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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Maximal moment inequalities for stochastic processes

Author: F. Móricz
Journal: Proc. Amer. Math. Soc. 120 (1994), 943-950
MSC: Primary 60E15; Secondary 60G60
MathSciNet review: 1216821
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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.

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Keywords: Stochastic process, maximal fluctuation, quasi-superadditivity, moment inequality, power type estimate, exponential estimate, continuous time analogue of the Rademacher-Men'shov inequality
Article copyright: © Copyright 1994 American Mathematical Society

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