 
Summary: Scaling of the Minimum of iid Random Variables
T. Y. AlNaffouri
Electrical Engineering Department, King Fahd University of Petroleum &
Minerals, Dhahran 31261, Saudi Arabia
Abstract
Several areas in signal processing and communications rely on various tools in order
statistics. Studying the scaling of the extreme values of iid random variables is of particular
interest as it is sometimes only possible to make meaningful statements in the large number
of variables case. This paper develops a new approach to finding the scaling of the minimum
of iid variables by studying the behavior of the CDF and its derivatives at one point, or
equivalently by studying the behavior of the characteristic function. The theory developed is
used to study the scaling of several types of random variables and is confirmed by simulations.
Keywords: scaling of random variables  extreme values  order statistics characteristic
function  initial value theorem
1 Introduction
Extreme value theory (EVT) is an important tool in statistics, signal processing, and commu
nications. EVT is concerned with the behavior of the maximum/minimum of a sequence of n
iid random variables when n becomes large (which is known as scaling analysis). This tool has
for example been used in abnormality detection in biomedical signal processing [1], for level
detection in hidden Markov models [2], and in characterization of sonar reverberations [3]. ETV
