[Paper Review] Convergence in distribution norms in the CLT for non identical distributed random variables

Vlad Bally, Lucia Caramellino+1 more|arXiv (Cornell University)|Jun 6, 2016

Random Matrices and Applications35 references18 citations

TL;DR

This paper establishes convergence in distribution norms for the Central Limit Theorem (CLT) under non-identically distributed, independent random variables with Doeblin-type regularity. Using Malliavin calculus and integration by parts, it derives explicit bounds on the error in terms of derivatives of test functions, enabling applications to invariance principles, small ball probabilities, and root counting for random trigonometric polynomials. The key contribution is a refined CLT error estimate in distribution norms, even when the sum law lacks a density, by controlling higher-order derivatives via regularity conditions on the summands.

ABSTRACT

We study the convergence in distribution norms in the Central Limit Theorem for non identical distributed random variables that is $$ \\varepsilon_{n}(f):={\\mathbb{E}}\\Big(f\\Big(\\frac 1{\\sqrt n}\\sum_{i=1}^{n}Z_{i}\\Big)\\Big)-{\\mathbb{E}}\\big(f(G)\\big)\ ightarrow 0 $$ where $Z_{i}$ are centred independent random variables and $G$ is a Gaussian random variable. We also consider local developments (Edgeworth expansion). This kind of results is well understood in the case of smooth test functions $f$. If one deals with measurable and bounded test functions (convergence in total variation distance), a well known theorem due to Prohorov shows that some regularity condition for the law of the random variables $Z_{i}$, $i\\in {\\mathbb{N}}$, on hand is needed. Essentially, one needs that the law of $ Z_{i}$ is locally lower bounded by the Lebesgue measure (Doeblin's condition). This topic is also widely discussed in the literature. Our main contribution is to discuss convergence in distribution norms, that is to replace the test function $f$ by some derivative $\\partial_{\\alpha }f$ and to obtain upper bounds for $\\varepsilon_{n}(\\partial_{\\alpha }f)$ in terms of the infinite norm of $f$. Some applications are also discussed: an invariance principle for the occupation time for random walks, small balls estimates and expected value of the number of roots of trigonometric polynomials with random coefficients.

Motivation & Objective

• To establish convergence in distribution norms for the CLT when the summands are non-identically distributed and may lack a density.
• To extend classical CLT and Edgeworth expansion results to test functions with low regularity, particularly bounded measurable functions.
• To provide quantitative error bounds in terms of the derivatives of test functions, not just their sup norms.
• To develop a framework for applications involving occupation time invariance principles, small ball probabilities, and expected number of roots of random trigonometric polynomials.
• To unify and generalize existing results on convergence in total variation and entropy distance by introducing a new error control mechanism via distribution norms.

Proposed method

• Uses abstract Malliavin calculus and integration by parts to control the error in the CLT for non-smooth test functions.
• Applies the Lindeberg method for Markov semigroups, inspired by parametrix techniques, to derive error estimates.
• Employs Nummelin's splitting to decompose the summands and ensure sufficient regularity for the Malliavin calculus framework.
• Derives explicit Edgeworth-type expansions up to order N by expressing coefficients as linear combinations of Hermite polynomials.
• Introduces distribution norms by replacing test functions f with their derivatives ∂αf and bounds the error εn(∂αf) in terms of ‖f‖∞.
• Establishes bounds on the density derivatives of the normalized sum via regularity conditions (Doeblin’s condition) and Malliavin matrix estimates.

Experimental results

Research questions

• RQ1 Can the classical CLT error be bounded in terms of the derivatives of the test function, even when the summands are non-identically distributed and the limiting law may have atoms?
• RQ2 What is the rate of convergence in distribution norms for the CLT when the summands satisfy Doeblin’s condition but are not necessarily absolutely continuous?
• RQ3 How can Edgeworth expansions be extended to non-smooth test functions using Malliavin calculus and integration by parts?
• RQ4 What are the implications of these distribution norm bounds for stochastic processes such as occupation times and random trigonometric polynomials?
• RQ5 Can convergence in total variation distance be recovered from distribution norm estimates under Doeblin-type regularity conditions?

Key findings

• The paper proves that for non-i.i.d. summands satisfying Doeblin’s condition, the error in the CLT for derivatives of bounded test functions is bounded by C(Y)/√n, where C(Y) depends on moments and regularity of the summands.
• For Edgeworth expansions up to order N, the error is bounded by CN / n^{(N+1)/2} ‖f‖_{b_N,∞}, with explicit coefficients expressed as combinations of Hermite polynomials.
• When the summands are i.i.d. and satisfy Doeblin’s condition, the convergence in total variation distance is guaranteed, extending Prohorov’s theorem to non-i.i.d. settings.
• The authors derive a new invariance principle for the occupation time of random walks by controlling the error in distribution norms.
• Small ball probabilities for the normalized sum are estimated using the distribution norm framework, yielding bounds that depend on the regularity of the summands.
• The expected number of real roots of random trigonometric polynomials with i.i.d. coefficients is shown to satisfy an invariance principle via the distribution norm error control.