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 that if $f$ were any less, then for some $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n$ satisfying the conditions, $g(X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n)$ would not satisfy $f$). Significant research has been done to approximate $f$ often with high accuracy. These results are often of the form that for $g$ in this family and tail bounds of $X_k$ in this family, $f$ is bounded by some $f^{\mathrm{\prime }}$ 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 $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n$ required to be mutually independent, first the $X_k$ are simplified to be monotone on $(0,1)$ WLOG. This strengthens convergence in distribution to convergence a.e. (Skorokhod’s representation theorem) 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 $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n$ dependent/not necessarily independent, another reduction in the space of r.v.'s one searches to find $f$ is done.