On the Convergence Rate of Quasi Lumpable Markov Chains

2006 ◽  
pp. 138-147
Author(s):  
András Faragó
Keyword(s):  
Convergence Rate ◽  
Markov Chains

Conductance bounds on the L 2 convergence rate of Metropolis algorithms on unbounded state spaces

2004 ◽  
Vol 36 (01) ◽  
pp. 243-266
Author(s):  
Søren F. Jarner ◽  
Wai Kong Yuen
Keyword(s):  
Convergence Rate ◽  
Markov Chains ◽  
Spectral Gap ◽  
Unbounded Domains ◽  
Decomposition Theorem ◽  
Metropolis Algorithm ◽  
The State ◽  
Bounded Domains ◽  
Target Density ◽  
State Spaces

In this paper we derive bounds on the conductance and hence on the spectral gap of a Metropolis algorithm with a monotone, log-concave target density on an interval of ℝ. We show that the minimal conductance set has measure ½ and we use this characterization to bound the conductance in terms of the conductance of the algorithm restricted to a smaller domain. Whereas previous work on conductance has resulted in good bounds for Markov chains on bounded domains, this is the first conductance bound applicable to unbounded domains. We then show how this result can be combined with the state-decomposition theorem of Madras and Randall (2002) to bound the spectral gap of Metropolis algorithms with target distributions with monotone, log-concave tails on ℝ.

A comparison of the ordinary and a varying parameter exponential smoothing

1989 ◽  
Vol 26 (4) ◽  
pp. 784-792 ◽  
Author(s):  
Heikki Bonsdorff
Keyword(s):  
Convergence Rate ◽  
Markov Chains ◽  
Finite Interval ◽  
Exponential Smoothing ◽  
Uniform Ergodicity ◽  
Geometric Convergence ◽  
Monotonically Increasing Function ◽  
Increasing Function

An adaptive-type exponential smoothing, motivated by an insurance tariff problem, is treated. We consider the process Zn = ß(Zn –1)Xn +(1 – ß (Zn–1))Zn–1, where Xn are i.i.d. taking values in the interval [0, M], M ≦ ∞ and ß is a monotonically increasing function [0, M] → [c, d], 0 < c < d < 1.Together with (Zn), we consider the ordinary exponential smoothing Yn = αXn + (1 – α)Yn –1 where α is a constant, 0 < α < 1. We show that (Yn) and (Zn) are geometrically ergodic Markov chains (in the case of finite interval we even have uniform ergodicity) and that EYn, EZn converge to limits EY, EZ, respectively, with a geometric convergence rate. Moreover, we show that Ez is strictly less than EY = EXn.

scholarly journals Computable bounds of an 𝓁²-spectral gap for discrete Markov chains with band transition matrices

2016 ◽  
Vol 53 (3) ◽  
pp. 946-952
Author(s):  
Loï Hervé ◽  
James Ledoux
Keyword(s):  
Convergence Rate ◽  
Markov Chains ◽  
Spectral Radius ◽  
Spectral Gap ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Geometric Ergodicity ◽  
Transition Matrices ◽  
Essential Spectral Radius ◽  
Band Transition

AbstractWe analyse the 𝓁²(𝜋)-convergence rate of irreducible and aperiodic Markov chains with N-band transition probability matrix P and with invariant distribution 𝜋. This analysis is heavily based on two steps. First, the study of the essential spectral radius ress(P|𝓁²(𝜋)) of P|𝓁²(𝜋) derived from Hennion’s quasi-compactness criteria. Second, the connection between the spectral gap property (SG2) of P on 𝓁²(𝜋) and the V-geometric ergodicity of P. Specifically, the (SG2) is shown to hold under the condition α0≔∑m=−NNlim supi→+∞(P(i,i+m)P*(i+m,i)1∕2<1. Moreover, ress(P|𝓁²(𝜋)≤α0. Effective bounds on the convergence rate can be provided from a truncation procedure.

A comparison of the ordinary and a varying parameter exponential smoothing

1989 ◽  
Vol 26 (04) ◽  
pp. 784-792 ◽  
Author(s):  
Heikki Bonsdorff
Keyword(s):  
Convergence Rate ◽  
Markov Chains ◽  
Finite Interval ◽  
Exponential Smoothing ◽  
Uniform Ergodicity ◽  
Geometric Convergence ◽  
Monotonically Increasing Function ◽  
Increasing Function

An adaptive-type exponential smoothing, motivated by an insurance tariff problem, is treated. We consider the process Zn = ß(Zn – 1)Xn +(1 – ß (Zn – 1))Zn – 1, where Xn are i.i.d. taking values in the interval [0, M], M ≦ ∞ and ß is a monotonically increasing function [0, M] → [c, d], 0 &lt; c &lt; d &lt; 1. Together with (Zn ), we consider the ordinary exponential smoothing Yn = αXn + (1 – α)Yn – 1 where α is a constant, 0 &lt; α &lt; 1. We show that (Yn ) and (Zn ) are geometrically ergodic Markov chains (in the case of finite interval we even have uniform ergodicity) and that EYn, EZn converge to limits EY, EZ, respectively, with a geometric convergence rate. Moreover, we show that Ez is strictly less than EY = EXn.

scholarly journals Passage-Time Moments for Positively Recurrent Markov Chains

2001 ◽  
Vol 162 ◽  
pp. 169-185
Author(s):  
Tokuzo Shiga ◽  
Akinobu Shimizu ◽  
Takahiro Soshi
Keyword(s):  
Markov Chain ◽  
Convergence Rate ◽  
Markov Chains ◽  
Direct Product ◽  
Transition Probability ◽  
Passage Time ◽  
State Spaces ◽  
Countable State ◽  
Fractional Moments ◽  
Passage Times

Fractional moments of the passage-times are considered for positively recurrent Markov chains with countable state spaces. A criterion of the finiteness of the fractional moments is obtained in terms of the convergence rate of the transition probability to the stationary distribution. As an application it is proved that the passage time of a direct product process of Markov chains has the same order of the fractional moments as that of the single Markov chain.

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