Maximal moment inequalities for stochastic processes

Abstract:

Let ${X_{a,b}}$ be nonnegative random variables with the property that ${X_{a,b}} \leqslant {X_{a,c}} + {X_{c,b}}$ for all $0 \leqslant a < c < b \leqslant T$, where $T > 0$ is fixed. We define ${M_{a,b}}: = \sup \{ {X_{a,c}}:a < c \leqslant b\}$ and establish bounds for $EM_{a,b}^\gamma$ and $E\exp (\lambda {M_{a,b}})$ in terms of assumed bounds for $EX_{a,b}^\gamma$ and $E\exp (\lambda {X_{a,b}})$, respectively, where $\gamma \geqslant 1$ and $\lambda$ runs through an interval $({\lambda _0},\infty )$ with fixed ${\lambda _0} \geqslant 0$. These bounds explicitly involve a nonnegative function $g(a,b)$ assumed to be quasi-superadditive with an index $Q$, which means that $g(a,c) + g(c,b) \leqslant Qg(a,b)$ for all $0 \leqslant a < c < b \leqslant T$, where $1 \leqslant Q < 2$ is fixed. Maximal inequalities obtained in this way can be applied to stochastic processes exhibiting long-range dependence. Among others, these applications may include certain self-similar processes such as fractional Brownian motion, stochastic processes occurring in linear time-series models, etc.