Related papers: Improved bounds for the total variation distance b…
We prove that the distribution density of any non-constant polynomial $f(\xi_1,\xi_2,\ldots)$ of degree $d$ in independent standard Gaussian random variables $\xi$ (possibly, in infinitely many variables) always belongs to the…
In this paper we study bounds for the total variation distance between two second degree polynomials in normal random variables provided that they essentially depend on at least three variables.
The paper provides an estimate of the total variation distance between distributions of polynomials defined on a space equipped with a logarithmically concave measure in terms of the $L^2$-distance between these polynomials.
The goal of this paper is to estimate the total variation distance between two general stochastic polynomials. As a consequence one obtains an invariance principle for such polynomials. This generalizes known results concerning the total…
New lower bounds on the total variation distance between the distribution of a sum of independent Bernoulli random variables and the Poisson random variable (with the same mean) are derived via the Chen-Stein method. The new bounds rely on…
We consider the approximation of a convolution of possibly different probability measures by (compound) Poisson distributions and also by related signed measures of higher order. We present new total variation bounds having a better…
Mixtures of high dimensional Gaussian distributions have been studied extensively in statistics and learning theory. While the total variation distance appears naturally in the sample complexity of distribution learning, it is analytically…
For arbitrary two probability measures on real d-space with given means and variances (covariance matrices), we provide lower bounds for their total variation distance. In the one-dimensional case, a tight bound is given.
We show that a measure on the real line that is the image of a log-concave measure under a polynomial of degree $d$ possesses a density from the Nikol'skii--Besov class of fractional order $1/d$. This result is used to prove an estimate of…
Upper bounds on the Kolmogorov distance (and, equivalently in this case, on the total variation distance) between the Student distribution with p degrees of freedom (SD_p) and the standard normal distribution are obtained. These bounds are…
We establish general upper bounds on the Kolmogorov distance between two probability distributions in terms of the distance between these distributions as measured with respect to the Wasserstein or smooth Wasserstein metrics. These bounds…
Motivated by problems on random differences in Szemer\'{e}di's theorem and on large deviations for arithmetic progressions in random sets, we prove upper bounds on the Gaussian width of point sets that are formed by the image of the…
Under Poincar\'e-type conditions, upper bounds are explored for the Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law. Based on improved concentration inequalities on high-dimensional…
Given two high-dimensional Gaussians with the same mean, we prove a lower and an upper bound for their total variation distance, which are within a constant factor of one another.
Let $X_1,X_2,...,X_n$ be a sequence of independent or locally dependent random variables taking values in $\mathbb{Z}_+$. In this paper, we derive sharp bounds, via a new probabilistic method, for the total variation distance between the…
We deal with stochastic differential equations with jumps. In order to obtain an accurate approximation scheme, it is usual to replace the "small jumps" by a Brownian motion. In this paper, we prove that for every fixed time $t$, the…
Labelled Markov chains (LMCs) are widely used in probabilistic verification, speech recognition, computational biology, and many other fields. Checking two LMCs for equivalence is a classical problem subject to extensive studies, while the…
Let $X_1,X_2,\ldots$ be a sequence of i.i.d. random variables, with mean zero and variance one. Let $W_n=(X_1+\ldots+X_n)/\sqrt{n}$. An old and celebrated result of Prohorov asserts that $W_n$ converges in total variation to the standard…
In the first part we study deviation of a polynomial from its mathematical expectation. This deviation can be estimated from above by Carbery--Wright inequality, so we investigate estimates of the deviation from below. We obtain such…
The total variation distance is a core statistical distance between probability measures that satisfies the metric axioms, with value always falling in $[0,1]$. This distance plays a fundamental role in machine learning and signal…