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Register a new userOn a characterization of infinitely divisible distributions with Gaussian component
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We give a necessary and sufficient condition for symmetric infinitely divisible distribution to have Gaussian component. The result can be applied to approximation the distribution of finite sums of random variables. Particularly, it shows that for a large class of distributions with finite variance stable approximation appears to be better than Gaussian. keywords: infinitely divisible distributions; Gaussian component; approximations of sums of random variables.
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A quasi-infinitely divisible distribution on $mathbb{R}^d$ is a probability distribution $mu$ on $mathbb{R}^d$ whose characteristic function can be written as the quotient of the characteristic functions of two infinitely divisible distributions on $mathbb{R}^d$. Equivalently, it can be characterised as a probability distribution whose characteristic function has a Levy--Khintchine type representation with a signed Levy measure, a so called quasi--Levy measure, rather than a Levy measure. A systematic study of such distributions in the univariate case has been carried out in Lindner, Pan and Sato cite{lindner}. The goal of the present paper is to collect some known results on multivariate quasi-infinitely divisible distributions and to extend some of the univariate results to the multivariate setting. In particular, conditions for weak convergence, moment and support properties are considered. A special emphasis is put on examples of such distributions and in particular on $mathbb{Z}^d$-valued quasi-infinitely divisible distributions.
A probability distribution $mu$ on $mathbb{R}^d$ is quasi-infinitely divisible if its characteristic function has the representation $widehat{mu} = widehat{mu_1}/widehat{mu_2}$ with infinitely divisible distributions $mu_1$ and $mu_2$. In cite[Thm. 4.1]{lindner2018} it was shown that the class of quasi-infinitely divisible distributions on $mathbb{R}$ is dense in the class of distributions on $mathbb{R}$ with respect to weak convergence. In this paper, we show that the class of quasi-infinitely divisible distributions on $mathbb{R}^d$ is not dense in the class of distributions on $mathbb{R}^d$ with respect to weak convergence if $d geq 2$.
We study fractional smoothness of measures on $mathbb{R}^k$, that are images of a Gaussian measure under mappings from Gaussian Sobolev classes. As a consequence we obtain Nikolskii--Besov fractional regularity of these distributions under some weak nondegeneracy assumption.
Given a domain G, a reflection vector field d(.) on the boundary of G, and drift and dispersion coefficients b(.) and sigma(.), let L be the usual second-order elliptic operator associated with b(.) and sigma(.). Under suitable assumptions that, in particular, ensure that the associated submartingale problem is well posed, it is shown that a probability measure $pi$ on bar{G} is a stationary distribution for the corresponding reflected diffusion if and only if $pi (partial G) = 0$ and $int_{bar{G}} L f (x) pi (dx) leq 0$ for every f in a certain class of test functions. Moreover, the assumptions are shown to be satisfied by a large class of reflected diffusions in piecewise smooth multi-dimensional domains with possibly oblique reflection.
Let $X$ be a second countable locally compact Abelian group containing no subgroup topologically isomorphic to the circle group $mathbb{T}$. Let $mu$ be a probability distribution on $X$ such that its characteristic function $hatmu(y)$ does not vanish and $hatmu(y)$ for some $n geq 3$ satisfies the equation $$ prod_{j=1}^{n} hatmu(y_j + y) = prod_{j=1}^{n}hatmu(y_j - y), quad sum_{j=1}^{n} y_j = 0, quad y_1,dots,y_n,y in Y. $$ Then $mu$ is a convolution of a Gaussian distribution and a distribution supported in the subgroup of $X$ generated by elements of order 2.
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