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.
Categories: math.PR