Asymptotic Behaviour of Stable Measures Near the Origin

Abstract

We investigate the lower tail of $q_r = (\sum^\infty_{i=1} | \alpha_i\theta_i|^r)^{1/r}$ seminorms on $R^\infty$, where $r \geq 1$ and $\theta_i$ are standard $p$-stable real random variables. We prove that for $p < r \leq 2$ we have $P\{q_r \leq t\} \geq \exp\{-ct^{-pr/(r-p)}\}$ in some neighbourhood of 0, where $c$ is a nonnegative constant. If $r \leq p$, then for any positive, increasing function $f$, we can find $q_r$ such that $P\{q_r \leq t\} \leq f(t)$ for $t \leq 1$. We also give a new characterization of Banach spaces of stable type $p$ in terms of the behaviour of $\mu\{\| \cdot \| \leq t\}$ near 0, where $\mu$ is a symmetric and $p$-stable measure.