DOI: https://doi.org/10.20535/1810-0546.2014.4.28220

### Estimates for Moments of Extreme Values of the Random Process with Ssuperadditive Moment Function

Тетяна Михайлівна Грозян

#### Abstract

This paper considers the stochastic process with superadditive moment function. The aim is to generalize the results of R. Serfling, which he received for a sequence of random variables with superadditive moment function. We have obtained the estimation for moments of supremum of a random process with the appropriate bounds for moments of this random process. We make no assumptions about the structure of the dependence of  increments of a random process, but only the estimation for moments of random process. The estimates for supremum of  the stochastic process with orthogonal increments and quasi-stationary process were obtained as a consequence of the main theorem. Also estimates for such random processes were considered under given estimates for moments. The technique of proof relies on the classical method of binary partitions that have been developed for orthogonal series and generalized to quasi-stationary sequences of random variables by R. Serfling. It should be mentioned, that unlike the case of random variables there appears a certain constant in the estimation of stochastic processes, but it has no significant impact on further research.

#### Keywords

Maximum estimates; Superadditive moment function; The process with orthogonal increments; Quasi-stationary process

PDF (Українська)

#### References

R.J. Serfling, “Moment inequalities for the maximum cumulative sum”, Ann. Math. Statist., vol. 41, pр. 1227–1234, 1970.

Алексич Г. Проблемы сходимости ортогональных рядов / Пер. с англ. – М.: Изд-во иностр. лит-ры, 1963. – 359 с.

F. Moricz, “Moment Inequalities and the Strong Laws of Large Numbers”, Z. Wahrscheinlichkeitstheorie Verw. Gebiete, vol. 35, pр. 299–314, 1976.

F.A. Móricz et al.,“Moment and probability bounds with quasi-superadditive structure for the maximum partial sum”, The Annals of Probability, vol. 10, no. 4, pр. 1032–1040, 1982.

F.A. Moricz, “A general moment inequality for the ma­ximum of the rectangular partial sums of multiple series”, Acta Math. Hung., vol. 41, no. 3-4, pр. 337–346, 1983.

T. Tómács, “A moment inequality for the maximum partial sums with a generalized superadditive structure”, Acta Acad. Paed. Agriensis, Sectio Mathematicae, vol. 26, pр. 75–79, 1999.

F. Moricz et al., “Strong Laws for Blockwise M-de­pen­dent Random Fields”, J. Theor. Probability, vol. 31, no. 3, pр. 660–671, 2008.

U. Stadtmuller and Le V. Thanh, “On the strong limit theorems for double arrays of blockwise M-dependent random variables”, Acta Math. Sinica, vol. 27, no. 10, pр. 1923–1934, 2011.

I. Fazekas and O. Klesov, “A general approach to the strong laws of large numbers”, Theory Probab. Appl., vol. 45, pр. 436–449, 2000.

Grozian T.M. Strong law of large numbers for random variables with superadditive moment function // Наукові вісті НТУУ “КПІ”. – 2012. – № 4. – С. 39–42.