Related papers: $H(X)$ vs. $H(f(X))$

For a continuous map $f$ from the real line (half-open interval $[0,1)$) into itself let ent(f) denote the supremum of topological entropies of $f|_K$, where $K$ runs over all compact $f$-invariant subsets of $\mathbb{R}$ ($[0,1)$,…

The entropy per coordinate in a log-concave random vector of any dimension with given density at the mode is shown to have a range of just 1. Uniform distributions on convex bodies are at the lower end of this range, the distribution with…

Entropic uncertainty relations place nontrivial lower bounds to the sum of Shannon information entropies for noncommuting observables. Here we obtain a novel lower bound on the entropy sum for general pairs of observables in…

We give an upper bound for the topological entropy of maps on inverse limit spaces in terms of their set-valued components. In a special case of a diagonal map on the inverse limit space $\underleftarrow{\lim}(I,f)$, where every diagonal…

Let $F^{*}$ be an approximation of a given $(a \times b)$ matrix $F$ derived by methods that are not randomized. We prove that for a given $F$ and $F^{*}$, $H$ and $T$ can be computed by randomized algorithm such that $(HT)$ is an…

Here is the simplest particular case of our main result: let $f:{\bf R}\to {\bf R}$ be a function of class $C^1$, with $\sup_{\bf R}f'>0$, such that $$\lim_{|\xi|\to +\infty}{{f(\xi)}\over {\xi}}=0\ .$$ Then, for each $\lambda>{{\pi^2}\over…

We provide finite sample upper and lower bounds on the Binomial tail probability which are a direct application of Sanov's theorem. We then use these to obtain high probability upper and lower bounds on the minimum of i.i.d. Binomial random…

A tight upper bound is given on the distribution of the maximum of a supermartingale. Specifically, it is shown that if $Y$ is a semimartingale with initial value zero and quadratic variation process $[Y,Y]$ such that $Y + [Y,Y]$ is a…

We derive the probability distribution of product of two independent random variables, each distributed according the one-dimensional stable law. We represent the density by its power series and its asymptotic expansions. As Fox's…

In this paper, we generalize and improve some fundamental concentration inequalities using information on the random variables' higher moments. In particular, we improve the classical Hoeffding's and Bennett's inequalities for the case…

For a certain moment, the information volume represented in a probability space can be accurately measured by Shannon entropy. But in real life, the results of things usually change over time, and the prediction of the information volume…

We consider the following frustrated optimization problem: given a prior probability distribution $q$, find the distribution $p$ minimizing the relative entropy with respect to $q$ such that $\textrm{mean}(p)$ is fixed and large. We show…

The zero bias distribution $W^*$ of $W$, defined though the characterizing equation $\mathit{EW}f(W)=\sigma^2Ef'(W^*)$ for all smooth functions $f$, exists for all $W$ with mean zero and finite variance $\sigma^2$. For $W$ and $W^*$ defined…

We consider two numerical entropy--type invariants for actions of $\Zk$, invariant under a choice of generators and well-adapted for smooth actions whose individual elements have positive entropy. We concentrate on the maximal rank case,…

We show that a proportionality between the entanglement Hamiltonian and the Hamiltonian of a subsystem exists near the limit of maximal entanglement under certain conditions. Away from that limit, solvable models show that the coupling…

We show that a lower bound for covariance of $\min(X_1,X_2)$ and $\max(X_1,X_2)$ is $\cov{X_1}{X_2}$ and an upper bound for variance of \\ $\min(X_2,\max(X,X_1))$ is $\var{X} + \var{X_1} +\var{X_2}$ generalizing previous results. We also…

The Fourier-Entropy Influence (FEI) Conjecture states that for any Boolean function $f:\{+1,-1\}^n \to \{+1,-1\}$, the Fourier entropy of $f$ is at most its influence up to a universal constant factor. While the FEI conjecture has been…

In this paper, we derive new probability bounds for Chebyshev's inequality if the supremum of the probability density function is known. This result holds for one-dimensional or multivariate continuous probability distributions with finite…

We study the probability of Boolean functions with small max influence to become constant under random restrictions. Let $f$ be a Boolean function such that the variance of $f$ is $\Omega(1)$ and all its individual influences are bounded by…

We consider upper bounds for the approximation error E|g(X)-g(\hat X)|^p, where X and \hat X are random variables such that \hat X is an approximation of X in the L_p-norm, and the function g belongs to certain function classes, which…