Exact inequalities for sums of asymmetric random variables, with applications

Abstract

Let ${{\rm BS}_{1},\dots,{\rm BS}_{n}}$ be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter ${p \in (0, 1)}$ . Let ${m_*(p):=(1 + p + 2 p^{2})/(2\sqrt{p - p^{2}} + 4 p^{2})}$ if ${0 < p \le \frac12}$ and ${m_{*}(p) := 1 if \frac12 \le p < 1}$ . Let ${m \ge m_{*}(p)}$ . Let f be such a function that f and f′′ are nondecreasing and convex. Then it is proved that for all nonnegative numbers ${c_{1},\dots,c_{n}}$ one has the inequality ${{\mathsf{E}} f(c_1{\mathrm{B\!S}}_1+\dots+c_n{\mathrm{B\!S}}_n)\le{\mathsf{E}} f\big(s^{(m)}({\mathrm{B\!S}}_1+\dots+{\mathrm{BS}}_{n})\big),}$ where ${s^{(m)}:=\big(\frac1n\,\sum_{i=1}^n c_i^{2m}\big)^\frac1{2m}}$ . The lower bound ${m_{*}(p)}$ on m is exact for each ${p \in (0,1)}$ . Moreover, ${{\operatorname{\mathsf{E}}} f(c_1{\mathrm{B\!S}}_1+\dots+c_n{\mathrm{B\!S}}_n)}$ is Schur-concave in ${(c_{1}^{2m},\ldots,c_{n}^{2m})}$ .

A number of corollaries are obtained, including upper bounds on generalized moments and tail probabilities of (super)martingales with differences of bounded asymmetry, and also upper bounds on the maximal function of such (super)martingales. Applications to generalized self-normalized sums and t-statistics are given.