t → ∞, for suitable subgeometric rate functions r(t), where ||·|| denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.

scholarly journals Foster-type criteria for Markov chains on general spaces

2006 ◽  
Vol 43 (4) ◽  
pp. 1194-1200 ◽  
Author(s):  
Brian H. Fralix
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Open Problem ◽  
Stopping Times ◽  
Geometric Ergodicity ◽  
General State ◽  
Harris Recurrence ◽  
General State Space ◽  
Drift Conditions

This paper establishes new Foster-type criteria for a Markov chain on a general state space to be Harris recurrent, positive Harris recurrent, or geometrically ergodic. The criteria are based on drift conditions involving stopping times rather than deterministic steps. Meyn and Tweedie (1994) developed similar criteria involving random-sized steps, independent of the Markov chain under study. They also posed an open problem of finding criteria involving stopping times. Our results essentially solve that problem. We also show that the assumption of ψ-irreducibility is not needed when stating our drift conditions for positive Harris recurrence or geometric ergodicity.

Semi-Markov processes on a general state space: α-theory and quasi-stationarity

1980 ◽  
Vol 30 (2) ◽  
pp. 187-200 ◽  
Author(s):  
E. Arjas ◽  
E. Nummelin ◽  
R. L. Tweedie
Keyword(s):  
State Space ◽  
Markov Processes ◽  
Limit Theorems ◽  
General State ◽  
General State Space ◽  
Quasistationary Distributions ◽  
Positive Recurrent ◽  
Semi Markov Processes

AbstractBy amalgamating the approaches of Tweedie (1974) and Nummelin (1977), an α-theory is developed for general semi-Markov processes. It is shown that α-transient, α-recurrent and α-positive recurrent processes can be defined, with properties analogous to those for transient, recurrent and positive recurrent processes. Limit theorems for α-positive recurrent processes follow by transforming to the probabilistic case, as in the above references: these then give results on the existence and form of quasistationary distributions, extending those of Tweedie (1975) and Nummelin (1976).

scholarly journals Foster-type criteria for Markov chains on general spaces

2006 ◽  
Vol 43 (04) ◽  
pp. 1194-1200 ◽  
Author(s):  
Brian H. Fralix
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Open Problem ◽  
Stopping Times ◽  
Geometric Ergodicity ◽  
General State ◽  
Harris Recurrence ◽  
General State Space ◽  
Drift Conditions

This paper establishes new Foster-type criteria for a Markov chain on a general state space to be Harris recurrent, positive Harris recurrent, or geometrically ergodic. The criteria are based on drift conditions involving stopping times rather than deterministic steps. Meyn and Tweedie (1994) developed similar criteria involving random-sized steps, independent of the Markov chain under study. They also posed an open problem of finding criteria involving stopping times. Our results essentially solve that problem. We also show that the assumption of ψ-irreducibility is not needed when stating our drift conditions for positive Harris recurrence or geometric ergodicity.

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