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We consider the limiting distribution of the quantity $X^s/(X+Y)^r$, where $X$ and $Y$ are two independent Binomial random variables with a common success probability and a number of trials $n$ and $m$, respectively, and $r,s$ are positive…

The Adaptive Multilevel Splitting algorithm is a very powerful and versatile iterative method to estimate the probability of rare events, based on an interacting particle systems. In an other article, in a so-called idealized setting, the…

We show that for every positive p, the L_p-norm of linear combinations (with scalar or vector coefficients) of products of i.i.d. random variables, whose moduli have a nondegenerate distribution with the p-norm one, is comparable to the…

Reducing the conditions under which a given set satisfies the stipulations of the subset sum proposition to a set of linear relationships, the question of whether a set satisfies subset sum may be answered in a polynomial number of steps by…

We prove limit theorems for the number of fixed points occurring in a random pattern-avoiding permutation distributed according to a one-parameter family of biased distributions. The bias parameter exponentially tilts the distribution…

This paper proposes a linear categorical random coefficient model, in which the random coefficients follow parametric categorical distributions. The distributional parameters are identified based on a linear recurrence structure of moments…

Symmetries occur naturally in CSP or SAT problems and are not very difficult to discover, but using them to prune the search space tends to be very challenging. Indeed, this usually requires finding specific elements in a group of…

A typical computational geometry problem begins: Consider a set P of n points in R^d. However, many applications today work with input that is not precisely known, for example when the data is sensed and has some known error model. What if…

Let $N$ be a finite set, let $p \in (0,1)$, and let $N_p$ denote a random binomial subset of $N$ where every element of $N$ is taken to belong to the subset independently with probability $p$ . This defines a product measure $\mu_p$ on the…

Linear systems are the bedrock of virtually all numerical computation. Machine learning poses specific challenges for the solution of such systems due to their scale, characteristic structure, stochasticity and the central role of…

Suppose $B_i:= B(p,r_i)$ are nested balls of radius $r_i$ about a point $p$ in a dynamical system $(T,X,\mu)$. The question of whether $T^i x\in B_i$ infinitely often (i. o.) for $\mu$ a.e.\ $x$ is often called the shrinking target problem.…

Let $\mathbb{F}_q$ be the finite field of order $q$, and $\mathcal{A}$ a non-empty proper subset of $\mathbb{F}_q$. Let $\mathbf{M}$ be a random $m \times n$ matrix of rank $r$ over $\mathbb{F}_q$ taken with uniform distribution. It was…

The aim of this paper is a quantitative analysis of the solution set of a system of polynomial nonlinear differential equations, both in the ordinary and partial case. Therefore, we introduce the differential counting polynomial, a common…

We want to select the best systems out of a given set of systems (or rank them) with respect to their expected performance. The systems allow random observations only and we assume that the joint observation of the systems has a…

Consider two problems about an unknown probability distribution $p$: 1. How many samples from $p$ are required to test if $p$ is supported on $n$ elements or not? Specifically, given samples from $p$, determine whether it is supported on at…

When a matrix A with n columns is known to be well approximated by a linear combination of basis matrices B_1,..., B_p, we can apply A to a random vector and solve a linear system to recover this linear combination. The same technique can…

In this article, we show that a linear combination $X$ of $n$ independent, unbiased Bernoulli random variables $\{X_k\}$ can match the first $2n$ moments of a random variable $Y$ which is uniform on an interval. More generally, for each $p…

We give a method for taking microscopic limits of normal matrix ensembles. We apply this method to study the behaviour near certain types of singular points on the boundary of the droplet. Our investigation includes ensembles without…

A non-classical formulation of the central limit theorem is given for sequences of independent random variables with finite second moments. Singular sequences whose members all have a degenerate or normal distribution are excluded from…

This paper considers the problem of testing whether there exists a non-negative solution to a possibly under-determined system of linear equations with known coefficients. This hypothesis testing problem arises naturally in a number of…