Related papers: Lowness for Integer-Valued Randomness

We prove that a real number a greater than or equal to 2 is the irrationality exponent of some computable real number if and only if a is the upper limit of a computable sequence of rational numbers. Thus, there are computable real numbers…

We are concerned with testing replicability hypotheses for many endpoints simultaneously. This constitutes a multiple test problem with composite null hypotheses. Traditional $p$-values, which are computed under least favourable parameter…

The notions of almost everywhere (a.e.) domination and its uniform version were introduced and studied in reverse mathematics. This paper studies these notions from a recursion-theoretic point of view and explore their connections to…

We consider the problem of exact low-rank matrix completion from a geometric viewpoint: given a partially filled matrix M, we keep the positions of specified and unspecified entries fixed, and study how the minimal completion rank depends…

Since the work of Ferrenberg et al.[PRL 69, (1992)] some pseudo random number generators are known to yield wrong results in cluster Monte Carlo simulations. In this contribution the fundamental mechanism behind this failure is discussed.…

We consider n by n real matrices whose entries are non-degenerate random variables that are independent but non necessarily identically distributed, and show that the probability that such a matrix is singular is O(1/sqrt{n}). The purpose…

In this paper we prove exponential inequalities (also called Bernstein's inequality) for fractional martingales. As an immediate corollary, we will discuss weak law of large numbers for fractional martingales under divergence assumption on…

A distinction is sometimes made between "statistical" and "subjective" probabilities. This is based on a distinction between "unique" events and "repeatable" events. We argue that this distinction is untenable, since all events are "unique"…

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…

Computing reachability probabilities is at the heart of probabilistic model checking. All model checkers compute these probabilities in an iterative fashion using value iteration. This technique approximates a fixed point from below by…

For a fixed $n\ge2$, consider an $n\times n$ matrix $M$ whose entries are random integers bounded by $k$ in absolute value. In this paper, we examine the probability that $M$ is singular (hence has eigenvalue 0), and the probability that…

The theory of random matrices contains many central limit theorems. We have central limit theorems for eigenvalues statistics, for the log-determinant and log-permanent, for limiting distribution of individual eigenvalues in the bulk, and…

We prove an elementary yet useful inequality bounding the maximal value of certain linear programs. This leads directly to a bound on the martingale difference for arbitrarily dependent random variables, providing a generalization of some…

We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In…

Turing's famous 'machine' framework provides an intuitively clear conception of 'computing with real numbers'. A recursive counterexample to a theorem shows that the theorem does not hold when restricted to computable objects. These…

We give the proof of a tight lower bound on the probability that a binomial random variable exceeds its expected value. The inequality plays an important role in a variety of contexts, including the analysis of relative deviation bounds in…

An index $e$ in a numbering of partial-recursive functions is called minimal if every lesser index computes a different function from $e$. Since the 1960's it has been known that, in any reasonable programming language, no effective…

We study minimum integer representations of weighted games, i.e., representations where the weights are integers and every other integer representation is at least as large in each component. Those minimum integer representations, if the…

A new class of rings, the class of left localizable rings, is introduced. A ring $R$ is left localizable if each nonzero element of $R$ is invertible in some left localization $S^{-1}R$ of the ring $R$. Explicit criteria are given for a…

In this brief paper the probability density of a random real, complex and quaternion determinant is rederived using singular values. The behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the…