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A completion of counterexamples to the classical central limit theorem for pair- and triplewise independent and identically distributed random variables

Martin Raič

TL;DR

This work removes a distributional restriction in prior central limit theorem counterexamples by showing that, for any real-valued $W$ with finite variance and not almost surely constant, there exists a sequence of triplewise independent variables $X_k$ with $X_k hicksim W$ for which the standardized sums $S_n$ fail to converge to the normal law. The authors adapt graph-based, $K$-tuplewise independence constructions and require only a two-component mixture representation of $W$, yielding a limit $S^{( ext{ell})} = ig(1- ho^2ig)^{1/2}Z + ho Y$ with a nonnormal $Y$, whenever the edge-sum limit $\xi^*_m$ converges to a nonnormal distribution. They establish the mixture decomposition for any $W$, including the atom-at-a-point case, ensuring finite variances of the mixture components, and demonstrate that existing nonnormal limits (e.g., variance-gamma) arising in the graph-based blocks propagate to the full sequence. The results significantly broaden the class of distributions and dependence structures that yield nonnormal limits, illuminating the fragile nature of the CLT under $K$-tuplewise independence and suggesting potential extensions to higher-order tuplewise independence.

Abstract

By the Lindeberg-Lévy central limit theorem, standardized partial sums of a sequence of mutually independent and identically distributed random variables converge in law to the standard normal distribution. It is known that mutual independence cannot be relaxed to pairwise and even not triplewise independence. Counterexamples have been constructed for most marginal distributions: a recent construction works under a condition which excludes certain probability distributions with atomic parts, in particular discrete distributions in the `general position.' In the present paper, we show that this condition can be lifted: for any probability distribution $ F $ on the real line, which has finite variance and is not concentrated in a single point, there exists a sequence of triplewise independent random variables with distribution $ F $, such that its standardized partial sums converge in law to a distribution which is not normal. There is also scope for extension to $ k $-tuplewise independence.

A completion of counterexamples to the classical central limit theorem for pair- and triplewise independent and identically distributed random variables

TL;DR

This work removes a distributional restriction in prior central limit theorem counterexamples by showing that, for any real-valued with finite variance and not almost surely constant, there exists a sequence of triplewise independent variables with for which the standardized sums fail to converge to the normal law. The authors adapt graph-based, -tuplewise independence constructions and require only a two-component mixture representation of , yielding a limit with a nonnormal , whenever the edge-sum limit converges to a nonnormal distribution. They establish the mixture decomposition for any , including the atom-at-a-point case, ensuring finite variances of the mixture components, and demonstrate that existing nonnormal limits (e.g., variance-gamma) arising in the graph-based blocks propagate to the full sequence. The results significantly broaden the class of distributions and dependence structures that yield nonnormal limits, illuminating the fragile nature of the CLT under -tuplewise independence and suggesting potential extensions to higher-order tuplewise independence.

Abstract

By the Lindeberg-Lévy central limit theorem, standardized partial sums of a sequence of mutually independent and identically distributed random variables converge in law to the standard normal distribution. It is known that mutual independence cannot be relaxed to pairwise and even not triplewise independence. Counterexamples have been constructed for most marginal distributions: a recent construction works under a condition which excludes certain probability distributions with atomic parts, in particular discrete distributions in the `general position.' In the present paper, we show that this condition can be lifted: for any probability distribution on the real line, which has finite variance and is not concentrated in a single point, there exists a sequence of triplewise independent random variables with distribution , such that its standardized partial sums converge in law to a distribution which is not normal. There is also scope for extension to -tuplewise independence.

Paper Structure

This paper contains 3 sections, 6 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Adaptation of construction
  3. Construction of mixture

Key Result

Proposition 1.3

For each $\tau \in (0, 1)$ and any real-valued random variable $W$ with finite expectation, which is not almost surely constant, there exist real-valued random variables $U$ and $V$ with different expectations, such that for all Borel sets $C \subseteq \mathbb{R}$. Moreover, if $W$ has finite variance, $U$ and $V$ can be chosen to have finite variances, too.

Theorems & Definitions (8)

  • Definition 1.1
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Remark 2.2
  • Corollary 2.3
  • Lemma 2.4
  • Corollary 2.5