Related papers: A quantitative central limit theorem for the effec…
We study central limit theorems for certain nonlinear sequences of random variables. In particular, we prove the central limit theorems for the bounded conductivity of the random resistor networks on hierarchical lattices.
We investigate the (generalized) Walsh decomposition of point-to-point effective resistances on countable random electric networks with i.i.d. resistances. We show that it is concentrated on low levels, and thus point-to-point effective…
We show a central limit theorem for random walk on a Galton-Watson tree, when the edges of the tree are assigned randomly uniformly elliptic conductances. When a positive fraction of edges is assigned a small conductance $\varepsilon$, we…
We prove a local limit theorem for nearest neighbours random walks in stationary random environment of conductances on Z without using any of both classic assumptions of uniform ellipticity and independence on the conductances. Besides the…
Inspired by the study of edge statistics of random band matrices, we investigate random walks on large $d$-dimensional periodic lattices, whose transition matrices are determined by discretized density functions. Under certain moment…
We consider the random walk among random conductances on Z^d. We assume that the conductances are independent, identically distributed and uniformly bounded away from 0 and infinity. We obtain a quantitative version of the central limit…
For a L\'evy basis $L$ on $\mathbb{R}^d$ and a suitable kernel function $f:\mathbb{R}^d \to \mathbb{R}$, consider the continuous spatial moving average field $X=(X_t)_{t\in \mathbb{R}^d}$ defined by $X_t = \int_{\mathbb{R}^d} f(t-s) \,…
We prove two theorems related to the Central Limit Theorem (CLT) for Martin-L\"of Random (MLR) sequences. Martin-L\"of randomness attempts to capture what it means for a sequence of bits to be "truly random". By contrast, CLTs do not make…
The Thouless conjecture states that the average conductance of a disordered metallic sample in the diffusive regime can be related to the sensitivity of the sample's spectrum to a change in the boundary conditions. Here we present results…
We consider a random conductance model on the $d$-dimensional lattice, $d\in[2,\infty)\cap\mathbb{N}$, where the conductances take values in $(0,\infty)$ and are however not assumed to be bounded from above and below. We assume that the law…
A central limit theorem for the integrated squared error of the directional-linear kernel density estimator is established. The result enables the construction and analysis of two testing procedures based on squared loss: a nonparametric…
Central limit theorems are established for the sum, over a spatial region, of observations from a linear process on a $d$-dimensional lattice. This region need not be rectangular, but can be irregularly-shaped. Separate results are…
We consider the damped/driven cubic NLS equation on the torus of a large period $L$ with a small nonlinearity of size $\lambda$, a properly scaled random forcing and dissipation. We examine its solutions under the subsequent limit when…
We consider a symmetric random walk on the $\nu$-dimensional lattice, whose exit probability from the origin is modified by an antisymmetric perturbation and prove the local central limit theorem for this process. A short-range correction…
The Central Limit Theorem for the random walk on a stationary random network of conductances has been studied by several authors. In one dimension, when conductances and resistances are integrable, and following a method of martingale…
Let $\Cal S$ be an abelian finitely generated semigroup of endomorphisms of a probability space $(\Omega, {\Cal A}, \mu)$, with $(T_1, ..., T_d)$ a system of generators in ${\Cal S}$. Given an increasing sequence of domains $(D_n) \subset…
We consider random resistor networks with nodes given by a point process on $\mathbb{R}^d$ and with random conductances. The length range of the electrical filaments can be unbounded. We assume that the randomness is stationary and ergodic…
A new method is developed for the study of transport properties of 1D models with random potentials. It is based on an exact transformation that reduces discrete Schr\"odinger equation in the tight-binding model to a two-dimensional…
Distance covariance is a popular dependence measure for two random vectors $X$ and $Y$ of possibly different dimensions and types. Recent years have witnessed concentrated efforts in the literature to understand the distributional…
We study the distribution of a fully connected neural network with random Gaussian weights and biases in which the hidden layer widths are proportional to a large constant $n$. Under mild assumptions on the non-linearity, we obtain…