An equivalent conjecture to Feige's Conjecture

TL;DR

This paper proposes an equivalent form of Feige's Conjecture involving bounds on the probability that the sum of non-negative independent variables is below a certain threshold, with applications in finance.

Contribution

It introduces a new conjecture equivalent to Feige's, providing a different perspective on probability bounds for sums of independent variables.

Findings

Sketch of the equivalent conjecture to Feige's Conjecture

Illustration of the conjecture's application in mathematical finance

Discussion of the conjecture's potential implications

Abstract

Let X1, ..., Xn be arbitrary non-negative independent random variables with respective expected values $\mu_{i}$ at most one. We sketch but do not prove an equivalent conjecture to Feige's Conjecture $\mathbb{P} \left( \sum_{i=1}^{n} X_{i} < \mu + 1 \right) \geq \exp \left(-1 \right)$, where $\mu$ is the expected value of the sum of the random variables. We show by a simple example how this inequality finds use in mathematical finance.