An Optimal Upper Bound on the Tail Probability for Sums of Random Variables

I. Pinelis
2019 Theory of Probability and its Applications  
Let s be any given real number. An explicit construction is provided of random variables (r.v.'s) X and Y for which sup P(X + Y s) is attained, where the sup is taken over all r.v.'s X and Y with given distributions. Let X and Y be random variables (r.v.'s) with given distributions. Let s be a real number. Then the tail probability P(X + Y s) for the sum X + Y of r.v.'s X and Y can be obviously bounded from above by the sum P(X x) + P(Y > s − x) of the "marginal" tail probabilities for X and Y
more » ... where x is any real number; the bound P(X x) + P(Y > s − x) can replaced here by P(X > x) + P(Y s − x). It seems plausible that one cannot get a better upper bound on P(X +Y s) without additional information on the joint distribution of X and Y . Indeed, using duality arguments -see e.g. [2, 1], it is not hard to show the following. 2010 Mathematics Subject Classification. 60E15.
doi:10.1137/s0040585x97t989635 fatcat:chnhejb2fbasxkrp6glcac7mnm