Self-Normalized Moderate Deviations for Degenerate U-Statistics

Abstract

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

32 pages