First-Passage Times for Random Partial Sums: Yadrenko’s Model for e and Beyond

Abstract

M. I. Yadrenko discovered that the expectation of the minimum number N₁ of independent and identically distributed uniform random variables on (0, 1) that have to be added to exceed 1 is e. For any threshold a > 0, K. G. Russell found the distribution, mean, and variance of the minimum number N_a of independent and identically distributed uniform random summands required to exceed a. Here we calculate the distribution and moments of N_a when the summands obey the negative exponential and Lévy distributions. The Lévy distribution has infinite mean. We compare these results with the results of Yadrenko and Russell for uniform random summands to see how the expected first-passage time $E(N_a),a>0$, and other moments of N_a depend on the distribution of the summand.

Keywords:

• Exponential distribution
• Lévy distribution
• Life length
• M. I. Yadrenko
• Napier’s number e
• Negative exponential distribution
• Reliability theory
• Uniform distribution

Acknowledgments

The author thanks Roseanne Benjamin for help during this work; and a referee, the Associate Editor, and the Editor, for generous comments and constructive suggestions.

Disclosure Statement

The author reports there are no competing interests to declare.