Let {Xα} be a family of random variables following a certain type of distributions with finite expectation \E[Xα]
and finite variance \Var(Xα), where α is a parameter. Motivated by the recent paper of Hollom and Portier (arXiv: 2306.07811v1), we study the anti-concentration function
(0,∞)∋y→inf and find its explicit expression.
We show that, for certain familiar families of distributions, including the uniform, exponential, non-degenerate Gaussian and student's t-distributions, the anti-concentration function is not identically zero, which means that the corresponding families of random variables
have some sort of anti-concentration property; while for some other familiar families of distributions, including the binomial, Poisson, negative binomial, hypergeometric, Gamma, Pareto, Weibull, log-normal and Beta distributions, the anti-concentration function is identically zero.
This paper is concerned with a class of
multi-dimensional reflected stochastic partial differential equations with elliptic operators, whose solutions are constrained on a bounded convex domain.
The aim of this paper is to establish the existence and uniqueness theorem of solutions for the reflected stochastic partial differential equations with the penalization method.