Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn a denseness result for quasi-infinitely divisible distributions
176
0
0.0
(
0
)
Authors
Merve Kutlu
Ask ChatGPT about the research
No Arabic abstract
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 $d geq 2$.
rate research
Read More
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.
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.
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 $X stackrel{d}{=} X$. This article surveys and extends some recent dynamic analogues of this result. For example, if $X$ and $Y$ are stochastic processes in discrete or continuous time, then, under a nonatomic assumption as well as a necessary and sufficient causality (or compatibility) condition, one can approximate $(X,Y)$ in law in path space by processes of the form $(X,Y)$, where $X$ is adapted to the filtration generated by $Y$. In addition, in finite discrete time, we can take $X$ to have the same law as $X$. A similar approximation is valid for randomized stopping times, without the first marginal fixed. Natural applications include relaxations of (mean field) stochastic control and causal optimal transport problems as well as new characterizations of the immersion property for progressively enlarged filtrations.
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, we show that it is possible for a species with a long-, but finite, range dispersal kernel to coexist with a superior competitor with nearest-neighbor dispersal in a model that includes deaths of blocks due to ``forest fires.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
