[Paper Review] The Central Limit Theorem for uniformly strong mixing measures

Nicolai Haydn|ArXiv.org|Mar 7, 2009

Mathematical Dynamics and Fractals14 references3 citations

TL;DR

This paper establishes the central limit theorem (CLT) and lognormal distribution for the information function $ I_n(x) = -\log\mu(A_n(x)) $ in uniformly strong mixing systems with countably infinite partitions, under minimal regularity assumptions. It proves polynomial rates of convergence when the fourth moment of the information function is finite, extending Ibragimov's results beyond finite partitions and eliminating the need for Gibbs or regularity conditions on conditional entropy.

ABSTRACT

The theorem of Shannon-McMillan-Breiman states that for every generating partition on an ergodic system, the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere (provided the entropy is finite). In this paper we prove that the measure of cylinder sets are lognormally distributed for strongly mixing systems and infinite partitions and show that the rate of convergence is polynomial provided the fourth moment of the information function is finite. Also, unlike previous results by Ibragimov and others which only apply to finite partitions, here we do not require any regularity of the conditional entropy function. We also obtain the law of the iterated logarithm and the weak invariance principle for the information function.

Motivation & Objective

• To generalize Ibragimov's CLT for the information function to systems with countably infinite partitions, where previous results required finite partitions and regularity of conditional entropy.
• To remove the need for Gibbs property or regularity conditions on the Radon-Nikodym derivative, which were required in earlier works on the CLT for information functions.
• To establish polynomial rates of convergence for the lognormal distribution of $ I_n(x) $ under the assumption that the fourth moment of the information function is finite.
• To prove the law of the iterated logarithm (LIL) and weak invariance principle for the information function in uniformly strong mixing systems.
• To provide a framework valid for general uniformly strong mixing measures, including Markov chains on infinite alphabets, without relying on Gibbs-type characterizations.

Proposed method

• Uses a coupling argument to decompose the information function $ I_n(x) $ into independent increments over time intervals, leveraging the strong mixing property to control dependence.
• Applies a truncation technique with a threshold $ \ell \sim n^\alpha $, $ \alpha < 1/2 $, to control tail probabilities and ensure convergence in distribution.
• Employs a mixing rate $ \psi(\Delta) $ to bound the dependence between separated blocks, with $ \psi(\Delta) $ decaying polynomially or exponentially.
• Introduces a random variable $ Y $ on product spaces to quantify dependence between blocks, enabling control of the variance of the sum of information increments.
• Uses the $ L^2 $-norm condition and exponential decay of correlations to bound the variance of the information function, particularly in the Markov chain case.
• Applies the almost sure invariance principle and functional central limit theorem techniques to derive weak convergence to Brownian motion for the normalized information process.

Experimental results

Research questions

• RQ1 Does the information function $ I_n(x) = -\log\mu(A_n(x)) $ converge in distribution to a lognormal law for uniformly strong mixing systems with countably infinite partitions?
• RQ2 Can the rate of convergence to the lognormal distribution be quantified as polynomial when the fourth moment of the information function is finite?
• RQ3 Is the central limit theorem for $ I_n(x) $ valid without requiring the Gibbs property or regularity of the conditional entropy function?
• RQ4 Can the law of the iterated logarithm and weak invariance principle be established for the information function in this general setting?
• RQ5 What is the asymptotic variance of $ I_n(x) $ for Markov chains on an infinite alphabet, and does it converge to a finite limit?

Key findings

• The information function $ I_n(x) $ is asymptotically lognormally distributed for uniformly strong mixing systems with countably infinite partitions, even without finite partition or regularity assumptions.
• The rate of convergence to the lognormal distribution is polynomial, specifically $ \mathcal{O}(n^{-\kappa}) $ for some $ \kappa > 0 $, provided the fourth moment of the information function is finite.
• The law of the iterated logarithm holds for the information function under the same conditions, confirming the almost sure growth rate of fluctuations.
• The weak invariance principle is established: the normalized process $ X_n(t,x) = (I_{[nt]}(x) - \mathbb{E}[I_{[nt]}]) / \sigma\sqrt{n} $ converges weakly to a standard Brownian motion.
• For Markov chains on an infinite alphabet, the asymptotic variance $ \sigma^2 $ is given by a convergent infinite series involving correlations of log-ratios of transition probabilities.
• The variance expression for Markov chains includes a main term from the squared log-ratio of transition probabilities and a correction term from long-range correlations, both of which converge due to exponential mixing.