Related papers: Random numbers from the tails of probability distr…

In this paper non-asymptotic exponential and moment estimates are derived for tail of distribution for discrete time martingale under norming sequence 1/n, as in the classical Law of Large Numbers (LLN), by means of martingale differences…

We prove simple general formulas for expectations of functions of a random walk and its running extremum. Under additional conditions, we derive analytical formulas using the inverse $Z$-transform, the Fourier/Laplace inversion and…

Let $X$ be the number of $k$-term arithmetic progressions contained in the $p$-biased random subset of the first $N$ positive integers. We give asymptotically sharp estimates on the logarithmic upper-tail probability $\log \Pr(X \ge E[X] +…

Deep generative models are routinely used in generating samples from complex, high-dimensional distributions. Despite their apparent successes, their statistical properties are not well understood. A common assumption is that with enough…

Recent works have shown that the family of probability distributions with moments given by the Fuss-Catalan numbers permit a simple parameterized form for their density. We extend this result to the Raney distribution which by definition…

Stirling numbers of the first kind are common in number theory and combinatorics; through Ewen's sampling formula, these numbers enter into the calculation of several population genetics statistics, such as Fu's Fs. In previous papers we…

We demonstrate extraction of randomness from spontaneous-emission events less than 36 ns in the past, giving output bits with excess predictability below $10^{-5}$ and strong metrological randomness assurances. This randomness generation…

Multivariate rapid variation describes decay rates of joint light tails of a multivariate distribution. We impose a local uniformity condition to control decay variation of distribution tails along different directions, and using…

We discuss non-Gaussian random matrices whose elements are random variables with heavy-tailed probability distributions. In probability theory heavy tails of the distributions describe rare but violent events which usually have dominant…

One method to generate random permutations involves using Gaussian elimination with partial pivoting (GEPP) on a random matrix $A$ and storing the permutation matrix factor $P$ from the resulting GEPP factorization $PA=LU$. We are…

To consider a high-dimensional random process, we propose a notion about stochastic tensor-valued random process (TRP). In this work, we first attempt to apply a generic chaining method to derive tail bounds for all p-th moments of the…

We consider continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates. We give explicit formulas for probability generating functions, and also for means, variances and state…

We establish some asymptotic expansions for infinite weighted convolution of distributions having regular varying tails. Various applications to statistics and probability are developed.

We derive in this preprint the moment and exponential tail estimates, sufficient conditions for the Non-Central Limit Theorem (NCLT) in the ordinary one-dimensional space as well as in the space of continuous functions for the properly…

In this paper, we consider a classic problem concerning the high excursion probabilities of a Gaussian random field $f$ living on a compact set $T$. We develop efficient computational methods for the tail probabilities $P(\sup_T f(t) > b)$…

We obtain decay rates of probabilities of tails of polynomials in several independent random variables with heavy tails and derive stable limit theorems for nonconventional sums of such polynomials

The authors present evidence for universality in numerical computations with random data. Given a (possibly stochastic) numerical algorithm with random input data, the time (or number of iterations) to convergence (within a given tolerance)…

This paper demonstrates the robustness of Lipschitz-regularized $\alpha$-divergences as objective functionals in generative modeling, showing they enable stable learning across a wide range of target distributions with minimal assumptions.…

We propose an analytical approach to the computation of tail probabilities of compound distributions whose individual components have heavy tails. Our approach is based on the contour integration method, and gives rise to a representation…

Random numbers are indispensable for a variety of applications ranging from testing physics foundation to information encryption. In particular, nonlocality tests provide a strong evidence to our current understanding of nature -- quantum…