An exponential inequality for martingales (Q1920084)

The martingales considered here take values in a Banach space whose norm \(|\cdot |\) admits three derivatives. For a martingale difference sequence \(\{Z_k, F_k\}\) corresponding to such a martingale and satisfying a condition of the form \(E\{|Z_k |^2 \mid F_{k-1}\} \leq a^2_k\), \(k=1,2, \dots,n\), an upper bound is given for the probability \(P(\max_{1\leq k\leq n} |\sum^k_{j=1} Z_j |\geq tB_n)\), where \(B^2_n = \sum^n_{k=1} a^2_k\). This bound is the sum of two terms, one of which is of the form \(\exp (-t^2/cG)\), where \(c\), \(G\) are certain constants, while the other does not depend on \(t\) and involves the moments \(E|Z_k |^3\) as well as the \(a_k\)'s and other constants.