Self-Normalized Moderate Deviations for Degenerate U-Statistics

TL;DR

This work establishes self-normalized moderate deviation and law of the iterated logarithm results for degenerate U-statistics of order $2$ with kernel$\;h(x,y)=\sum_{l=1}^{\infty} \lambda_l g_l(x) g_l(y)$ under $\sum_l \lambda_l<\infty$ and DoA-normal conditions for $g_l(X)$. By a careful truncation scheme, decomposition into component sums $Y_{n,l}$ and variances $V_{n,l}^2$, and a suite of exponential inequalities (including Bernstein-type, Einmahl-type, and de la Peña–Lai–Shao bounds), the authors show that for $x_n\to\infty$ with $x_n=o(\sqrt{n})$ the self-normalized tail satisfies $\log P\left( \frac{\sum_{i\neq j} h(X_i,X_j)}{\max_{l} \lambda_l V_{n,l}^2} \ge x_n^2 \right) \sim -\frac{x_n^2}{2}$, and they obtain a corresponding LIL with constant $2$. The results extend self-normalized deviation theory to degenerate U-statistics, requiring relatively mild moment-type conditions and providing precise tail asymptotics with potential applications in high-order U-statistic inference.

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 $h(x,y)=\sum_{l=1}^{\infty} λ_l g_l (x) g_l(y)$, where $λ_l > 0$, $E g_l(X_1)=0$, and $g_l (X_1)$ is in the domain of attraction of a normal law for all $l \geq 1$. Under the condition $\sum_{l=1}^{\infty}λ_l<\infty$ 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}λ_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.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Proposition 2.1
• Proposition 2.2
• Lemma 2.1
• Proposition 2.3
• Lemma 2.2
• Proposition 2.4
• Lemma 3.1