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Covariance parameter estimation of Gaussian processes is analyzed in an asymptotic framework. The spatial sampling is a randomly perturbed regular grid and its deviation from the perfect regular grid is controlled by a single scalar…
This paper establishes a CLT for linear statistics of the form $\langle \mathbf{q},\boldsymbol{\sigma} \rangle$ with quantitative Berry-Esseen bounds, where $\boldsymbol{\sigma}$ is an observation from an exponential family with a quadratic…
The paper presents a systematic theory for asymptotic inference of autocovariances of stationary processes. We consider nonparametric tests for serial correlations based on the maximum (or ${\cal L}^\infty$) and the quadratic (or ${\cal…
This paper studies the limits of a spatial random field generated by uniformly scattered random sets, as the density $\lambda$ of the sets grows to infinity and the mean volume $\rho$ of the sets tends to zero. Assuming that the volume…
Spontaneous stochasticity is a modern paradigm for turbulent transport at infinite Reynolds numbers. It suggests that tracer particles advected by rough turbulent flows and subject to additional thermal noise, remain non-deterministic in…
A lot of efforts have been devoted in the last decade to the investigation of the high-frequency behaviour of geometric functionals for the excursion sets of random spherical harmonics, i.e., Gaussian eigenfunctions for the spherical…
The last few years have seen a surge in excitement about measurements of statistics of the primordial fluctuations beyond the power spectrum. New ideas for precision tests of Gaussianity and statistical isotropy in the data are developing…
Analytic treatment of a non-equilibrium random system with large degrees of freedoms is one of most important problems of physics. However, little research has been done on this problem as far as we know. In this paper, we propose a new…
We study distributions $F$ on $[0,\infty)$ such that for some $T\le\infty$, $F^{*2}(x,x+T]\sim 2 F(x,x+T]$. The case $T=\infty$ corresponds to $F$ being subexponential, and our analysis shows that the properties for $T<\infty$ are, in fact,…
Mesoscopic effects associated with wave propagation in spacetime with metric stochasticity are studied. We show that the scalar and spinor waves in a stochastic spacetime behave similarly to the electrons in a disordered system. Viewing…
Order parameter fluctuations for the two dimensional Ising model in the region of the critical temperature are presented. A locus of temperatures T*(L) and of magnetic fields B*(L) are identified, for which the probability density function…
This paper surveys Abelian and Tauberian theorems for long-range dependent random fields. We describe a framework for asymptotic behaviour of covariance functions or variances of averaged functionals of random fields at infinity and…
Information about the physical processes that generate the primordial fluctuations in the early universe can be gained by testing the Gaussian nature of the fluctuations through cosmic microwave background radiation (CBR) temperature…
In expanding FRW spacetimes, it is usually the case that homogeneous scalar fields redshift and their amplitudes approach limiting values: Hubble friction usually ensures that the field relaxes to its minimum energy configuration, which is…
Area fluctuations of a Brownian excursion are described by the Airy distribution, which found applications in different areas of physics, mathematics and computer science. Here we generalize this distribution to describe the area…
A higher-order multiscale analysis of spatial anisotropy in inertial range magnetohydrodynamic turbulence is presented using measurements from the STEREO spacecraft in fast ambient solar wind. We show for the first time that, when measuring…
We consider a particle moving in a one dimensional potential which has a symmetric deterministic part and a quenched random part. We study analytically the probability distributions of the local time (spent by the particle around its mean…
We reconsider Ising spins in a Gaussian random field within the replica formalism. The corresponding continuum model involves several coupling constants beyond the single one which was considered in the standard $\phi^4$ theory approach.…
We study quantitative asymptotics of planar random walks that are spatially non-homogeneous but whose mean drifts have some regularity. Specifically, we study the first exit time $\tau_\alpha$ from a wedge with apex at the origin and…
Random field excursions is an increasingly vital topic within data analysis in medicine, cosmology, materials science, etc. This work is the first detailed study of their Betti numbers in the so-called `sparse' regime. Specifically, we…