Related papers: The Maximal Probability that k-wise Independent Bi…

Given a random binary sequence $X^{(n)}$ of random variables, $X_{t},$ $t=1,2,...,n$, for instance, one that is generated by a Markov source (teacher) of order $k^{*}$ (each state represented by $k^{*}$ bits). Assume that the probability of…

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…

Randomness expansion where one generates a longer sequence of random numbers from a short one is viable in quantum mechanics but not allowed classically. Device-independent quantum randomness expansion provides a randomness resource of the…

Problem 1.5.7 from Pitman's Saint-Flour lecture notes: Does there exist for each n a fragmentation process (\Pi_{n,k}, 1 \leq k \leq n) taking values in the space of partitions of {1,2,...,n} such that \Pi_{n,k} is distributed like the…

We consider the problem of distilling uniform random bits from an unknown source with a given $p$-entropy using linear hashing. As our main result, we estimate the expected $p$-divergence from the uniform distribution over the ensemble of…

A 1-independent bond percolation model on a graph $G$ is a probability distribution on the spanning subgraphs of $G$ in which, for all vertex-disjoint sets of edges $S_1$ and $S_2$, the states of the edges in $S_1$ are independent of the…

Let $k$ be any positive integer and $G$ a compact (Hausdorff) group. Let $\mf{np}_k(G)$ denote the probability that $k+1$ randomly chosen elements $x_1,\dots,x_{k+1}$ satisfy $[x_1,x_2,\dots,x_{k+1}]=1$. We study the following problem: If…

We give highly efficient algorithms, and almost matching lower bounds, for a range of basic statistical problems that involve testing and estimating the L_1 distance between two k-modal distributions $p$ and $q$ over the discrete domain…

Let $P = \{p(i)\}$ be a measure of strictly positive probabilities on the set of nonnegative integers. Although the countable number of inputs prevents usage of the Huffman algorithm, there are nontrivial $P$ for which known methods find a…

A partial $(k-1)$-spread in $\operatorname{PG}(n-1,q)$ is a collection of $(k-1)$-dimensional subspaces with trivial intersection, i.e., each point is covered at most once. So far the maximum size of a partial $(k-1)$-spread in…

We investigate the problem of testing the equivalence between two discrete histograms. A {\em $k$-histogram} over $[n]$ is a probability distribution that is piecewise constant over some set of $k$ intervals over $[n]$. Histograms have been…

This note considers a variation of the full-information secretary problem where the random variables to be observed are independent and identically distributed. Consider $X_1,\dots,X_n$ to be an independent sequence of random variables, let…

The property of perfectness plays an important role in the theory of Bayesian networks. First, the existence of perfect distributions for arbitrary sets of variables and directed acyclic graphs implies that various methods for reading…

We consider the problem of estimating the mean of a symmetric log-concave distribution under the constraint that only a single bit per sample from this distribution is available to the estimator. We study the mean squared error as a…

The maximum entropy principle is a powerful tool for solving underdetermined inverse problems. This paper considers the problem of discretizing a continuous distribution, which arises in various applied fields. We obtain the approximating…

The computation of two Bayesian predictive distributions which are discrete mixtures of incomplete beta functions is considered. The number of iterations can easily become large for these distributions and thus, the accuracy of the result…

The following anticoncentration property is proved. The probability that the $k$-order statistic of an arbitrarily correlated jointly Gaussian random vector $X$ with unit variance components lies within an interval of length $\varepsilon$…

Suppose a customer is faced with a sequence of fluctuating prices, such as for airfare or a product sold by a large online retailer. Given distributional information about what price they might face each day, how should they choose when to…

We study the problem of computing the tightest upper and lower bounds on the probability that the sum of $n$ dependent Bernoulli random variables exceeds an integer $k$. Under knowledge of all pairs of bivariate distributions denoted by a…

In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems having optimal solutions. The bound is polynomial of the number of constraints,…