Related papers: Uniform tail approximation of homogenous functiona…

In a variety of problems in pure and applied probability, it is of relevant to study the large exceedance probabilities of the perpetuity sequence $Y_n := B_1 + A_1 B_2 + \cdots + (A_1 \cdots A_{n-1}) B_n$, where $(A_i,B_i) \subset…

For a centered $d$-dimensional Gaussian random vector $\xi =(\xi_1,\ldots,\xi_d)$ and a homogeneous function $h:R^d\to R$ we derive asymptotic expansions for the tail of the Gaussian chaos $h(\xi)$ given the function $h$ is sufficiently…

Let $X(t),t\in \mathbb{R}$ be a stochastically continuous stationary max-stable process with Fr\'{e}chet marginals $\Phi_\alpha, \alpha>0$ and set $M_X(T)=\sup_{t \in [0,T]} X(t),T>0$. In the light of the seminal articles [1,2], it follows…

Under K.-T. Sturm's formulation, we obtain a Gaussian upper bound for tail probability of mean value of independent, identically distributed random variables with values in $\mathbb{R}$-trees and Hadamard manifolds.

We study probability inequalities leading to tail estimates in a general semigroup $\mathscr{G}$ with a translation-invariant metric $d_{\mathscr{G}}$. (An important and central example of this in the functional analysis literature is that…

This paper presents precise large deviation estimates for solutions to stochastic fixed point equations of the type V =_d f(V), where f(v) = Av + g(v) for a random function g(v) = o(v) a.s. as v tends to infinity. Specifically, we provide…

We generalize the proof of Karamata's Theorem by the method of approximation by polynomials to the operator case. As a consequence, we offer a simple proof of \emph{uniform dual ergodicity} for a very large class of dynamical systems with…

This note presents a proof that the non-tangential maximal function of the Ornstein-Uhlenbeck semigroup is bounded almost surely by the Gaussian Hardy-Littlewood maximal function. In particular this entails improvement on a result by Pineda…

Let $\{\xi_1,\xi_2,\ldots\}$ be a sequence of independent but not necessarily identically distributed random variables. In this paper, the sufficient conditions are found under which the tail probability…

We consider the Gumbel or extreme value statistics describing the distribution function p_G(x_max) of the maximum values of a random field x within patches of fixed size. We present, for smooth Gaussian random fields in two and three…

Let $X$ be a standard Gaussian random variable. For any $p \in (1, \infty)$, we prove the existence of a universal constant $C_{p}>0$ such that the inequality $$(\mathbb{E} |h'(X)|^{p})^{1/p} \geq C_{p} \sqrt{d} (\mathbb{E}…

We study the tail of $p(U)$, the probability distribution of $U=\vert\psi(0,L)\vert^2$, for $\ln U\gg 1$, $\psi(x,z)$ being the solution to $\partial_z\psi -\frac{i}{2m}\nabla_{\perp}^2 \psi =g\vert S\vert^2\, \psi$, where $S(x,z)$ is a…

We derive upper bounds for probabilities of the form $P(g(\mathbf{X})\geq t)$ using the southwest boundary (recently introduced in our previous work) $\partial_{\mathrm{SW}} Q(g^{-1}[t,\infty))$, where $Q$ is a reflection to the first…

Random matrices acting on structured sets play a fundamental role in high-dimensional geometry, compressed sensing, and randomized algorithms. Existing results primarily focus on subgaussian models, when random matrices act as…

An infinite convergent sum of independent and identically distributed random variables discounted by a multiplicative random walk is called perpetuity, because of a possible actuarial application. We give three disjoint groups of sufficient…

Exponential tail bounds for sums play an important role in statistics, but the example of the $t$-statistic shows that the exponential tail decay may be lost when population parameters need to be estimated from the data. However, it turns…

For an $N \times T$ random matrix $X(\beta)$ with weakly dependent uniformly sub-Gaussian entries $x_{it}(\beta)$ that may depend on a possibly infinite-dimensional parameter $\beta\in \mathbf{B}$, we obtain a uniform bound on its operator…

In this paper, we study estimates on tail probabilities $\mathbb{P}(S_r \ge t)$ of several classes of subordinators under mild assumptions on the tail of its L\'evy measure. As an application of that result, we obtain two-sided estimates…

We prove that for Gaussian random normal matrices the correlation function has universal behavior. Using the technique of orthogonal polynomials and identities similar to the Christoffel-Darboux formula, we find that in the limit, as the…

Under the Ornstein-Uhlenbeck semigroup $\{U_t\}$, any non-negative measurable $f : \mathbb R^n \to \mathbb R_+$ exhibits a uniform tail bound better than that implied by Markov's inequality and conservation of mass: For every $\alpha \geq…