Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2501.03915 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
Table des matières:
- 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.