Overshoot in the Case of Normal Variables: Chernoff's Integral, Latta's Observation, and Wijsman's Sum

Abstract

In a recent article, Robert A. Wijsman (2004) derived a new formula for the limiting expected excess over a boundary for the sum of independent and identically distributed normal variables with unit variance, as their positive mean approaches zero. This formula was expressed in the form of an infinite series, and evaluated numerically to an accuracy of six decimal places. This refines a computation by Siegmund (1985) based upon an integral representation originating from Chernoff (1965). We show that both representations evaluate to , where ζ(s) is the Riemann zeta-function, in agreement with an observation by Gordon Latta.

Keywords:

• Excess over boundary
• Expected overshoot
• Riemann zeta-function

Subject Classifications:

• 60G50
• 11M06
• 11Y60

Notes

Recommended by N. Mukhopadhyay