Fluctuations of shapes of large areas under paths of random walks

Summary We discuss statistical properties of random walks conditioned by fixing a large area under their paths. We prove the functional central limit theorem (invariance principle) for these conditional distributions. The limiting Gaussian measure coincides with the conditional probability distribution of certain timenonhomogeneous Gaussian random process obtained by an integral transformation of the white noise. From the point of view of statistical mechanics the studied problem is the problem of describing the fluctuations of the phase boundary in the one-dimensional SOS-model.     

On testing equality of pairwise rank correlations in a multivariate random vector

Spearman’s rank-correlation coefficient (also called Spearman’s rho) represents one of the best-known measures to quantify the degree of dependence between two random variables. As a copula-based dependence measure, it is invariant with respect to the distribution’s univariate marginal distribution functions. In this paper, we consider statistical tests for the hypothesis that all pairwise Spearman’s rank correlation coefficients in a multivariate random vector are equal. The tests are nonparametric and their asymptotic distributions are derived based on the asymptotic behavior of the empirical copula process. Only weak assumptions on the distribution function, such as continuity of the marginal distributions and continuous partial differentiability of the copula, are required for obtaining the results. A nonparametric bootstrap method is suggested for either estimating unknown parameters of the test statistics or for determining the associated critical values. We present a simulation study in order to investigate the power of the proposed tests. The results are compared to a classical parametric test for equal pairwise Pearson’s correlation coefficients in a multivariate random vector. The general setting also allows the derivation of a test for stochastic independence based on Spearman’s rho.     

On the recurrence of edge-reinforced random walk on ℤ×G

Let G be a finite tree. It is shown that edge-reinforced random walk on ℤ×G with large initial weights is recurrent. This includes recurrence on multi-level ladders of arbitrary width. For edge-reinforced random walk on {0,1, . . . ,n}×G, it is proved that asymptotically, with high probability, the normalized edge local times decay exponentially in the distance from the starting level. The estimates are uniform in n. They are used in the recurrence proof.     

On the equivalence of the static and dynamic points of view for diffusions in a random environment

We study the equivalence of the static and dynamic points of view for diffusions in a random environment in dimension one. First we prove that the static and dynamic distributions are equivalent if and only if either the speed in the law of large numbers does not vanish, or b/a$b/a$ is a.s. the gradient of a stationary function, where a$a$ and b$b$ are the covariance coefficient resp. the local drift attached to the diffusion. We moreover show that the equivalence of the static and dynamic points of view is characterized by the existence of so-called “almost linear coordinates”.     

Moderate deviations for a random walk in random scenery

We investigate the cumulative scenery process associated with random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramér’s condition. We prove moderate deviation principles in dimensions d≥2$d\ge 2$, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. For the case d≥4$d\ge 4$ we even obtain precise asymptotics for the probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. For d≥3$d\ge 3$, an important ingredient in the proofs are new concentration inequalities for self-intersection local times of random walks, which are of independent interest, whilst for d=2$d=2$ we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.     

Sharp critical behavior for pinning models in a random correlated environment

This article investigates the effect for random pinning models of long range power-law decaying correlations in the environment. For a particular type of environment based on a renewal construction, we are able to sharply describe the phase transition from the delocalized phase to the localized one, giving the critical exponent for the (quenched) free-energy, and proving that at the critical point the trajectories are fully delocalized. These results contrast with what happens both for the pure model (i.e., without disorder) and for the widely studied case of i.i.d. disorder, where the relevance or irrelevance of disorder on the critical properties is decided via the so-called Harris Criterion (Harris, 1974) [21].     

Valleys and the Maximum Local Time for Random Walk in Random Environment

Let ξ (n, x) be the local time at x for a recurrent one-dimensional random walk in random environment after n steps, and consider the maximum ξ^*(n) = max _x ξ(n, x). It is known that lim sup is a positive constant a.s. We prove that lim inf is a positive constant a.s. this answers a question of P. Révész [5]. The proof is based on an analysis of the valleys in the environment, defined as the potential wells of record depth. In particular, we show that almost surely, at any time n large enough, the random walker has spent almost all of its lifetime in the two deepest valleys of the environment it has encountered. We also prove a uniform exponential tail bound for the ratio of the expected total occupation time of a valley and the expected local time at its bottom.     

On the Brownian-directed polymer in a Gaussian random environment

In this paper, we introduce a model of Brownian polymer in a continuous random environment. The asymptotic behavior of the partition function associated to this polymer measure is studied, and we are able to separate a weak and strong disorder regime under some reasonable assumptions on the spatial covariance of the environment. Some further developments, concerning some concentration inequalities for the partition function, are given for the weak disorder regime.     

