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Weak dependence and optimal quantitative self-normalized central limit theorems

Moritz Jirak

TL;DR

The paper addresses the problem of obtaining optimal quantitative Gaussian approximations for self-normalized sums of weakly dependent, stationary sequences with polynomial decay of autocovariances, showing a Berry-Esseen rate of $n^{-1/2}$ under mild assumptions ($\mathfrak{a}>\tfrac{13}{6}$). A key technique is a simple linearization that removes the stochastic variability in the self-normalizing denominator, combined with concentration of the empirical long-run variance and variance expansions; this reveals that minimaxly optimal estimators of the long-run variance are generally suboptimal and that simple oversmoothing achieves the optimal rate. The results apply across a wide range of models, including Banach-space linear processes, SDE functionals, left random walks on $GL_d(\mathbb{R})$, iterative random systems, products of positive random matrices, GARCH processes, and infinite-order Markov chains, showcasing the broad practicality of the approach. The work also shows that oversmoothing, rather than adaptive, model-selection procedures for $\hat{\sigma}_{nb}^2$, suffices to retain the $n^{-1/2}$ rate for self-normalized CLTs, with consequences for Kolmogorov and Wasserstein distances and related $L^q$-norm deviations, thereby contributing a conceptual shift in long-run variance estimation under weak dependence.

Abstract

Consider a stationary, weakly dependent sequence of random variables. Given only mild conditions, allowing for polynomial decay of the autocovariance function, we show a Berry-Esseen bound of optimal order $n^{-1/2}$ for studentized (self-normalized) partial sums, both for the Kolmogorov and Wasserstein (and $L^p$) distance. The results show that, in general, (minimax) optimal estimators of the long-run variance lead to suboptimal bounds in the central limit theorem, that is, the rate $n^{-1/2}$ cannot be reached. This can be salvaged by simple methods: In order to maintain the optimal speed of convergence $n^{-1/2}$, simple over-smoothing within a certain range is necessary and sufficient. The setup contains many prominent dynamical systems and time series models, including random walks on the general linear group, products of positive random matrices, functionals of Garch models of any order, functionals of dynamical systems arising from SDEs, iterated random functions and many more.

Weak dependence and optimal quantitative self-normalized central limit theorems

TL;DR

The paper addresses the problem of obtaining optimal quantitative Gaussian approximations for self-normalized sums of weakly dependent, stationary sequences with polynomial decay of autocovariances, showing a Berry-Esseen rate of under mild assumptions (). A key technique is a simple linearization that removes the stochastic variability in the self-normalizing denominator, combined with concentration of the empirical long-run variance and variance expansions; this reveals that minimaxly optimal estimators of the long-run variance are generally suboptimal and that simple oversmoothing achieves the optimal rate. The results apply across a wide range of models, including Banach-space linear processes, SDE functionals, left random walks on , iterative random systems, products of positive random matrices, GARCH processes, and infinite-order Markov chains, showcasing the broad practicality of the approach. The work also shows that oversmoothing, rather than adaptive, model-selection procedures for , suffices to retain the rate for self-normalized CLTs, with consequences for Kolmogorov and Wasserstein distances and related -norm deviations, thereby contributing a conceptual shift in long-run variance estimation under weak dependence.

Abstract

Consider a stationary, weakly dependent sequence of random variables. Given only mild conditions, allowing for polynomial decay of the autocovariance function, we show a Berry-Esseen bound of optimal order for studentized (self-normalized) partial sums, both for the Kolmogorov and Wasserstein (and ) distance. The results show that, in general, (minimax) optimal estimators of the long-run variance lead to suboptimal bounds in the central limit theorem, that is, the rate cannot be reached. This can be salvaged by simple methods: In order to maintain the optimal speed of convergence , simple over-smoothing within a certain range is necessary and sufficient. The setup contains many prominent dynamical systems and time series models, including random walks on the general linear group, products of positive random matrices, functionals of Garch models of any order, functionals of dynamical systems arising from SDEs, iterated random functions and many more.

Paper Structure

This paper contains 18 sections, 30 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Kolmogorov distance
  4. Wasserstein distance
  5. Examples
  6. Banach space valued linear processes
  7. SDEs
  8. Left random walk on $GL_d(\mathbb{R})$
  9. Iterative random systems
  10. Products of positive random matrices
  11. GARCH$(\mathfrak{p},\mathfrak{q})$ processes
  12. Markov chains of infinite order
  13. Proofs
  14. Kolmogorov distance
  15. Concentration of the empirical long-run variance
  16. ...and 3 more sections

Key Result

Theorem 2.3

Grant Assumption ass_main_dependence. Then there exists a constant $C > 0$, only depending on $\Theta_{\mathfrak{a} 6}$ and $\sigma^2$, such that where $\sigma_{b}^2$ is defined in defn:sigma:nb.

Theorems & Definitions (45)

  • Remark 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Proposition 3.1
  • Remark 3.2
  • Corollary 3.3
  • Proposition 3.4
  • ...and 35 more