Journal of Theoretical Probability

List of Papers (Total 51)

From intersection local time to the Rosenblatt process

The Rosenblatt process was obtained by Taqqu (Z. Wahr. Verw. Geb. 31:287–302, 1975) from convergence in distribution of partial sums of strongly dependent random variables. In this paper, we give a particle picture approach to the Rosenblatt process with the help of intersection local time and white noise analysis, and discuss measuring its long-range dependence by means of a ...

$U$ -Statistics of Ornstein–Uhlenbeck Branching Particle System

We consider a branching particle system consisting of particles moving according to the Ornstein–Uhlenbeck process in $\mathbb {R}^d$ and undergoing a binary, supercritical branching with a constant rate $\lambda >0$. This system is known to fulfill a law of large numbers (under exponential scaling). Recently the question of the corresponding central limit theorem (CLT) has ...

Characterizations of Some Free Random Variables by Properties of Conditional Moments of Third Degree Polynomials

We investigate Laha–Lukacs properties of noncommutative random variables (processes). We prove that some families of free Meixner distributions can be characterized by the conditional moments of polynomial functions of degree 3. We also show that this fact has consequences in describing some free Lévy processes. The proof relies on a combinatorial identity. At the end of this paper ...

On Hidden Markov Processes with Infinite Excess Entropy

We investigate stationary hidden Markov processes for which mutual information between the past and the future is infinite. It is assumed that the number of observable states is finite and the number of hidden states is countably infinite. Under this assumption, we show that the block mutual information of a hidden Markov process is upper bounded by a power law determined by the ...

Stochastic Representation of Weak Solutions of Viscous Conservation Laws: A BSDE Approach

We consider the Cauchy problem for systems of viscous conservation laws. We obtain three different but related stochastic representations of weak solutions of the problem: in terms of solutions to systems of usual backward stochastic differential equations, in terms of solutions to some stochastic backward systems, and in terms of solutions to some forward-backward stochastic ...

On Time-Dependent Functionals of Diffusions Corresponding to Divergence Form Operators

We consider processes of the form [s,T]∋t↦u(t,X t ), where (X,P s,x ) is a multidimensional diffusion corresponding to a uniformly elliptic divergence form operator. We show that if $u\in{\mathbb{L}}_{2}(0,T;H_{\rho }^{1})$ with $\frac{\partial u}{\partial t} \in{\mathbb{L}}_{2}(0,T;H_{\rho }^{-1})$ then there is a quasi-continuous version $\tilde{u}$ of u such that ...

Convergence of Semigroups of Complex Measures on a Lie Group

A theorem of Siebert in its essential part asserts that if μ n (t) are semigroups of probability measures on a Lie group G, and P n are the corresponding generating functionals, then $$\bigl \langle \mu_n(t),f \bigr \rangle \ \xrightarrow[n]{}\ \bigl \langle \mu_0(t),f \bigr \rangle , \quad f\in C_b(G), \ t>0,$$ implies \langle \pi_{P_n}u,v\rangle \ \xrightarrow[n]{}\ \langle ...

Random Walks on the Affine Group of a Homogeneous Tree in the Drift-Free Case

The affine group of a homogeneous tree is the group of all its isometries fixing an end of its boundary. We consider a random walk with law μ on this group and the associated random processes on the tree and its boundary. In the drift-free case there exists on the boundary of the tree a unique μ-invariant Radon measure. In this paper we describe its behaviour at infinity.

Current Fluctuations for Independent Random Walks in Multiple Dimensions

Consider a system of particles evolving as independent and identically distributed (i.i.d.) random walks. Initial fluctuations in the particle density get translated over time with velocity $\vec{v}$, the common mean velocity of the random walks. Consider a box centered around an observer who starts at the origin and moves with constant velocity $\vec{v}$. To observe ...

A Combinatorial Analysis of Interacting Diffusions

We attempt to unify the analysis of several families of naturally occurring multidimensional stochastic processes by studying the underlying combinatorics involved. At equilibrium, the behavior of these processes is determined by the properties of a randomly chosen point of a corresponding polyhedron. How such a randomly chosen point behaves is a difficult question which is ...

The Real Powers of the Convolution of a Gamma Distribution and a Bernoulli Distribution

In this paper, we essentially compute the set of x,y>0 such that the mapping $z\longmapsto(1-r+re^{z})^{x}(\frac{\lambda}{\lambda-z})^{y}$ is a Laplace transform. If X and Y are two independent random variables which have respectively Bernoulli and Gamma distributions, we denote by μ the distribution of X+Y. The above problem is equivalent to finding the set of x>0 such that μ *x ...

Initial Enlargement of Filtrations and Entropy of Poisson Compensators

Let μ be a Poisson random measure, let $\mathbb{F}$ be the smallest filtration satisfying the usual conditions and containing the one generated by μ, and let $\mathbb{G}$ be the initial enlargement of $\mathbb{F}$ with the σ-field generated by a random variable G. In this paper, we first show that the mutual information between the enlarging random variable G and the ...

Distribution of Global Measures of Deviation Between the Empirical Distribution Function and Its Concave Majorant

We investigate the distribution of some global measures of deviation between the empirical distribution function and its least concave majorant. In the case that the underlying distribution has a strictly decreasing density, we prove asymptotic normality for several L k -type distances. In the case of a uniform distribution, we also establish their limit distribution together with ...