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Register a new userOn multivariate quasi-infinitely divisible distributions
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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.
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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 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.
In this paper we show that the family P_d of probability distributions on R^d with log-concave densities satisfies a strong continuity condition. In particular, it turns out that weak convergence within this family entails (i) convergence in total variation distance, (ii) convergence of arbitrary moments, and (iii) pointwise convergence of Laplace transforms. Hence the nonparametric model P_d has similar properties as parametric models such as, for instance, the family of all d-variate Gaussian distributions.
The Gaussian correlation inequality for multivariate zero-mean normal probabilities of symmetrical n-rectangles can be considered as an inequality for multivariate gamma distributions (in the sense of Krishnamoorthy and Parthasarathy [5]) with one degree of freedom. Its generalization to all integer degrees of freedom and sufficiently large non-integer degrees of freedom was recently proved in [10]. Here, this inequality is partly extended to smaller non-integer degrees of freedom and in particular - in a weaker form - to all infinitely divisible multivariate gamma distributions. A further monotonicity property - sometimes called more PLOD (positively lower orthant dependent) - for increasing correlations is proved for multivariate gamma distributions with integer or sufficiently large degrees of freedom.
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