The average position of the first maximum in a sample of geometric random variables

Margaret Archibald,Arnold Knopfmacher

doi:10.46298/dmtcs.3523

Abstract

We consider samples of n geometric random variables $(Γ _1, Γ _2, \dots Γ _n)$ where $\mathbb{P}\{Γ _j=i\}=pq^{i-1}$, for $1≤j ≤n$, with $p+q=1$. The parameter we study is the position of the first occurrence of the maximum value in a such a sample. We derive a probability generating function for this position with which we compute the first two (factorial) moments. The asymptotic technique known as Rice's method then yields the main terms as well as the Fourier expansions of the fluctuating functions arising in the expected value and the variance.

Highlights

We consider samples of n geometric random variables (Γ1, Γ2, . . . Γn) where P{Γj = i} = pqi−1, for 1 ≤ j ≤ n, with p + q = 1. Such samples have been studied in their own right, as well as with reference to skiplists and probabilistic counting algorithms
Questions relating to the maximum value of such a sample have attracted quite a lot of attention
We look at the alternating sum labelled (b) which has a double pole at z = 1 and a simple pole at z = 2

Summary

Introduction

In this paper we study the mean and variance of the position at which the maximum value first occurs. Previous statistics studied for maxima of geometric samples, the mean and variance do not converge as n → ∞ but instead exhibit small fluctuations. (To count the first maximum as well in the position, just add 1 to the mean value below; the variance remains unchanged.). Theorem 1 The average position En of the first (left-most) occurrence of the maximum in a sample of geometric random variables is given by. There are negligibly small contributions from the fluctuating terms These are calculated only for the dominant term of the variance, and are given by n2([δE2 1]0 + δv(logQ n)). Note that [δE2 1]0 is an extremely small quantity, typically of size 10−12

Method

The expected value

The variance