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

Abstract

Let $X_1,\ldots,X_n$ be an i.i.d. sample from symmetric stable distribution with stability parameter $\alpha$ and scale parameter $\gamma$. Let $\varphi_n$ 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 $$P\big(\sup_{u>0:r(u)\leq \bar{r}}|r(u)-\hat{r}(u)|\geq \epsilon\big)\leq p,$$ where $r(u)=(u\gamma)^{\alpha}$ and $\hat{r}(u)=-\ln|\varphi_n(u)|$. As an applications of the result, we show how it can be used in estimation unknown stability parameter $\alpha$.