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Egor Kosov

Publications and source records attributed to Egor Kosov.

14 recordsLinked to original sources
arXiv2026

Absolute continuity of two-dimensional polynomial random vectors

Let $X=\{X_j\}_{j=1}^\infty$ be a sequence of independent random variables whose densities and moments of order $2d$ are uniformly bounded. For a random vector $f(X)=(f_1(X),f_2(X))$ whose components are polynomial functionals of degree at most $d$, we prove that \[ [[f]]_{\mu,\infty}^{\frac1{2d-1}}\mu(f\in A) \le C\bigl(\lambda_2(A)\bigr)^{\frac1{2d-1}} \] for every Borel set $A\subset\mathbb R^2$, where $C$ depends only on $d$ and the uniform density and moment bounds, and $\lambda_2$ denotes the Lebesgue measure on $\mathbb R^2$. Here $[[f]]_{\mu,\infty}$ measures the failure of proportionality of the highest-order orthogonal-chaos components of $f_1$ and $f_2$ with respect to the law $\mu$ of $X$. Consequently, whenever these components are not proportional, the law of $f$ admits a density in the weak Lorentz space $L^{\frac{2d-1}{2d-2},\infty}(\mathbb R^2)$. This recovers the dichotomy established by Nualart and Tudor for two-dimensional Wiener chaos vectors and extends it beyond the Gaussian setting. We also obtain the lower bound \[ \int_{\mathbb R^\infty}\Delta_f\,d\mu \ge C[[f]]_{\mu,\infty}^2, \] where $\Delta_f$ is the determinant of the Gram matrix of $\nabla f_1$ and $\nabla f_2$. In the special case of Gaussian measures, this gives a relaxed version of the estimate conjectured by Nourdin, Nualart, and Poly.

math.PR↗
arXiv2026

Nondegeneracy and regularity of polynomial pushforwards

Let $\mu$ be a log-concave probability measure on $\mathbb R^n$ and let $f\colon\mathbb R^n\to\mathbb R^k$ be a polynomial mapping of degree at most $d$. We show that \[ \mu(f\in A) \le C\bigl(\lambda_k(A)\bigr)^{\frac{1}{k(d-1)+1}} \] for every Borel set $A\subset\mathbb R^k$ whenever the image measure $\mu\circ f^{-1}$ is absolutely continuous. The constant $C$ is independent of the dimension $n$, and the exponent $\frac{1}{k(d-1)+1}$ is sharp. This extends the scalar Carbery--Wright inequality and answers, in the log-concave setting, a question raised by Avni, Glazer, and Larsen. In addition, we show that the density of $\mu\circ f^{-1}$, whenever it exists, belongs to the Nikolskii--Besov space $B^{\frac{1}{k(d-1)+1}}_{1,\infty}(\mathbb R^k)$, with a dimension-free bound for the corresponding norm. A central difficulty in passing from scalar polynomials to vector-valued polynomial mappings is the lack of a suitable nondegeneracy parameter quantifying absolute continuity of $\mu\circ f^{-1}$, as the variance does in the scalar case. Natural candidates such as the covariance matrix or the Jacobian matrix either fail to characterize this property or do not lead to dimension-free estimates. We identify such a parameter and define it to be the covariance matrix of the vector formed by the monomials of degree up to $d^{k-1}$ in the normalized components of $f$. The dimension-free nature of our results allows us to extend Kusuoka's absolute continuity criterion for Gaussian polynomial random vectors to the log-concave setting. Moreover, in this setting, we obtain estimates relating convergence in distribution to convergence in total variation for polynomial random vectors.

math.PR↗
arXiv2026

Estimates of the total variation distance between laws of Sobolev mappings on Gaussian spaces

Under small-ball bounds for the Malliavin determinants of two $\mathbb R^k$-valued Sobolev mappings on a Gaussian space, we estimate the total variation distance between their laws both in terms of the Kantorovich--Rubinstein distance and in terms of the distance between the mappings in the corresponding Sobolev space. In particular, our results yield new total variation distance estimates for distributions of random vectors whose components belong to finite sums of Wiener chaoses, with exponents improved by an asymptotic factor of two. The proof is based on fractional regularity estimates for distributions of Sobolev mappings. Namely, we show that if an $\mathbb R^k$-valued mapping has components in $W^{2,p}(\gamma)$ and the determinant of the corresponding Malliavin matrix satisfies a small-ball bound of order $\varkappa\in(0,1]$, then the law of the mapping has fractional regularity of order \[ \frac{\varkappa}{1+(2k-1)\varkappa p^{-1}}. \] In particular, for large $p$, this gives regularity of order $\varkappa$ up to an $O(p^{-1})$ loss.

