On Sharpest Tail Bounds for Functions of Tail Bounded Random Variables

Consider n real/complex, independent/dependent random variables with respective tail bounds and g a measurable function of the r.v.’s. Consider f the “sharpest” tail bound of g (sharpest in the sense, if f were any less, then for some X1, ..., Xn satisfying the conditions, g(X1, ..., Xn) would not satisfy the tail f). Significant research has been done to approximate f often with high accuracy. These results are often of the form, for g in this family, and tail bounds of Xk in this family, f is bounded by some f′ with high accuracy. However, the question “what would it take to find f exactly?” has received little attention, apparently even for simple cases. This is the question we try to answer. For X1, ..., Xn required to be mutually ind., first Xk are simplified to be monotone on (0, 1) WLOG. This strengthens convergence in distribution to convergence a.e., and allows defining shift operators, which help reduce the space of r.v.’s one searches to find f and/or the maximum measure of a subset. We do find f in some special cases, however f rarely has a closed form. For X1, ..., Xn dependent/not necessarily independent, another reduction in the space of r.v.’s one searches to find f is done.