Eigenvectors for a random walk on a hyperplane arrangement

We find explicit eigenvectors for the transition matrix of the Bidigare–Hanlon–Rockmore random walk, from Bidigare et al. (1999) [1]. This is accomplished by using Brown and Diaconis? (1998) analysis in [3] of the stationary distribution, together with some combinatorics of functions on the face lattice of a hyperplane arrangement, due to Gel?fand and Varchenko (1987) [10].     

Representations for the extreme zeros of orthogonal polynomials

We establish some representations for the smallest and largest zeros of orthogonal polynomials in terms of the parameters in the three-terms recurrence relation. As a corollary we obtain representations for the endpoints of the true interval of orthogonality. Implications of these results for the decay parameter of a birth-death process (with killing) are displayed.     

Poisson type approximations for the Markov binomial distribution

The Markov binomial distribution is approximated by the Poisson distribution with the same mean, by a translated Poisson distribution and by two-parametric Poisson type signed measures. Using an adaptation of Le Cam’s operator technique, estimates of accuracy are proved for the total variation, local and Wasserstein norms. In a special case, asymptotically sharp constants are obtained. For some auxiliary results, we used Stein’s method.     

Absolute continuity under flows generated by SDE with measurable drift coefficients

We consider the Itô SDE with a non-degenerate diffusion coefficient and a measurable drift coefficient. Under the condition that the gradient of the diffusion coefficient and the divergences of the diffusion and drift coefficients are exponentially integrable with respect to the Gaussian measure, we show that the stochastic flow leaves the reference measure absolutely continuous.     

Shift-coupling and a zero-one law for random walk in random environment

This paper discusses several aspects of shift-coupling for random walk in random environment.     

On the local time of random walk on the 2-dimensional comb

We study the path behaviour of general random walks, and that of their local times, on the 2-dimensional comb lattice C² that is obtained from Z² by removing all horizontal edges off the x-axis. We prove strong approximation results for such random walks and also for their local times. Concentrating mainly on the latter, we establish strong and weak limit theorems, including Strassen-type laws of the iterated logarithm, Hirsch-type laws, and weak convergence results in terms of functional convergence in distribution.     

Strong approximations of stochastic differential equations with jumps

This paper is a survey of strong discrete time approximations of jump-diffusion processes described by stochastic differential equations (SDEs). It also presents new results on strong discrete time approximations for the specific case of pure jump SDEs.     

Random orderings of the integers and card shuffling

In this paper we study random orderings of the integers with a certain invariance property. We describe all such orders in a simple way. We define and represent random shuffles of a countable set of labels and then give an interpretation of these orders in terms of a class of generalized riffle shuffles.     

Stationary max-stable processes with the Markov property

We prove that the class of discrete time stationary max-stable process satisfying the Markov property is equal, up to time reversal, to the class of stationary max-autoregressive processes of order 1. A similar statement is also proved for continuous time processes.     

Upper escape rate of Markov chains on weighted graphs

We obtain an upper escape rate function for a continuous time minimal symmetric Markov chain defined on a locally finite weighted graph. This upper rate function, which has the same form as the manifold setting, is given in terms of the volume growth with respect to an adapted path metric. Our approach also gives a weak form of Folz’s theorem on the conservativeness as a consequence.     

Random walks on quasisymmetric functions

Conditions are provided under which an endomorphism on quasisymmetric functions gives rise to a left random walk on the descent algebra which is also a lumping of a left random walk on permutations. Spectral results are also obtained. Several important random walks are now realized this way: Stanley's QS-distribution results from endomorphisms given by evaluation maps, a-shuffles result from the ath convolution power of the universal character, and the Tchebyshev operator of the second kind introduced recently by Ehrenborg and Readdy yields traditional riffle shuffles. A conjecture of Ehrenborg regarding the spectra for a family of random walks on ab-words is proven. A theorem of Stembridge from the theory of enriched P-partitions is also recovered as a special case.     

White noise driven SPDEs with reflection: Existence,uniqueness and large deviation principles

In the first part of this paper, we prove the uniqueness of the solutions of SPDEs with reflection, which was left open in the paper [C. Donati-Martin, E. Pardoux, White noise driven SPDEs with reflection, Probab. Theory Related Fields 95 (1993) 1–24]. We also obtain the existence of the solution for more general coefficients depending on the past with a much shorter proof. In the second part of the paper, we establish a large deviation principle for SPDEs with reflection. The weak convergence approach is proven to be very efficient on this occasion.     

Fractional integrals on compact Riemannian symmetric spaces of rank one

In this paper we study mixed norm boundedness for fractional integrals related to Laplace–Beltrami operators on compact Riemannian symmetric spaces of rank one. The key point is the analysis of weighted inequalities for fractional integral operators associated to trigonometric Jacobi polynomials expansions. In particular, we find a novel sharp estimate for the Jacobi fractional integral kernel with explicit dependence on the type parameters.