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

Fix any n≥1. Let X~1,…,X~n be independent random variables. For each 1≤j≤n, X~j is transformed in a canonical manner into a random variable Xj. The Xj inherit independence from the X~j. Let sy and s∗y denote the upper 1y th −−− quantile of Sn=∑nj=1Xj and S∗n=sup1≤k≤nSk, respectively. We construct a computable quantity Q−−y based on the marginal distributions of X1,…,Xn to produce upper and lower bounds for sy and s∗y. We prove that for y≥8
6−1γ3y/16Q−−3y/16≤s∗y≤Q−−y
where
γy=12wy+1
and wy is the unique solution of
(wyeln(yy−2))wy=2y−4
for wy>ln(yy−2), and for y≥37
19γu(y)Q−−u(y)<sy≤Q−−y
where
u(y)=3y32(1+1−643y−−−−−−√).
The distribution of Sn is approximately centered around zero in that P(Sn≥0)≥118 and P(Sn≤0)≥165. The results extend to n=∞ if and only if for some (hence all) a>0
∑j=1∞E{(X~j−mj)2∧a2}<∞.