Approximation of stationary solutions of Gaussian driven stochastic differential equations

We study sequences of empirical measures of Euler schemes associated to some non-Markovian SDEs: SDEs driven by Gaussian processes with stationary increments. We obtain the functional convergence of this sequence to a stationary solution to the SDE. Then, we end the paper by some specific properties of this stationary solution. We show that, in contrast to Markovian SDEs, its initial random value and the driving Gaussian process are always dependent. However, under an integral representation assumption, we also obtain that the past of the solution is independent of the future of the underlying innovation process of the Gaussian driving process.     

Limiting behavior of a diffusion in an asymptotically stable environment

Let V be a two sided random walk and let X denote a real valued diffusion process with generator . This process is the continuous equivalent of the one-dimensional random walk in random environment with potential V. Hu and Shi (1997) described the Lévy classes of X in the case where V behaves approximately like a Brownian motion. In this paper, based on some fine results on the fluctuations of random walks and stable processes, we obtain an accurate image of the almost sure limiting behavior of X when V behaves asymptotically like a stable process. These results also apply for the corresponding random walk in random environment.     

Asymptotic properties of random matrices and pseudomatrices

We study the asymptotics of sums of matricially free random variables, called random pseudomatrices, and we compare it with that of random matrices with block-identical variances. For objects of both types we find the limit joint distributions of blocks and give their Hilbert space realizations, using operators called ‘matricially free Gaussian operators’. In particular, if the variance matrices are symmetric, the asymptotics of symmetric blocks of random pseudomatrices agrees with that of symmetric random blocks. We also show that blocks of random pseudomatrices are ‘asymptotically matricially free’ whereas the corresponding symmetric random blocks are ‘asymptotically symmetrically matricially free’, where symmetric matricial freeness is obtained from matricial freeness by an operation of symmetrization. Finally, we show that row blocks of square, block-lower-triangular and block-diagonal pseudomatrices are asymptotically free, monotone independent and boolean independent, respectively.     

Crossing velocities for an annealed random walk in a random potential

We consider a random walk in an i.i.d. non-negative potential on the d-dimensional integer lattice. The walk starts at the origin and is conditioned to hit a remote location y on the lattice. We prove that the expected time under the annealed path measure needed by the random walk to reach y grows only linearly in the distance from y to the origin. In dimension 1 we show the existence of the asymptotic positive speed.     

Slepian’s inequality with respect to majorization

In this paper, we obtain some sufficient conditions for Slepian’s inequality with respect to majorization for two Gaussian random vectors.     

Intermittency for stochastic partial differential equations driven by strongly inhomogeneous space–time white noises

In this paper, the main topic is to investigate the intermittent property of the one-dimensional stochastic heat equation driven by an inhomogeneous Brownian sheet, which is a noise deduced from the study of the catalytic super-Brownian motion. Under some proper conditions on the catalytic measure of the inhomogeneous Brownian sheet, we show that the solution is weakly full intermittent based on the estimates of moments of the solution. In particular, it is proved that the second moment of the solution grows at the exponential rate. The novelty is that the catalytic measure relative to the inhomogeneous noise is not required to be absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$.     

Representation of Gaussian isotropic spin random fields

We develop a technique for the construction of random fields on algebraic structures. We deal with two general situations: random fields on homogeneous spaces of a compact group and in the spin line bundles of the 2$2$-sphere. In particular, every complex Gaussian isotropic spin random field can be represented in this way. Our construction extends P. Lévy’s original idea for the spherical Brownian motion.     

Generalized Markov fields and Dirichlet forms

We prove that Gaussian measure-indexed random fields, of which the covariance functional is given by the dual form of a transient Dirichlet form, have the global Markov property (where global here means w.r.t. arbitrary, not necessarily open sets), if and only if the Dirichlet form has the local property. Applications to Nelson's free Euclidean field of quantum theory and to Rozanov's generalized random functions are given.     

One-dimensional Schrödinger operators with random potentials: A survey

Acta Applicandae Mathematicae - We present and discuss recent results on the spectral properties of random Schrödinger operators. This survey paper deals only with the one-dimensional case....     

On the concentration of Sinai’s walk

We consider Sinai’s random walk in a random environment. We prove that for an interval of time [1,n]$[1,n]$ Sinai’s walk sojourns in a small neighborhood of the point of localization for the quasi-totality of this amount of time. Moreover the local time at the point of localization normalized by n$n$ converges in probability to a well defined random variable of the environment. From these results we get applications to the favorite sites of the walk and to the maximum of the local time.     

Concordance measures for multivariate non-continuous random vectors

A notion of multivariate concordance suitable for non-continuous random variables is defined and many of its properties are established. This allows the definition of multivariate, non-continuous versions of Kendall’s tau, Spearman’s rho and Spearman’s footrule, which are concordance measures. Since the maximum values of these association measures are not +1 in general, a special attention is given to the computation of upper bounds. The latter turn out to be multivariate generalizations of earlier findings made by Nešlehová (2007) [9] and Denuit and Lambert (2005) [2]. They are easy to compute and can be estimated from a data set of (possibly) discontinuous random vectors. Corrected versions are considered as well.