Analytic and asymptotic properties of multivariate generalized Linnik's probability densities

الأصل · EN

This paper studies the properties of the probability density function pα,ν, ₙ(x) of the n-variate generalized Linnik distribution whose characteristic function φα,ν,ₙ(t) is given by φα,ν,ₙ(t)=1/(1+α)ν, α∈ (0,2], ν>0, t∈ Rⁿ, where is the Euclidean norm of tⁿ. Integral representations of pα,ν, ₙ(x) are obtained and used to derive the asymptotic expansions of pα,ν, ₙ(x) when → 0 and → ∞ respectively. It is shown that under certain conditions which are arithmetic in nature, pα,ν, ₙ(x) can be represented in terms of entire functions.