Random Walks in the Quarter-Plane: Advances in Explicit Criterions for the Finiteness of the Associated Group in the Genus 1 Case

In the book [FIM], original methods were proposed to determine the invariant measure of random walks in the quarter plane with small jumps, the general solution being obtained via reduction to boundary value problems. Among other things, an important quantity, the so-called group of the walk, allows to deduce theoretical features about the nature of the solutions. In particular, when the \emph{order} of the group is finite, necessary and sufficient conditions have been given in [FIM] for the solution to be rational or algebraic. In this paper, when the underlying algebraic curve is of genus $1$, we propose a concrete criterion ensuring the finiteness of the group. It turns out that this criterion can be expressed as the cancellation of a determinant of a matrix of order 3 or 4, which depends in a polynomial way on the coefficients of the walk.Comment: 2 figure

Letter to editors

International audienceThis letter is to let you know that we found serious errors in the paper [1], recently published in QUEST

Conditions for some non stationary random walks in the quarter plane to be singular or of genus 0

International audienceWe analyze the kernel K(x,y,t) of the basic functional equation associated with the tri-variate counting generating function (CGF) of walks in the quarter plane. In this short paper, taking t ∈]0, 1[, we provide the conditions on the jump probabilities {pi,j ’s} to decide whether walks are singular or regular, as defined in [3, Section 2.3]. These conditions are independent of t ∈]0, 1[ and given in terms of step set configurations. We also find the configurations for the kernel to be of genus 0, knowing that the genus is always ≤1. All these conditions are very similar to that of the stationary case considered in [3]. Our results extend the work [2], which considers only the special situation where t ∈]0, 1[ is a transcendental number over the field Q(pi,j). In addition, when p(0,0) = 0, our classification holds for all t ∈]0, +∞]

Asymptotic optimality of the Bayes estimator on Differentiable in Quadratic Mean models

This paper deals with the study of the Bayes estimator's asymptotic properties on Differentiable in Quadratic Mean (DQM) models in the case of independent and identically distributed observations. The investigation is led in order to define weak assumptions on the model under which this estimator is asymptotically efficient, regular and asymptotically of minimal risk. The results of the paper are applied to models based on a mixture distribution, the Cauchy with location and scale parameter's and the Weibull'

Random Walks in the Quarter Plane: Algebraic Methods, Boundary Value Problems, Applications to Queueing Systems and Analytic Combinatorics

The first edition was published in 1999International audienceThis monograph aims to promote original mathematical methods to determine the invari-ant measure of two-dimensional random walks in domains with boundaries. Such processesare encountered in numerous applications and are of interest in several areas of mathemat-ical research like Stochastic Networks, Analytic Combinatorics, Quantum Physics. Thissecond edition consists of two parts.Part I is a revised upgrade of the rst edition (1999), with additional recent results onthe group of the random walk. The theoretical approach given therein has been developedby the authors since the early 1970s. By using Complex Function Theory, BoundaryValue Problems, Riemann Surfaces, Galois Theory, completely new methods are proposedfor solving functional equations of two complex variables, which can also be applied tocharacterize the Transient Behavior of the walks, as well as to nd explicit solution tothe one-dimensional Quantum Three-Body Problem, or to tackle a new class of IntegrableSystems.Part II borrows specic case-studies from queueing theory (in particular the famousproblem of Joining the Shorter of Two Queues), and enumerative combinatorics (Counting,Asymptotics)

Algebraic generating functions for two-dimensional random walks

SIGLEAvailable at INIST (FR), Document Supply Service, under shelf-number : 14802 E, issue : a.1990 n.1184 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc