Self-Normalized Moderate Deviations for Degenerate U-Statistics

TL;DR

This paper establishes self-normalized moderate deviation principles for degenerate U-statistics of order 2, providing asymptotic probabilities and a law of the iterated logarithm under specific conditions.

Contribution

It introduces new self-normalized deviation results for degenerate U-statistics with infinite kernel expansions, extending existing moderate deviation theory.

Findings

Asymptotic equivalence of deviation probabilities to a normal tail

Conditions under which the law of the iterated logarithm holds

Applicability to kernels with infinite spectral decompositions

Abstract

In this paper, we study self-normalized moderate deviations for degenerate { $U$}-statistics of order $2$. Let $\{X_i, i \geq 1\}$ be i.i.d. random variables and consider symmetric and degenerate kernel functions in the form $h(x,y)=\sum_{l=1}^{\infty} \lambda_l g_l (x) g_l(y)$, where $\lambda_l > 0$, $E g_l(X_1)=0$, and $g_l (X_1)$ is in the domain of attraction of a normal law for all $l \geq 1$. Under the condition $\sum_{l=1}^{\infty}\lambda_l<\infty$ and some truncated conditions for $\{g_l(X_1): l \geq 1\}$, we show that $ \text{log} P({\frac{\sum_{1 \leq i \neq j \leq n}h(X_{i}, X_{j})} {\max_{1\le l<\infty}\lambda_l V^2_{n,l} }} \geq x_n^2) \sim - { \frac {x_n^2}{ 2}}$ for $x_n \to \infty$ and $x_n =o(\sqrt{n})$, where $V^2_{n,l}=\sum_{i=1}^n g_l^2(X_i)$. As application, a law of the iterated logarithm is also obtained.