Extending the Support Theorem to Infinite Dimensions

Functional central limit theorems for occupancies and missing mass process in infinite urn models

We study the infinite urn scheme when the balls are sequentially distributed over an infinite number of urns labelled 1,2,... so that the urn $j$ at every draw gets a ball with probability $p_j$, $\sum_j p_j=1$. We prove functional central limit theorems for discrete time and the poissonised version for the urn occupancies process, for the odd-occupancy and for the missing mass processes extending the known non-functional central limit theorems.

Sum rules for effective resistances in infinite graphs

Extending work of Foster, Doyle, and others, we show how the Foster Theorems, a family of results concerning effective resistances on finite graphs, can in certain cases be extended to infinite graphs. A family of sum rules is then obtained, which allows one to easily calculate the sum of the resistances over all paths of a given length. The results are illustrated with some of the most common grids in the plane, including the square, triangular, and hexagonal grids.

Limit theory for some positive, stationary processes with infinite mean

We prove distributional limit theorems and one-sided laws of the iterated logarithm for a class of positive, mixing, stationary, stochastic processes which contains those obtained from non-integrable observables over certain piecewise expanding maps. This is done by extending Darling-Kac theory to a suitable family of infinite measure preserving transformations.

无限维随机微分方程中持有时性的一个紧支撑性质

A compact support property for infinite-dimensional SDEs with Hölder continuous coefficients

本文研究了无限维随机微分方程中Hölder连续噪声系数的非负解的紧支撑性质，证明当Hölder指数小于1/2时，解在几乎所有时间具有紧支撑，且展示了超过程在空间运动不连续时的支持传播特性。

A support property for infinite dimensional interacting diffusion processes

The Dirichlet form associated with the intrinsic gradient on Poisson space is known to be quasi-regular on the complete metric space $\ddotΓ=$ $\{Z_+$-valued Radon measures on $\IR^d\}$. We show that under mild conditions, the set $\ddotΓ\setminusΓ$ is $\e$-exceptional, where $Γ$ is the space of locally finite configurations in $\IR^d$, that is, measures $γ\in\ddotΓ$ satisfying $\sup_{x\in\IR^d}γ(\{x\})\leq 1$. Thus, the associated diffusion lives on the smaller space $Γ$. This result also holds for Gibbs measures with superstable interactions.

A new non-negative distribution with both finite and infinite support

The Tukey-$λ$ distribution has interesting properties including (i) for some parameters values it has finite support, and for others infinite support, and (ii) it can mimic several other distributions such that parameter estimation for the Tukey distribution is a method for identifying an appropriate class of distribution to model a set of data. The Tukey-$λ$ is, however, symmetric. Here we define a new class of {\em non-negative} distribution with similar properties to the Tukey-$λ$ distribution. As with the Tukey-$λ$ distribution, our distribution is defined in terms of its quantile function…