Approximate evaluation of sums of a large number of terms
We often need to evaluate approximately expressions involving sums of infinitely many terms. Such sums may arise for example in evaluating partition functions, or as total probabilities of certain events.
We discuss approximate evaluation procedures for deterministic sums and for stochastic sums. Some of the methods come under the heading of the method of Laplace. These are discussed separately. Here, we present another approximate method for the deterministic sums in the next section. Some known methods for sums composed of random terms are also collected, to justify a procedure for approximating sums by the maximum terms in the sums.
Darling then shows that the expression in the square bracket goes to zero as β goes to infinity if the tail of the distribution is slowly varying. This is more or less the definition of the notion of slow variation. See Bingham et al. (1987) on slowly varying functions.
6.7 Approximations of error functions
6.7.2 Example
McFadden calls these generalized extreme value (GEV) functions.