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 N 1 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 Na of independent and identically distributed uniform random summands required to exceed a. Here we calculate the distribution and moments of Na 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 Na depend on the distribution of the summand.