562 research outputs found
A mixture representation of the Linnik distribution
Cataloged from PDF version of article.Linnik distribution with the characteristic function
~o,(tl = 1/(1 + Itl=), 0 < ~ < 2,
is shown to possess the following property.
Let X,,Xp be random variables possessing the Linnik distribution with parameters ~ and
(0 < ~ < fl ~< 2). Denote by Y~ an independent of X~ non-negative random variable with the density
,q(s;~,fl) = sin 1 + s 2~ + 2s cos~
Then
X, -'- X e Y~p,
fl respectively
where - denotes the equality in the sense of distributions.
Infinite divisibility of mixtures of Linnik distributions with respect to the parameter ~ and scale is obtained as
a corollary.
AMS 1980 Subject Classification." Primary 62H05, 60El0; Secondary 33A4
Analytic and asymptotic properties of generalized Linnik probability densities
Cataloged from PDF version of article.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 where {norm of matrix}t{norm of matrix} is the Euclidean norm of t ∈ ℝn. Integral representations of pα,ν,n(x) are obtained and used to derive the asymptotic expansions of pα,ν,n(x) when {norm of matrix}x{norm of matrix}→0 and {norm of matrix}x{norm of matrix}→∞ 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. © 2009 Birkhäuser Boston
Modulational instability in nonlocal Kerr-type media with random parameters
Modulational instability of continuous waves in nonlocal focusing and
defocusing Kerr media with stochastically varying diffraction (dispersion) and
nonlinearity coefficients is studied both analytically and numerically. It is
shown that nonlocality with the sign-definite Fourier images of the medium
response functions suppresses considerably the growth rate peak and bandwidth
of instability caused by stochasticity. Contrary, nonlocality can enhance
modulational instability growth for a response function with negative-sign
bands.Comment: 6 pages, 12 figures, revTeX, to appear in Phys. Rev.
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