Related papers: Connection Coefficients for Higher-order Bernoulli…
Let $X = (X_1, X_2)$ be a 2-dimensional random variable and $X(n), n \in \mathbb{N}$ a sequence of i.i.d. copies of $X$. The associated random walk is $S(n)= X(1) + \cdots +X(n)$. The corresponding absorbed-reflected walk $W(n), n \in…
We consider (random) walks in a multidimensional orthant. Using the idea of universality in probability theory, one can associate a unique polyhedral domain to any given walk model. We use this connection to prove two sets of new results.…
In this paper, we exploit the r-Stirling numbers of both kinds in order to give explicit formulae for the values of the high order Bernoulli numbers and polynomials of both kinds at integers. We give also some identities linked the…
The problem of detecting a few anomalous processes among a large number of data streams is considered. At each time, aggregated observations can be taken from a chosen subset of the processes, where the chosen subset conforms to a given…
We investigate semi-classical generalizations of the Charlier and Meixner polynomials, which are discrete orthogonal polynomials that satisfy three-term recurrence relations. It is shown that the coefficients in these recurrence relations…
Random walks are studied on disordered cellular networks in 2-and 3-dimensional spaces with arbitrary curvature. The coefficients of the evolution equation are calculated in term of the structural properties of the cellular system. The…
We revisit in a probabilistic framework the umbral approach of Bernoulli, Euler and Carlitz Hermite polynomials by Gessel [1].
We study the properties of discrete-time random walks on networks formed by randomly interconnected cliques, namely, random networks of cliques. Our purpose is to derive the parameters that define the network structure -- specifically, the…
This paper is concerned with Random walk approximations of the Brownian motion on the Affine group Aff(R). We are in particular interested in the case where the innovations are discrete. In this framework, the return probability of the walk…
Based on a determinantal formula for the higher derivative of a quotient of two functions, we first present the determinantal expressions of Eulerian polynomials and Andre polynomials. In particular, we discover that the Euler number…
In this note we prove convergence of Green functions with Neumann boundary conditions for the random walk to their continuous counterparts. Also a few Beurling type hitting estimates are obtained for the random walk on discretizations of…
We prove several identities expressing polynomials counting permutations by various descent statistics in terms of Eulerian polynomials, extending results of Stembridge, Petersen, and Br\"and\'en. Additionally, we find $q$-exponential…
In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.
We initiate the study of what we refer to as random walk labelings of graphs. These are graph labelings that are obtainable by performing a random walk on the graph, such that the labeling occurs increasingly whenever an unlabeled vertex is…
Random walks on regular bounded degree expander graphs have numerous applications. A key property of these walks is that they converge rapidly to the uniform distribution on the vertices. The recent study of expansion of high dimensional…
We find particular relations which we call "Bernoulli-type" in some noncommutative polynomial ring with a single nontrivial relation. More precisely, our ring is isomorphic to the universal enveloping algebra of a two-dimensional…
Consider a d-dimensional Brownian motion in a random potential defined by attaching a nonnegative and polynomially decaying potential around Poisson points. We introduce a repulsive interaction between the Brownian path and the Poisson…
The purpose of this paper is to provide an exact formula for the second moment of the empirical correlation of two independent Gaussian random walks as well as implicit formulas for higher moments. The proofs are based on a symbolically…
Arratia, and later T\'oth and Werner, constructed random processes that formally correspond to coalescing one-dimensional Brownian motions starting from every space-time point. We extend their work by constructing and characterizing what we…
In this review paper, we first discuss some open problems related to two-dimensional self-avoiding paths and critical percolation. We then review some closely related results (joint work with Greg Lawler and Oded Schramm) on critical…