We consider the white-noise driven stochastic heat equation on $[0,1]$ with Lipschitz-continuous drift and diffusion coefficients. We derive an inequality for the $L^1([0,1])$-norm of the difference between two solutions. Using some martingale arguments, we show that this inequality provides some…

In this paper, we provide the sharp asymptotics for the quantization radius (maximal radius) for a sequence of optimal quantizers for random variables $X$ in $(\mathbb{R}^d,\|\,\cdot\,\|)$ with radial exponential tails. This result sharpens and generalizes the results developed for the…

Let $Y$ be a nonnegative random variable with mean $\mu$ and finite positive variance $\sigma^2$, and let $Y^s$, defined on the same space as $Y$, have the $Y$ size biased distribution, that is, the distribution characterized by $$ E[Yf(Y)]=\mu E…

By means of a simple conditioning/comparison argument, we derive the chance of a long lifetime for Brownian motion in a horn-shaped domain.

We study a class of adaptive Markov Chain Monte Carlo (MCMC) processes which aim at behaving as an ``optimal'' target process via a learning procedure. We show, under appropriate conditions, that the adaptive MCMC chain and the ``optimal'' (nonadaptive) MCMC…

Central limit theorems for Markov chains are considered, and in particular the relationships between various expressions for asymptotic variance known from the literature. These turn out to be equal under fairly general conditions, although not always. We also investigate the…

This paper proves that the scaling limit of nearest-neighbour senile reinforced random walk is Brownian Motion when the time T spent on the first edge has finite mean. We show that under suitable conditions, when T has heavy tails the…

This paper introduces and studies a family of new classes of infinitely divisible distributions on $\mathbb{R}^d$ with two parameters. Depending on parameters, these classes connect the Goldie-Steutel-Bondesson class and the class of generalized type $G$ distributions, connect the Thorin class…

We investigate the asymptotic behavior of weighted sums of independent standardized random variables with uniformly bounded third moments. The sequence of weights is given by a family of rectangular matrices with uniformly small entries and approximately orthogonal rows. We prove…

In this note we give semi-explicit solutions for $r$-balayages of matrix-exponential-Lévy processes. To this end, we turn to an identity for the joint Laplace transform of the first entry time and the undershoot and a semi-explicit solution of the negative…

For a coordinate symmetric random vector $(Y_1,\ldots,Y_n)={\bf Y} \in \mathbb{R}^n$, that is, one satisfying $(Y_1,\ldots,Y_n)=_d(e_1Y_1,\ldots,e_nY_n)$ for all $(e_1,\ldots,e_n) \in \{-1,1\}^n$, for which $P(Y_i=0)=0$ for all $i=1,2,\ldots,n$, the following Berry Esseen bound to the cumulative standard normal $\Phi$ for the standardized…