Analytic and Asymptotic Properties of Multivariate Generalized Linnik’s Probability Densities

Abstract

This paper studies the properties of the probability density function p α,ν,n (x) of the n-variate generalized Linnik distribution whose characteristic function ϕ α,ν,n (t) is given bywhere t is the Euclidean norm of t ∈ R n . Integral representations of p α,ν,n (x) are obtained and used to derive the asymptotic expansions of p α,ν,n (x) when x → 0 and x → ∞ respectively. It is shown that under certain conditions which are arithmetic in nature, p α,ν,n (x) can be represented in terms of entire functions.

Interpolation of spatial data – A stochastic or a deterministic problem?

“…Then the native space of Φ coincides with the Sobolev space W τ 2 (R d ) as a vector space, and the native space norm and the Sobolev norm are equivalent. Table 1 shows examples (see [49,30,45,82] for details) of positive definite functions Φ commonly used in geostatistics and approximation theory. For Φ ∈ L 2 (R d ) their Fourier transforms are defined in the way stated in Theorem 2.1.…”

Bochner subordination, logarithmic diffusion equations, and blind deconvolution of hubble space telescope imagery and other scientific data

“…Indeed, such tail behavior is directly related to the smoothness of the characteristic function near the origin in Fourier (ξ, η) space. The following important results may be found in [26].ĥ Applied to the M51 image in Figure 3, where δ = 1.07, β = 0.15, σ = 0.86, γ = 0.09, and λ = 0.821, this leads to…”

“…The present method is based on otfs in the form of generalized Linnik characteristic functions [16], [26], [32], and the use of time-reversed diffusion equations involving the logarithm of the negative Laplacian plus the identity. We show that this results in higher quality reconstructions than previously obtained.…”

Matrix-variate statistical distributions and fractional calculus

“…One can establish a path of going from a Mittag-Leffler variable to a positive Lévy variable. Replace a by a(q − 1) and η by η/(q − 1) in the Laplace transform (22) of the generalized Mittag-Leffler density in (8) and (18). That is,…”

“…These ideas are generalized to the matrix-variate cases recently, see Mathai [27]. Some multivariable, not matrix variable, analogues may be seen from Lim and Teo [18], Pakes [37] and Lin [19].…”