We study convergence in law of partial sums of linear processes with heavy-tailed innovations. In the case of summable coefficients, necessary and sufficient conditions for the finite dimensional convergence to an \(\alpha \)-stable Lévy Motion are given. The conditions lead to new, tractable sufficient conditions in the case \(\alpha \le 1\). In the functional setting, we...

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...

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...

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...

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...

We introduce a method of proving maximal inequalities for Hilbert- space-valued differentially subordinate local martingales. As an application, we prove that if \(X=(X_t)_{t\ge 0},\, Y=(Y_t)_{t\ge 0}\) are local martingales such that \(Y\) is differentially subordinate to \(X\), then $$\begin{aligned} ||Y||_1\le \beta ||\sup _{t\ge 0}|X_t|\;||_1, \end{aligned}$$where \(\beta =2...

We prove the Wiener–Hopf factorization for Markov additive processes. We derive also Spitzer–Rogozin theorem for this class of processes which serves for obtaining Kendall’s formula and Fristedt representation of the cumulant matrix of the ladder epoch process. Finally, we also obtain the so-called ballot theorem.

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...

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...

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...

We relate various concepts of fractal dimension of the limiting set \(\mathcal{C}\) in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in \(\mathcal{C}\) (the “dust”). In two dimensions, we also show that the set consisting of connected components larger than one point is almost surely the union of non...

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.

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...

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...

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...

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...

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...