Moment and Probability Bounds with Quasi-Superadditive Structure for the Maximum Partial Sum

Abstract

Let $X_1, \cdots, X_n$ be arbitrary random variables and put $S(i, j) = X_i + \cdots + X_j$ and $M(i, j) = \max \{|S(i, i)|, |S(i, i + 1)|, \cdots, |S(i, j)|\}$ for $1 \leq i \leq j \leq n$. Bounds for $E\{\exp tM (1, n)\}, E M^\gamma(1, n)$ and $P\{M(1, n) \geq t\}$ are established in terms of assumed bounds for $E \{\exp t|S(i, j)|\}, E|S(i, j)|^\gamma$ and $P\{|S(i, j)| \geq t\}$, respectively. The bounds explicitly involve a nonnegative function $g(i, j)$ assumed to be quasi-superadditive with index $Q(1 \leq Q \leq 2): g(i, j) + g(j + 1, k) \leq Q g(i, k)$, all $1 \leq i \leq j < k \leq n$. Results previously established for the case $Q = 1$ are improved and are extended to the case $1 < Q < 2$. When $g(i, j)$ is given by $\operatorname{Var} S(i, j)$, applications of the case $Q > 1$ include sequences $\{X_i\}$ exhibiting long-range dependence, in particular certain self-similar processes such as fractional Brownian motion.