ABSTRACT
Approximations are often based on statistical estimation, and our reliance on the supremum norm as the measure of approximation will usually require the characterization of the maximum value of many random quantities. Suppose we are given a sequence of random variables X1,X2, . . . ,Xm, and assume that at least the means and variances E[Xi]=µi, var[Xi]= σ2i are finite. We will use the following notation:
M = max 1≤i≤m
Xi,
M¯ = E[M] , µmax = max
1≤i≤m µi,
σ2max = max1≤i≤m σ 2 i ,
µ¯ = m−1 m∑ i=1
µi,
µ¯2 = m−1 m∑ i=1
ν¯ = µ¯2 − µ¯2
σ¯2 = m−1 m∑ i=1
var[Xi −Xj], j=1, . . .m.
