A Probability Inequality of General Nature for the Maximum of Partial Sums

Let $$ \{ {X_k}\} _{k = 1}^\infty $$ be an arbitrary sequence of random variables and define $$ S(b,n) = \sum\limits_{k = b + 1}^{b + n} {{X_k}} $$ and $$ M(b,n) = \mathop {\max }\limits_{l \leqslant k \leqslant n} \left| {S(b,k)} \right|,{\text{ all }}b = 0,1, \ldots ;n = 1,2, \ldots $$ . Our goal is to establish bounds for the tail distribution P{M(b,n) ≥t} in terms of corresponding similar bounds assumed for the tail distribution P{|S(b,k)| ≥t}. These bounds will be related in specific ways to the variables S(b,k) through some function g(b,k) assumed to be nonnegative, nondecreasing in k for each fixed b, and Q-superadditive with an index 1≤Q≤2. The latter property is defined by the requirement that g(b,k)++g(b+k,l)≤Qg(b,k+l), all b≥O and k,l≥1. The case Q=1 corresponds to the usual notion of superadditivity.