Related papers: Invariance principles for operator-scaling Gaussia…
We establish the discrete approximation to Brownian motion with varying dimension (BMVD in abbreviation) by random walks. The setting is very similar to that in [11], but here we use a different method allowing us to get rid the…
We obtain sharp upper and lower bounds for the moderate deviations of the volume of the range of a random walk in dimension five and larger. Our results encompass two regimes: a Gaussian regime for small deviations, and a stretched…
The computation of multifractal scaling properties associated with a critical field theory involves non-local operators and remains an open problem using conventional techniques of field theory. We propose a new description of Gaussian…
Consider a one dimensional simple random walk $X=(X_n)_{n\geq0}$. We form a new simple symmetric random walk $Y=(Y_n)_{n\geq0}$ by taking sums of products of the increments of $X$ and study the two-dimensional walk…
In the paper we present simple examples of linear random fields defined on $\ZZ^2$ and $\ZZ^3$ which exhibit the scaling transition phenomenon. These examples lead to more general definition of the scaling transition and allow to understand…
We introduce oscillatory analogues of fractional Brownian motion, sub-fractional Brownian motion and other related long range dependent Gaussian processes, we discuss their properties, and we show how they arise from particle systems with…
We prove the existence of scaling limits for the projection on the backbone of the random walks on the Incipient Infinite Cluster and the Invasion Percolation Cluster on a regular tree. We treat these projected random walks as randomly…
We obtain an elementary invariance principle for multi-dimensional Brownian sheet where the underlying random fields are not necessarily independent or stationary. Possible applications include unit-root tests for spatial as well as panel…
We introduce a variation of the step-reinforced random walk with general memory. For the diffusive regime, we establish a functional invariance principle and show that, given suitable conditions on the memory sequence, the arising limiting…
We prove large deviations principles (LDPs) for the perimeter and the area of the convex hull of a planar random walk with finite Laplace transform of its increments. We give explicit upper and lower bounds for the rate function of the…
In this paper the multi-dimensional random walk models governed by distributed fractional order differential equations and multi-term fractional order differential equations are constructed. The scaling limits of these random walks to a…
We study pathwise invariances of centred random fields that can be controlled through the covariance. A result involving composition operators is obtained in second-order settings, and we show that various path properties including…
Real-world road networks have an approximate scale-invariance property; can one devise mathematical models of random networks whose distributions are {\em exactly} invariant under Euclidean scaling? This requires working in the continuum…
Under certain mild conditions, some limit theorems for functionals of two independent Gaussian processes are obtained. The results apply to general Gaussian processes including fractional Brownian motion, sub-fractional Brownian motion and…
We prove that the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian. A similar statement is proven for the $\lambda \phi^4$…
Some probabilistic aspects of the number variance statistic are investigated. Infinite systems of independent Brownian motions and symmetric alpha-stable processes are used to construct new examples of processes which exhibit both divergent…
We consider possible scale-dependence of the non-linearity parameter f_NL in local and quasi-local models of non-Gaussian primordial density perturbations. In the simplest model where the primordial perturbations are a quadratic local…
We obtain a complete description of anisotropic scaling limits of random grain model on the plane with heavy tailed grain area distribution. The scaling limits have either independent or completely dependent increments along one or both…
In this article we prove large deviations principles for high minima of Gaussian processes with nonnegatively correlated increments on arbitrary intervals. Furthermore, we prove large deviations principles for the increments of such…
We study large deviations for random walks on stratified (Carnot) Lie groups. For such groups, there is a natural collection of vectors which generates their Lie algebra, and we consider random walks with increments in only these…