math.PR↗
arXiv2026

Dimension-free Markov--Bernstein inequalities for product measures

For even integer exponents $p$, we obtain dimension-free Markov--Bernstein inequalities for polynomials with respect to product probability measures $\mu=\mu_1\otimes\cdots\otimes\mu_n$. When the measures $\mu_j$ have unimodal densities, we prove \[ \|\nabla f\|_{L^p(\mu)} \le C(p) d^2 \|f\|_{L^p(\mu)} \] for polynomials $f$ of degree at most $d$. The dependence on $d$ is optimal already for the uniform distribution on the unit cube. For products of one-dimensional Freud measures with densities proportional to $e^{-|t|^{2m}}$, the factor $d^2$ can be replaced by $d^{1-\frac{1}{2m}}$. In the Gaussian case, for all $p\ge4$, we prove that \[ \|\nabla f\|_{L^p(\gamma^n)} \le C(p)d^{\frac12+\theta_p}\|f\|_{L^p(\gamma^n)} \] for every polynomial $f$ of degree at most $d$, where $\theta_p\le \frac{2}{3p}$ and $\theta_p=0$ whenever $p$ is an even integer. Thus, in the even-integer case, we establish the sharp dependence on the degree conjectured by Eskenazis--Ivanisvili. For general $p\ge4$, the estimate improves upon their dimension-free inequality.

math.CA↗
arXiv2026

Empirical Approximation of $L_p$ Norms

We study empirical $L_p$ moments of a random vector $\pmb\varphi$ based on its i.i.d.\ copies $\pmb\varphi^1,\ldots,\pmb\varphi^m$, that is, $\frac1m\sum_{j=1}^m |\langle \pmb\varphi^j,y\rangle|^p$. Our main result is a new estimate for the expected uniform deviation \[ \mathbb{E}\sup_{y\in D}\biggl| \frac1m\sum_{j=1}^m |\langle \pmb\varphi^j,y\rangle|^p -\mathbb{E}|\langle \pmb\varphi,y\rangle|^p \biggr| \] over an arbitrary index set $D$. The proof is based on a new bound for Talagrand's $\gamma$-functional, sharper than the standard Dudley-type entropy estimate. We then apply this estimate to the following two problems. First, for $p>2$, we study Marcinkiewicz-type discretization of $L_p$ norms on an $N$-dimensional subspace $X_N\subset B(\Omega)$ of bounded functions on a probability space $(\Omega,\mu)$. We obtain bounds in terms of the norm of the embedding $ (X_N,\|\cdot\|_{L_p(\mu)})\hookrightarrow B(\Omega). $ In particular, we prove that when this norm is of order $N^{1/p}$ and \[ m \ge C(p)\, N\log N\,(\log\log N)^{p-1}, \] then $m$ random samples suffice to approximate the $L_p(\mu)$ norm uniformly on $X_N$ by the sampled discrete $L_p$ norm. This substantially improves the previously known bound in this setting $ m \ge C(p)\, N(\log N)^{\min\{p,3\}}, $ and is optimal up to the factor $(\log\log N)^{p-1}$ in the random-sampling setting. Second, for $1\le p<2$, we obtain an $L_p$ analogue of the restricted isometry property via random sampling for bounded orthogonal systems and, more generally, for $N$-element systems $\mathcal D_N$ satisfying a Riesz-type condition. We prove that when \[ m \ge C(p)\, s\log N\,(\log s)^2\,\log\log s, \] then $m$ random samples suffice to guarantee an $L_p$ restricted isometry-type property uniformly over the class of all $s$-sparse functions generated by $\mathcal D_N$.

math.FA↗
arXiv2025

Oscillatory integrals with polynomial phase and regularity of distributions

We obtain dimension-free estimates for the modulus of continuity of densities of polynomial images of $s$-concave and product measures. As a consequence, we settle a conjecture of A. Carbery and J. Wright (2001) on sharp upper bounds for oscillatory integrals over convex sets with polynomial phase.

math.FA↗
arXiv2024

Sampling discretization in Orlicz spaces

We obtain new sampling discretization results in Orlicz norms on finite dimensional spaces. As applications, we study sampling recovery problems, where the error of the recovery process is calculated with respect to different Orlicz norms. In particular, we are interested in the recovery by linear methods in the norms close to $L^2$.

math.FA↗
arXiv2023

Improved bounds for the total variation distance between stochastic polynomials

The paper studies upper bounds for the total variation distance between two polynomials of a special form in random vectors satisfying the Doeblin-type condition. Our approach is based on the recent results concerning Nikolskii--Besov-type smoothness of distribution densities of polynomials in logarithmically concave random vectors. The main results of the paper improve previously obtained estimates of Nourdin--Poly and Bally--Caramellino.

math.PR↗
arXiv2021

Marcinkiewicz-type discretization of $L^p$-norms under the Nikolskii-type inequality assumption

The paper studies the sampling discretization problem for integral norms on subspaces of $L^p(μ)$. Several close to optimal results are obtained on subspaces for which certain Nikolskii-type inequality is valid. The problem of norms discretization is connected with the probabilistic question about the approximation with high probability of marginals of a high dimensional random vector by sampling. As a byproduct of our approach we refine the result of O. Gu$\acute{e}$don and M. Rudelson concerning the approximation of marginals. In particular, the obtained improvement recovers a theorem of J. Bourgain, J. Lindenstrauss, and V. Milman concerning embeddings of finite dimensional subspaces of $L^p[0, 1]$ into $\ell_p^m$. The proofs in the paper use the recent developments of the chaining technique by R. van Handel.

math.FA↗