Related papers: Supremum of Random Dirichlet Polynomials with Sub-…

We show that for a random polynomial \[ F(X) = \sum_{n=1}^{N} f(n) X^{n-1}, \] where $f(n)$ is a random completely multiplicative function taking values in $\{\pm 1\}$, one has \[ \limsup_{N \to \infty} \mathbb{P}\big[F(X) \text{ is…

Maximal inequalities refer to bounds on expected values of the supremum of averages of random variables over a collection. They play a crucial role in the study of non-parametric and high-dimensional estimators, and especially in the study…

In this work, we derive sharp non-asymptotic deviation bounds for weighted sums of Dirichlet random variables. These bounds are based on a novel integral representation of the density of a weighted Dirichlet sum. This representation allows…

We take an $L_1$-dense class of functions $\Cal F$ on a measurable space $(X,\Cal X)$ together with a sequence of independent, identically distributed $X$-space valued random variables $\xi_1,\dots,\xi_n$ and give a good estimate on the…

The Dirichlet process mixture (DPM) is a ubiquitous, flexible Bayesian nonparametric statistical model. However, full probabilistic inference in this model is analytically intractable, so that computationally intensive techniques such as…

We study moderate deviations of suprema of parametrized sequences of sample bounded Gaussian processes $\{X _x(t), t\in T _x\}$, and first present recent sharp bounds in simple cases. In the almost periodic case, we prove an approximation…

A great challenge in the analysis of the discrepancy function D_N is to obtain universal lower bounds on the L-infty norm of D_N in dimensions d \geq 3. It follows from the average case bound of Klaus Roth that the L-infty norm of D_N is at…

In the paper I study properties of random polynomials with respect to a general system of functions. Some lower bounds for the mathematical expectation of the uniform and recently introduced integral-uniform norms of random polynomials are…

For a fixed unit vector a=(a_1,a_2,...,a_n) in S^{n-1}, i.e. sum_{i=1}^n a_i^2=1, we consider the 2^n sign vectors epsilon=(epsilon_1,epsilon_2,...,epsilon_n) in {-1,1}^n and the corresponding scalar products a.epsilon=sum_{i=1}^n a_i…

As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a {\em stochastic maximal inequality} derived by using the formula for…

Motivated by a question of V. Bergelson and F. K. Richter (2017), we obtain asymptotic formulas for the number of relatively prime tuples composed of positive integers $n\le N$ and integer parts of polynomials evaluated at $n$. The error…

For each $\alpha \in (0, 1)$, we construct a bounded monotone deterministic sequence $(c_k)_{k \geq 0}$ of real numbers so that the number of real roots of the random polynomial $f_n(z) = \sum_{k=0}^n c_k \varepsilon_k z^k$ is $n^{\alpha +…

There is a natural pluripotential-theoretic extremal function V_{K,Q} associated to a closed subset K of C^m and a real-valued, continuous function Q on K. We define random polynomials H_n whose coefficients with respect to a related…

We consider the problem of generating n-dimensional vectors with a fixed sum, with the goal of generating a uniform distribution of vectors over a valid region. This means that each possible vector has an equal probability of being…

Let $x \in \mathbb{R}$ be arbitrary and consider the `greedy' approximation of $x$ by signed harmonic sums: given $a_n = \sum_{k \leq n} \varepsilon_k/k$ with $\varepsilon_k \in \left\{-1,1\right\}$, we set $\varepsilon_{n+1} = 1$ if $a_n…

Relying on random matrix theory (RMT), this paper studies asymmetric order-$d$ spiked tensor models with Gaussian noise. Using the variational definition of the singular vectors and values of (Lim, 2005), we show that the analysis of the…

Let $G, G_1,\dots,G_N$ be independent copies of a standard gaussian random vector in $\mathbb{R}^d$ and denote by $\Gamma = \sum_{i=1}^N \langle G_i,\cdot\rangle e_i$ the standard gaussian ensemble. We show that, for any set $A\subset…

Subset sum is a very old and fundamental problem in theoretical computer science. In this problem, $n$ items with weights $w_1, w_2, w_3, \ldots, w_n$ are given as input and the goal is to find out if there is a subset of them whose weights…

For functions $f$ in Dirichlet-type spaces we study how to determine constructively optimal polynomials $p_n$ that minimize $\|p f-1\|_\alpha$ among all polynomials $p$ of degree at most $n$. Then we give upper and lower bounds for the rate…

We consider the problem of maximizing the sum of squares of the leading coefficients of polynomials $P_{i_1}(x),\ldots ,P_{i_m}(x)$ (where $P_j(x)$ is a polynomial of degree $j$) under the restriction that the sup-norm of $\sum_{j=1}^m…