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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 $d geq 2$.
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 $
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
Suppose that $X$ is a subcritical superprocess. Under some asymptotic conditions on the mean semigroup of $X$, we prove the Yaglom limit of $X$ exists and identify all quasi-stationary distributions of $X$.
It is well known that any pair of random variables $(X,Y)$ with values in Polish spaces, provided that $Y$ is nonatomic, can be approximated in joint law by random variables of the form $(X,Y)$ where $X$ is $Y$-measurable and $X stackrel{d}{=} X$. Th
Neuhauser [Probab. Theory Related Fields 91 (1992) 467--506] considered the two-type contact process and showed that on $mathbb{Z}^2$ coexistence is not possible if the death rates are equal and the particles use the same dispersal neighborhood. Here