On inequalities for sums of bounded random variables. (English) Zbl 1156.60015
Summary: A new upper bound on \(P(a_1\eta_1+a_2\eta_2+ \dots \geq x)\) is obtained, where \(\eta_1,\eta_2,\dots\) are independent zero-mean random variables such that \(|\eta|\leq 1\) for all \(i\). A multidimensional analogue of this result and extensions to (super)martingales are presented, as well as an application to self-normalized sums (or, equivalently, to \(t\)-statistics).
MSC:
60E15 | Inequalities; stochastic orderings |
60G50 | Sums of independent random variables; random walks |
60G42 | Martingales with discrete parameter |
60G48 | Generalizations of martingales |
26A48 | Monotonic functions, generalizations |
26D10 | Inequalities involving derivatives and differential and integral operators |