Related papers: Differentiating the absolutely continuous invarian…

Let $G_1,\dots,G_m$ be independent copies of the standard gaussian random vector in $\mathbb{R}^d$. We show that there is an absolute constant $c$ such that for any $A \subset S^{d-1}$, with probability at least $1-2\exp(-c\Delta m)$, for…

We consider the differential entropy of probability measures absolutely continuous with respect to a given $\sigma$-finite reference measure on an arbitrary measurable space. We state the asymptotic equipartition property in this general…

Let's consider a control system described by the implicit equation $F(x,\dot x) = 0$. If this system is differentially flat, then the following criterion is satisfied : For some integer $r$, there exists a function $\varphi(y_0, y_1,…

Let $\mu$ be a Borel probability measure on a compact path-connected metric space $(X, \rho)$ for which there exist constants $c,\beta>1$ such that $\mu(B) \geq c r^{\beta}$ for every open ball $B\subset X$ of radius $r>0$. For a class of…

Consider the case that we observe $n$ independent and identically distributed copies of a random variable with a probability distribution known to be an element of a specified statistical model. We are interested in estimating an infinite…

Let $f:M\to M$ be a homeomorphism over a compact Riemannian manifold, ergodic with respect to a measure $\mu$ defined on the completion of the Borel $\sigma$-algebra and $\mathcal F$ a $f$-invariant one dimensional continuous foliation of…

Let $X$ be a Harris recurrent strong Markov process in continuous time with general Polish state space $E,$ having invariant measure $\mu .$ In this paper we use the regeneration method to derive non asymptotic deviation bounds for $$P_{x}…

Let $f$ be a homeomorphism of the closed annulus $A$ isotopic to the identity, and let $X\subset {\rm Int}A$ be an $f$-invariant continuum which separates $A$ into two domains, the upper domain $U_+$ and the lower domain $U_-$. Fixing a…

We construct SRB measures for endomorphisms satisfying conditions far weaker than the non-uniformly expansion. As a consequence, the definition of non-uniformly expanding map can be weakened. We also prove the existence of an absolutely…

Let $S(0,1)$ be the $*$-algebra of all classes of Lebesgue measurable functions on the unit interval $(0,1)$ and let $(\mathcal{A},\left\|\cdot \right\|_\mathcal{A})$ be a complete symmetric $\Delta$-normed $*$-subalgebra of $S(0,1)$, in…

Let $\psi$ be a continuous decreasing function defined on all large positive real numbers. We say that a real $m\times n$ matrix $A$ is $\psi$-Dirichlet if for every sufficiently large real number $t$ one can find $\boldsymbol{p} \in…

A generalization of the classical Sard theorem in the plane is the following. Let $f$ be a function defined on a subset $A\subset{\mathbb R}^2$. If $f$ has modulus of continuity $\omega(r)\lesssim r^2$, then $f(A)\subset{\mathbb R}$ has…

Let $X$ be a locally compact Polish space. Let $\mathbb K(X)$ denote the space of discrete Radon measures on $X$. Let $\mu$ be a completely random discrete measure on $X$, i.e., $\mu$ is (the distribution of) a completely random measure on…

Consider a symmetric function $\mathcal{C}(x,y)$ on $[0,1]\times[0,1]$ which is twice continuously differentiable up to the boundary, and which satisfies $ \mathcal{C}(x,y)=\mathcal{C}(1-x,1-y)$. Let $A^{(n)} = \big(a^{(n)}_{i,j}\, :\, i,j…

We prove that there is a universal constant $C>0$ with the following property. Suppose that $n\in \mathbb{N}$ and that $\mathsf{A}=(a_{ij})\in M_n(\mathbb{R})$ is a symmetric stochastic matrix. Denote the second-largest eigenvalue of…

For piecewise $C^1$ interval maps possibly containing critical points and discontinuities with negative Schwarzian derivative, under two summability conditions on the growth of the derivative and recurrence along critical orbits, we prove…

Let $M$ be a hyperbolic Riemann surface with the first eigenvalue $\lambda_1(M)>0$. Let $\rho$ denote the distance from a fixed point $x_0\in{M}$ and $r_x$ the injectivity radius at $x$. We show that there exists a numerical constant…

We compute analytically the joint probability density of eigenvalues and the level spacing statistics for an ensemble of random matrices with interesting features. It is invariant under the standard symmetry groups (orthogonal and unitary)…

If $f:[a,b]\to \mathbb{R}$, with $a<b$, is continuous and such that $a$ and $b$ are mapped in opposite directions by $f$, then $f$ has a fixed point in $I$. Suppose that $f:\mathbb{C}\to\mathbb{C}$ is map and $X$ is a continuum. We extend…

We compute spectra of sample auto-covariance matrices of second order stationary stochastic processes. We look at a limit in which both the matrix dimension $N$ and the sample size $M$ used to define empirical averages diverge, with their…