Estimating the logarithm of characteristic function and stability parameter for symmetric stable laws

Abstract

Let $X_1,\dots ,X_n$ be an i.i.d. sample from symmetric stable distribution with stability parameter $\alpha$ and scale parameter $\gamma$. Let ${\phi }_n$ be the empirical characteristic function. We prove an uniform large deviation inequality: given preciseness $\epsilon >0$ and probability $p\in (0,1)$, there exists universal (depending on $\epsilon$ and $p$ but not depending on $\alpha$ and $\gamma$) constant $\bar{r}>0$ so that $$P(\underset{u>0:r(u)\le \bar{r}}{\sup }|r(u)-\hat{r}(u)|\ge \epsilon )\le p,$$ where $r(u)=(u\gamma {)}^{\alpha }$ and $\hat{r}(u)=-\ln |{\phi }_n(u)|$. As an applications of the result, we show how it can be used in estimation unknown stability parameter $\alpha$.