Related papers: $q$-Chaos

Consider the ensemble of Gaussian random potentials $\{V^L(q)\}_{L=1}^\infty$ on the $d$-dimensional torus where, essentially, $V^L(q)$ is a real-valued trigonometric polynomial of degree $L$ whose coefficients are independent standard…

Let T : Lp --> Lp be a positive contraction, with p strictly between 1 and infinity. Assume that T is analytic, that is, there exists a constant K such that \norm{T^n-T^{n-1}} < K/n for any positive integer n. Let q strictly betweeen 2 and…

A necessary condition is given for a sequence of identically distributed and pairwise positively quadrant dependent random variables obeying the strong laws of large numbers with respect to the normalising constants $n^{1/p}$ $(1 \leqslant…

The $L^p$ ($1<p<\infty$) and weak-$L^1$ estimates for the variation for Calder\'on-Zygmund operators with smooth odd kernel on uniformly rectifiable measures are proven. The $L^2$ boundedness and the corona decomposition method are two key…

We investigate conditions for logarithmic complete monotonicity of product ratios of gamma and q-gamma functions whose arguments are linear functions of the variable. We give necessary and sufficient conditions in terms of nonnegativity of…

Under certain conditions, the rate of increase of the statistical entropy of a simple, fully chaotic, conservative system is known to be given by a single number, characteristic of this system, the Kolmogorov-Sinai entropy rate. This…

Suppose $X$ is a real or complexified Banach space containing a complemented copy of $\ell_p$, $p\in(1,2)$, and a copy (not necessarily complemented) of either $\ell_q$, $q\in(p,\infty)$, or $c_0$. Then $\mathcal{L}(X)$ and…

Consider the problem of estimating a random variable $X$ from noisy observations $Y = X+ Z$, where $Z$ is standard normal, under the $L^1$ fidelity criterion. It is well known that the optimal Bayesian estimator in this setting is the…

We give an alternate proof of one of the inequalities proved recently for martingales (=sums of martingale differences) in a non-commutative $L_p$-space, with $1<p<\infty$, by Q. Xu and the author. This new approach is restricted to $p$ an…

In the present paper we investigate some geometrical properties of the norms in Banach function spaces. Particularly there is shown that if exponent $1/p(\cdot)$ belongs to $BLO^{1/\log}$ then for the norm of corresponding variable exponent…

In a recent article A.M. Kagan and G.J.Sz\'ekely introduced a notion of Q-independent and Q-identical distributed random variables. We give a complete description of polynomials which appear in these definitions.

We show that, for every $1 \leq p < +\infty$ and for every Borel probability measure $\mathbb{P}$ over $\mathbb{R}$, every element of $L^{p}(\mathbb{R}, \mathscr{B}_{\mathbb{R}}, \mathbb{P})$ is the $L^{p}$-limit of some sequence of bounded…

The subject of this paper is a generalized Camassa-Holm equation under random perturbation. We first establish local existence and uniqueness results as well as blow-up criteria for pathwise solutions in the Sobolev spaces $H^s$ with…

We consider the problem of testing mutual independence among the components of a high-dimensional random vector. Building on the rank-based max-sum framework, we introduce fixed finite-$L_q$ power-sum statistics under three general classes…

We consider a measurable stationary Gaussian stochastic process. A criterion for testing hypotheses about the covariance function of such a process using estimates for its norm in the space $L_p(\mathbb {T}),\,p\geq1$, is constructed.

We consider multilinear Littlewood polynomials, polynomials in $n$ variables in which a specified set of monomials $U$ have $\pm 1$ coefficients, and all other coefficients are $0$. We provide upper and lower bounds (which are close for $U$…

We study parameter estimation in linear Gaussian covariance models, which are $p$-dimensional Gaussian models with linear constraints on the covariance matrix. Maximum likelihood estimation for this class of models leads to a non-convex…

We prove matching direct and inverse theorems for (algebraic) polynomial approximation with doubling weights $w$ having finitely many zeros and singularities (i.e., points where $w$ becomes infinite) on an interval and not too ``rapidly…

Let \psi(x) be a polynomial with rational coefficients. Suppose that \psi has the positive leading coefficient and zero constant term. Let A be a set of positive integers with the positive upper density. Then there exist x,y\in A and a…

In this paper, we establish $L_p$ estimates and solvability for time fractional divergence form parabolic equations in the whole space when leading coefficients are merely measurable in one spatial variable and locally have small mean…