Asymptotic behavior of increments of random fields

Author:
O. E. Shcherbakova

Translated by:
The author

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **68** (2003).

Journal:
Theor. Probability and Math. Statist. **68** (2004), 173-186

MSC (2000):
Primary 60F15; Secondary 60K05

DOI:
https://doi.org/10.1090/S0094-9000-04-00599-X

Published electronically:
May 25, 2004

MathSciNet review:
2000647

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Some results on the asymptotic behavior of increments of a -dimensional random field are proved. Let and be fixed and let be the maximum increment of a -dimensional random field of independent identically distributed random variables evaluated for -dimensional rectangles such that and . Denote also by the maximum increment evaluated for rectangles such that .

We determine the asymptotic almost sure behavior of random variables and . Steinebach (1983) proved a similar result for the case of rectangles belonging to the cube (of volume ) and under the condition that as for *all* . Note that the sequence is monotone in this case.

We also consider the cases where or .

**1.**J. Steinebach,*On the increments of partial sum processes with multidimensional indices*, Z. Wahrcsheinlichkeitstheorie verw. Gebiete**63**(1983), 59-70. MR**84i:60043****2.**D. Plachky and J. Steinebach,*A theorem about probabilities of large deviation with application to queuing theory*, Period. Math. Hungar.**6**(1975), 343-345. MR**53:14613****3.**M. Csörgo and P. Révész,*Strong Approximation in Probability and Statistics*, Akadémiai Kiadó, Budapest, 1981. MR**84d:60050****4.**K. Prachar,*Primzahlverteilung*, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957. MR**19:393b****5.**O. I. Klesov,*The Háek-Rényi inequality for random fields and the strong law of large numbers*, Teor. Veroyatnost. i Mat. Statist.**22**(1980), 58-66; English transl. in Theory Probab. Math. Statist.**22**(1981), 63-72. MR**82i:60050****6.**E. K. Titchmarsh,*The Theory of Riemann Zeta-Function*, 2nd ed., Oxford University Press, Oxford, 1951. MR**13:741c****7.**I. P. Natanson,*Theory of Functions of a Real Variable*, ``Lan'', St. Petersburg, 1999; English transl. of the 2nd ed., Ungar, New York, 1961. MR**26:6309****8.**M. I. D'yachenko and P. L. Ul'yanov,*Measure and Integral*, ``Faktorial Press'', Moscow, 2002. (Russian)

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Additional Information

**O. E. Shcherbakova**

Affiliation:
Chair of Mathematics, Department of Physics and Mechanics, St. Petersburg State Technical University, Politekhnitcheskaya Street 29, St. Petersburg 195251, Russia

Email:
helgagold_@pochtamt.ru, helga_scher@mailru.com

DOI:
https://doi.org/10.1090/S0094-9000-04-00599-X

Received by editor(s):
April 4, 2002

Published electronically:
May 25, 2004

Additional Notes:
Supported in part by the Ministry of Education of the Russian Federation under grants N E00-1.0-82 and “Leading scientific school” # 00-15-96019.

Article copyright:
© Copyright 2004
American Mathematical Society