[Paper Review] Law of the iterated logarithm for U-statistics of weakly dependent observations

Herold Dehling, Martin Wendler|arXiv (Cornell University)|Dec 14, 2009

Bayesian Methods and Mixture Models28 references10 citations

TL;DR

This paper extends the law of the iterated logarithm (LIL) from partial sums of weakly dependent processes to nondegenerate U-statistics under strong mixing or absolutely regular dependence structures. By leveraging functional central limit theorem approximations and moment inequalities for U-statistics, the authors establish almost sure functional limits that generalize Philipp’s classical LIL results to U-statistics of weakly dependent data.

ABSTRACT

The law of the iterated logarithm for partial sums of weakly dependent processes was intensively studied by Walter Philipp in the late 1960s and 1970s. In this paper, we aim to extend these results to nondegenerate U-statistics of data that are strongly mixing or functionals of an absolutely regular process.

Motivation & Objective

• To generalize Walter Philipp’s law of the iterated logarithm for partial sums to nondegenerate U-statistics of weakly dependent processes.
• To address the lack of LIL results for U-statistics in the context of weak dependence, particularly strong mixing or absolutely regular processes.
• To establish almost sure functional limits for normalized U-statistics under weak dependence, extending classical asymptotic theory.
• To provide a theoretical foundation for inference using U-statistics when dependence is present but weakly mixing.

Proposed method

• Utilizes functional central limit theorem approximations for U-statistics under weak dependence.
• Applies moment inequalities tailored for U-statistics of strongly mixing or absolutely regular processes.
• Employs a coupling argument to relate U-statistics to partial sums of weakly dependent random variables.
• Applies the theory of empirical processes and weak convergence to derive functional laws of the iterated logarithm.
• Establishes almost sure convergence rates for normalized U-statistics using maximal inequalities.
• Adapts techniques from Philipp’s work on partial sums to the U-statistic setting via martingale approximation and entropy methods.

Experimental results

Research questions

• RQ1 Can the law of the iterated logarithm be extended from partial sums to nondegenerate U-statistics under weak dependence?
• RQ2 What are the almost sure functional limits of normalized U-statistics when the underlying data are strongly mixing or absolutely regular?
• RQ3 How do the growth rates of U-statistics compare to those of partial sums under weak dependence?
• RQ4 What moment and dependence conditions ensure the validity of the LIL for U-statistics in weakly dependent settings?

Key findings

• The law of the iterated logarithm holds for nondegenerate U-statistics of strongly mixing or absolutely regular processes under mild moment and mixing conditions.
• The almost sure functional limit of the normalized U-statistic process is characterized by a Brownian motion scaled by a logarithmic factor.
• The convergence rate of the U-statistic process is shown to be of order sqrt(n log log n), consistent with the classical LIL for partial sums.
• The results extend Philipp’s LIL for partial sums to the U-statistic setting, preserving the same logarithmic scaling.
• The dependence structure is controlled via strong mixing or absolute regularity coefficients, ensuring the applicability to a broad class of weakly dependent processes.
• The functional limit theorem for U-statistics is established under weaker moment assumptions than previously known, enhancing its practical relevance.