Sharp estimates on the tail behavior of a multistable distribution

Multistable stochastic integrals on RR, have been introduced quite recently in Falconer and Liu (2012); they are defined through their characteristic functions. Roughly speaking, in a neighborhood of an arbitrary point x∈Rx∈R, such an integral can be viewed as a usual stable stochastic integral, with a stability parameter α(x)α(x) depending on the location xx.Let YY be an arbitrary symmetric αα-stable random variable of scale parameter σ>0σ>0, an important classical result concerning the heavy-tailed behavior of its distribution (see e.g. Samorodnitsky and Taqqu, 1994), is that, there exists an explicit constant C(α)>0C(α)>0, only depending on α∈(0,2)α∈(0,2), such that limλ→+∞(C(α)σαλ−α)−1P(|Y|>λ)=1limλ→+∞(C(α)σαλ−α)−1P(|Y|>λ)=1. In this article, by using basic methods of Fourier analysis, we show that the latter result can be extended to the setting of random variables defined as multistable stochastic integrals.