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 a 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

$$P\big (\sup _{u>0:r(u)\le \bar{r}}|r(u)-\hat{r}(u)|\ge \epsilon \big )\le p,$$

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 the unknown stability parameter $\alpha$.