An asymptotic evaluation of the tail of a multiple symmetric \(\alpha\)- stable integral. (English) Zbl 0689.60051

Samorodnitsky, Gennady; Szulga, Jerzy

Ann. Probab. 17, No. 4, 1503-1520 (1989).

Let Z be a symmetric Lévy \(\alpha\)-stable process on [0,1] with characteristic function \[ E \exp \{i\lambda Z(u)\}=\exp \{-u| \lambda |^{\alpha}\},\quad 0<\alpha <2, \] and let f be a real function on \([0,1]^ n\) vanishing on diagonals. Let \[ I_ n(f)=\int...\int f(t_ 1,...,t_ n)dZ(t_ 1)...dZ(t_ n) \] be a multiple stable integral (MSI). It is shown that under certain integrability conditions, \[ 2\lim_{x\to \infty}x^{\alpha}(\ln x)^{1-n}P(| I_ n(f)| >x)=\lim_{x\to \infty}x^{\alpha}(\ln x)^{1-n}P(I_ n(f)>x)= \]
\[ =n\alpha^{n-1}(n!)^{\alpha -2}s^{- n}\int...\int | f(t_ 1,...,t_ n)|^{\alpha}dt_ 1...dt_ n \] where \(s=\int^{\infty}_{0}x^{-\alpha}\sin x x\). The proof is based on a LePage-type representation of MSI by multiple series of transformed arrival times of a Poisson process.

Reviewer: Yu.M.Kabanov

MSC:

60H05	Stochastic integrals
60F10	Large deviations
60G57	Random measures
60E05	Probability distributions: general theory

Keywords:

stable Lévy process; multiple stable integral; LePage-type representation; Poisson process