Moments of a symmetric stable distribution

Question

If ${X_n}$ are iid with characteristic function $\exp\left(-c|t|^\alpha\right)$ then when are the moments $|X|^p $ , where $p$ is a real , defined ? If possible , can we evaluate them ?

Answer 1

I will show that $p^{th}$ moment exists for $p<\alpha$. $\newcommand{\P}{\mathbb{P}} \newcommand{\E}{\mathbb{E}}$

1. Recall/prove following inequality that is being used in the proof of Levy-Cramer's/Levy's continuity Theorem $$\P(|X|\geq t) \leq At\int_{-1/t}^{1/t}(1-\varphi(s))ds$$

2. Use it to bound tails $$ \P(|X|\geq t) \leq At\int_{-1/t}^{1/t}(1-e^{-c|s|^\alpha})ds \leq 2At\int_0^{1/t}cs^\alpha ds = \frac{2Ac}{\alpha+1}t^{-\alpha} =Bt^{-\alpha} $$

3. Use derived bound to show that $p^{th}$ moment exists $$ \E|X|^p = \int_0^\infty pt^{p-1}\P(|X|\geq t)dt < \infty \Leftrightarrow\\ \int_1^\infty pt^{p-1}\P(|X|\geq t)dt < pB\int_1^\infty t^{p-1}t^{-\alpha}dt < \infty \Leftrightarrow p-1-\alpha < -1 \Leftrightarrow \\ p < \alpha $$