Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] J. L. Doob, Stochastic processes, Wiley, New York, 1953. MR 0058896 (15:445b)
  • [2] L. Gerencsér, On a class of mixing processes, Stochastics 26 (1989), 165-191. MR 1018543 (90m:60058)
  • [3] E. J. Hannan, The asymptotic theory of linear time-series models, J. Appl. Probab. 10 (1973), 130-145. MR 0365960 (51:2212)
  • [4] E. J. Hannan and M. Deistler, The statistical theory of linear systems, Wiley, New York, 1988. MR 940698 (89m:60085)
  • [5] F. Móricz, Moment inequalities and the strong laws of large numbers, Z. Wahrsch. Verw. Gebiete 35 (1976), 299-314. MR 0407950 (53:11717)
  • [6] -, Probability inequalities of exponential type and laws of the iterated logarithm, Acta Sci. Math. (Szeged) 38 (1976), 325-341. MR 0433568 (55:6543)
  • [7] F. A. Móricz, R. J. Serfling, and W. F. Stout, Moment and probability bounds with quasi-superadditive structure for the maximum partial sum, Ann. Probab. 10 (1982), 1032-1040. MR 672303 (84c:60071)
  • [8] J. Rissanen, Stochastic complexity in statistical inquiry, World Sci. Publ., Singapore and Teaneck, NJ, 1989. MR 1082556 (92f:68076)
  • [9] M. S. Taqqu, Self-similar processes and related ultraviolet and infrared catastrophes, Random Fields: Rigorous Results in Statistical Mechanics and Quantum Field Theory, North-Holland, Amsterdam, 1980. MR 712727 (84m:82028)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60E15, 60G60

Retrieve articles in all journals with MSC: 60E15, 60G60

Additional Information

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

American Mathematical Society