Estimating the Logarithm of Characteristic Function and Stability Parameter for Symmetric Stable Laws

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