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

Let X₁, ⋯, Xₙ be arbitrary random variables and put S(i, j) = Xᵢ + ⋯ + Xⱼ and M(i, j) = max \|S(i, i)|, |S(i, i + 1)|, ⋯, |S(i, j)|\ for 1 ≤ i ≤ j ≤ n. Bounds for Eexp tM (1, n)\, E M^γ(1, n) and P(1, n) ≥ t\ are established in terms of assumed bounds for E exp t|S(i, j)|\, E|S(i, j)|^γ and P\|S(i, j)| ≥ t\, respectively. The bounds explicitly involve a nonnegative function $g(i, j)$ assumed to be quasi-superadditive with index Q(1 ≤ Q ≤ 2): g(i, j) + g(j + 1, k) ≤ Q g(i, k), all 1 ≤ i ≤ j < k ≤ 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 Var S(i, j), applications of the case $Q > 1$ include sequences ᵢ\ exhibiting long-range dependence, in particular certain self-similar processes such as fractional Brownian motion.