On some conjectures of Samuels and Feige

TL;DR

This paper investigates conjectures by Samuels and Feige regarding bounds on the probability that a weighted sum of independent non-negative random variables falls below a threshold, establishing that Samuels' conjecture implies Feige's.

Contribution

It demonstrates that the conjecture of Samuels implies the conjecture of Feige, linking two important probabilistic bounds.

Findings

Samuels' conjecture implies Feige's conjecture.

Provides a theoretical connection between two probabilistic bounds.

Advances understanding of bounds on sums of independent random variables.

Abstract

Let $\mu_1 \ge \dotsc \ge \mu_n > 0$ and $\mu_1 + \dotsm + \mu_n = 1$. Let $X_1, \dotsc, X_n$ be independent non-negative random variables with $EX_1 = \dotsc = EX_n = 1$, and let $Z = \sum_{i=1}^n \mu_i X_i$. Let $M = \max_{1 \le i \le n} \mu_i = \mu_1$, and let $\delta > 0$ and $T = 1 + \delta$. Both Samuels and Feige formulated conjectures bounding the probability $P(Z < T)$ from above. We prove that Samuels' conjecture implies a conjecture of Feige.