VARIANCE BOUNDING MARKOV CHAINS, L2-UNIFORM MEAN ERGODICITY AND THE CLT

2011 ◽  
Vol 11 (01) ◽  
pp. 81-94 ◽  
Author(s):  
YVES DERRIENNIC ◽  
MICHAEL LIN
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Markov Chains ◽  
Additive Functional ◽  
Finite Variance ◽  
Geometric Ergodicity ◽  
The Central Limit Theorem ◽  
Harris Recurrence ◽  
Mean Ergodicity ◽  
Compact Abelian Groups

We prove that variance bounding Markov chains, as defined by Roberts and Rosenthal [31], are uniformly mean ergodic in L2 of the invariant probability. For such chains, without any additional mixing, reversibility, or Harris recurrence assumptions, the central limit theorem and the invariance principle hold for every centered additive functional with finite variance. We also show that L2-geometric ergodicity is equivalent to L2-uniform geometric ergodicity. We then specialize the results to random walks on compact Abelian groups, and construct a probability on the unit circle such that the random walk it generates is L2-uniformly geometrically ergodic, but is not Harris recurrent.

THE CENTRAL LIMIT THEOREM FOR STATIONARY MARKOV CHAINS UNDER INVARIANT SPLITTINGS

2004 ◽  
Vol 04 (01) ◽  
pp. 15-30 ◽  
Author(s):  
MIKHAIL GORDIN ◽  
HAJO HOLZMANN
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Markov Chains ◽  
Central Limit ◽  
Homogeneous Spaces ◽  
Markov Operators ◽  
The Central Limit Theorem ◽  
Integrable Functions ◽  
Compact Abelian Groups ◽  
Square Integrable Functions

The central limit theorem (CLT) for stationary ergodic Markov chains is investigated. We give a short survey of related results on the CLT for general (not necessarily Harris recurrent) chains and formulate a new sufficient condition for its validity. Furthermore, Markov operators are considered which admit invariant orthogonal splittings of the space of square-integrable functions. We show how conditions for the CLT can be improved if this additional structure is taken into account. Finally we give examples of this situation, namely endomorphisms of compact Abelian groups and random walks on compact homogeneous spaces.

Heavy traffic limit theorems in fluctuation theory

1978 ◽  
Vol 10 (04) ◽  
pp. 852-866
Author(s):  
A. J. Stam
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
The Central Limit Theorem ◽  
Heavy Traffic Limit ◽  
Oscillating Random Walk

Let be a family of random walks with For ε↓0 under certain conditions the random walk U (∊) n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M ∊ = max {U (∊) n , n ≧ 0}, v 0 = min {n : U (∊) n = M ∊}, v 1 = max {n : U (∊) n = M ∊}. The joint limiting distribution of ∊2σ∊ –2 v 0 and ∊σ∊ –2 M ∊ is determined. It is the same as for ∊2σ∊ –2 v 1 and ∊σ–2 ∊ M ∊. The marginal ∊σ–2 ∊ M ∊ gives Kingman's heavy traffic theorem. Also lim ∊–1 P(M ∊ = 0) and lim ∊–1 P(M ∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U (∊) n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.

GALTON'S QUINCUNX: RANDOM WALK OR CHAOS?

2007 ◽  
Vol 17 (12) ◽  
pp. 4463-4469 ◽  
Author(s):  
KEVIN JUDD
Keyword(s):  
Dynamical System ◽  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Mechanical Device ◽  
Dynamical Models ◽  
The Central Limit Theorem ◽  
Random Events ◽  
Low Dimensional ◽  
Deterministic Dynamical System

In 1873 Francis Galton had constructed a simple mechanical device where a ball is dropped vertically through a harrow of pins that deflect the ball sideways as it falls. Galton called the device a quincunx, although today it is usually referred to as a Galton board. Statisticians often employ (conceptually, if not physically) the quincunx to illustrate random walks and the central limit theorem. In particular, how a Binomial or Gaussian distribution results from the accumulation of independent random events, that is, the collisions in the case of the quincunx. But how valid is the assumption of "independent random events" made by Galton and countless subsequent statisticians? This paper presents evidence that this assumption is almost certainly not valid and that the quincunx has the richer, more predictable qualities of a low-dimensional deterministic dynamical system. To put this observation into a wider context, the result illustrates that statistical modeling assumptions can obscure more informative dynamics. When such dynamical models are employed they will yield better predictions and forecasts.

Discrete time methods for simulating continuous time Markov chains

1976 ◽  
Vol 8 (04) ◽  
pp. 772-788 ◽  
Author(s):  
Arie Hordijk ◽  
Donald L. Iglehart ◽  
Rolf Schassberger
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Markov Chains ◽  
Confidence Intervals ◽  
Discrete Time ◽  
Continuous Time ◽  
Statistical Efficiency ◽  
Numerical Examples ◽  
The Central Limit Theorem ◽  
Continuous Time Markov Chains

This paper discusses several problems which arise when the regenerative method is used to analyse simulations of Markov chains. The first problem involves calculating the variance constant which appears in the central limit theorem used to obtain confidence intervals. Knowledge of this constant is very helpful in evaluating simulation methodologies. The second problem is to devise a method for simulating continuous time Markov chains without having to generate exponentially distributed holding times. Several methods are presented and compared. Numerical examples are given to illustrate the computional and statistical efficiency of these methods.

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