Analytic and Asymptotic Properties of Generalized Linnik Probability Densities

Abstract

[No abstract available

Weak Limits for Multivariate Random Sums

“…If the sum (1) is deterministic, then the limiting distributions are stable laws (see [20] and references therein). If & p is a geometric random variable with mean 1Âp then the limit is a geometric stable (GS) law (see [1,5,8,11,15,16,18]). Random summation appears in applied problems in many fields, including physics, biology, economics, insurance mathematics, reliability and queuing theories (see, [7] and the references therein).…”

“…Thus, GS distributions are regular &-stable. A more general class of generalized Linnik distributions is obtained when & p has a negative binomial distribution, in which case & has a Gamma distribution (see [5] for the univariate case).…”

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

“…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.…”

“…Given the deconvolution problem h(x, y) ⊗ f (x, y) = g(x, y), with a known otf h(ξ, η) in the form of one the three types in (8), we viewĥ(ξ, η) as the Green's functionĥ(ξ, η, t) at time t = 1, in the corresponding forward evolution equation w t = −Lw, in one of (14), (15), or (16). Deconvolution is mathematically equivalent to solving w t = −Lw backwards in time, given the noisy blurred image g(x, y) as data at time t = 1.…”

“…whereĥ(ξ, η, t) is the Green's function for the corresponding diffusion equation in one of (14), (15), or (16). Using this in (22), together with g(x, y) as data at t = 1, great benefit derives from the ability to perform the deconvolution in slow motion by marching backwards in time in the diffusion equation w t = −Lw.…”