Exact expressions for the maximal probability that all $k$-wise independent bits are 1

TL;DR

This paper derives exact formulas for the maximum probability that all bits are 1 in a k-wise independent Bernoulli distribution, addressing a long-standing open problem in probability theory.

Contribution

It provides closed-form expressions for M(n,k,p) when p is near 0 or 1, advancing the understanding of k-wise independence probabilities.

Findings

Exact formulas for M(n,k,p) near p=0 and p=1

Addresses a long-standing open problem

Enhances understanding of k-wise independent distributions

Abstract

Let $M(n, k, p)$ denote the maximum probability of the event $X_1 = X_2 = \cdots = X_n=1$ under a $k$-wise independent distribution whose marginals are Bernoulli random variables with mean $p$. A long-standing question is to calculate $M(n, k, p)$ for all values of $n,k,p$. This question has been partially addressed by several authors, primarily with the goal of answering asymptotic questions. The present paper focuses on obtaining exact expressions for this probability. To this end, we provide closed-form formulas of $M(n,k,p)$ for $p$ near 0 as well as $p$ near 1.