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

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