Maximal Probability Inequalities for Stochastic Processes

✍ Scribed by F. Móricz

• Publisher: John Wiley and Sons
• Year: 1995
• Tongue: English
• Weight: 346 KB
• Volume: 172
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19951720117

✦ Synopsis

Abstract

Let X~a,b~ be nonnegative random variables with the property that X~a,b~ ≦ X~a,c~ + X~c.b~ for all 0__≦ a < c < b ≦ T__, where T > 0 is fixed. We define M~a,b~ = sup {X~a,c~: a < c ≦ h} and establish bounds for P[M~a,b~ ≧ λ] in terms of given bounds for P[X~a,b~ ≧ λ], where λ runs through some interval (0, λ~o~), 0 < λ~o~ ≦ ∞ fixed. These bounds explicitly involve a nonnegative function g(a, b) assumed to be quasi‐superadditive with an index Q, i.e., g(a, c) + g(c, b) ≦ Qg(a, b) for all 0__≦ a < c < b < T__, where 1 ≦ 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.