On the Markov chain central limit theorem

By Galin L. Jones. 2004

468 citations recorded by the source corpus.

The goal of this expository paper is to describe conditions which guarantee a central limit theorem for functionals of general state space Markov chains. This is done with a view towards Markov chain Monte Carlo settings and hence the focus is on the connections between drift and mixing conditions and their implications. In particular, we consider three commonly cited central limit theorems and discuss their relationship to classical results for mixing processes. Several motivating examples are given which range from toy one-dimensional settings to complicated settings encountered in Markov chain Monte Carlo.

Subfields: Control and Systems Engineering, Statistics and Probability

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Citations in the corpus

Efficiently Breaking the Curse of Horizon in Off-Policy Evaluation with Double Reinforcement Learning (2019, methodology): “When ˆ q ( i,t ) = q , ˆ w ( i,t ) = w , we have The conditions above are used in order to invoke the Markov chain central limit theorem (MC-CLT; see Jones, 2004, Corollary 2).”
Batch means and spectral variance estimators in Markov chain Monte Carlo (2008, unknown): “4 in Jones et al. (2006) we have that |σ̂2 BM − σ2 g σ̃2 BM | → 0 w.p. 1 as n → ∞. An application of the generalized majorized convergence theorem (pp 1015 Zeidler (1990)) implies that, as n→∞, E [∣∣σ̂2 BM − σ2 g σ̃2 BM ∣∣]→ 0.”
Fixed-Width Output Analysis for Markov Chain Monte Carlo (2006, background): “See Jones (2004) for details on conditions 1 and 2.”
Stationarity and geometric ergodicity of BEKK multivariate GARCH models (2011, background): “cline exponentially fast (which is the case for all geometrically ergodic Markov chains), needs a stronger order moment condition than the second order ones obtained in Theorem 2.4 (see [19] and also [20, 22]). In [14] (see also [22, Corollary 3]) it has been shown that for a positive Harris recurre”
Mod-ϕ Convergence (2016, background): “[Cog72, IL71, GL78], and [Jon04] for a survey), mixing sequences (see [Ros56, Phi69, Dav73, LL96], and [Bra05] for a survey on mixing conditions), lattice models ([GJL75, EN78, New80]), m-dependence (cf.”
Markov chain Monte Carlo: Can we trust the third significant figure? (2007, background): “On the other hand, Jones et al. (2006) showed that if the batch size and the number of batches are allowed to increase as the overall length of the simulation increases by setting bn = n and an = n/bn , then σ̂ 2 g → σ 2 g with probability 1 as n → ∞.”
Efficiently Breaking the Curse of Horizon: Double Reinforcement Learning in Infinite-Horizon Processes (2019, methodology): “When ˆ q ( i ) = q , ˆ w ( i ) = w , we have The conditions above are used in order to invoke the Markov chain central limit theorem (MC-CLT; see Jones, 2004, Corollary 2).”
ASYMPTOTIC THEORY FOR A FACTOR GARCH MODEL (2009, background): “Jones (2004) . Furthermore, these limit theorems, and the existence of”
A Practical Sequential Stopping Rule for High-Dimensional Markov Chain Monte Carlo (2016, background): “Under certain regularity conditions (Chan and Geyer 1994; Tierney 1994; Jones 2004; Roberts and Rosenthal 2004), we can obtain a marginal Markov chain central limit theorem (CLT) for the sampling distribution of an unknown MCSE.”
Multivariate output analysis for Markov chain Monte Carlo (2015, methodology): “We also implement the uBM methods described in Jones et al. (2006) to estimate σ2 i , which captures the autocorrelation in each component while ignoring the cross-correlation.”
Revisiting the Gelman–Rubin Diagnostic (2018, methodology): “g the mean of a function g: X !Rp. Almost all of the methods require the estimation of the limiting variance ˝2 1:= lim n!1nVar(X i), which is nite if a Markov chain central limit theorem holds (see Jones, 2004, for conditions). That is, a Markov chain central limit theorem holds if there exists ˝2 ”
Bridging the gap between constant step size stochastic gradient descent and Markov chains (2017, unknown): “t; 0 when k tends to +∞ In this Section, we analyze the convergence of θ¯(γ) k to θ¯γ, when k → ∞, and the convergence of E θ¯(γ) k −θ¯γ 2 to 0. Under suitable conditions (Meyn and Tweedie, 1993; Jones, 2004), θ¯(γ) k satisﬁes a central limit theorem: √ k θ¯(γ) k −θ¯γ → Nd (0,σ2 ϕ), where σ 2 ≥ ”
Optimal Proposal Distributions and Adaptive MCMC (2008, background): “…Weak Law of Large Numbers (WLLN), Strong Law of Large Numbers (SLLN), and Central Limit Theorem (CLT), in order to guarantee good asymptotic estimates and estimate standard errors (see e.g. Tierney, 1994; Jones and Hobert, 2001; Hobert et al., 2002; Roberts and Rosenthal, 2004; and Jones, 2004).”