Uniform Bounds on the Relative Error in the Approximation of Upper Quantiles for Sums of Arbitrary Independent Random Variables

摘要

Fix any n >= 1. Let (X) over tilde (1), ..., (X) over tilde (n) be independent random variables. For each 1 <= j <= n, (X) over tilde (j) is transformed in a canonical manner into a random variable X-j. The X j inherit independence from the (X) over tilde (j). Let s(y) and s(y)* denote the upper 1y (th) under bar quantile of S-n = Sigma(n)(j=1) X-j and S-n* = sup(1 <= k <= n) S-k, respectively. We construct a computable quantity Q(y) based on the marginal distributions of X-1, ..., X-n to produce upper and lower bounds for s(y) and s(y)*. We prove that for y >= 8