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 $α$ and scale parameter $γ$. Let $φ_n$ be the empirical characteristic function. We prove an uniform large deviation inequality: given preciseness $ε>0$ and probability $p\in (0,1)$, there exists universal (depending on $ε$ and $p$ but not depending on $α$ and $γ$) constant $\bar{r}>0$ so that $P\big(\sup_{u>0:r(u)\leq \bar{r}}|r(u)-\hat{r}(u)|\geq ε\big)\leq p,$ where $r(u)=(uγ)^α$ and $\hat{r}(u)=-\ln|φ_n(u)|$. As an applications of the result, we show how it can be used in estimation unknown stability parameter $α$.

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