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Theory of Probability and Mathematical Statistics

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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
Published electronically: May 25, 2004
MathSciNet review: 2000647
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Abstract: Some results on the asymptotic behavior of increments of a $d$-dimensional random field are proved. Let $N$ and $a_N\in\{1,2,\dots \}$ be fixed and let $S_N^{\star}$ be the maximum increment of a $d$-dimensional random field of independent identically distributed random variables evaluated for $d$-dimensional rectangles $(i,j]=\{k\colon i<k\leq j\}$ such that $\vert j\vert\leq N$ and $\vert j-i\vert=a_N$. Denote also by $S_N$ the maximum increment evaluated for rectangles such that $\vert j-i\vert\leq a_N$.

We determine the asymptotic almost sure behavior of random variables $S_N$ and $S_N^{\star}$. Steinebach (1983) proved a similar result for the case of rectangles belonging to the cube $(0,N^{1/d}]$ (of volume $N$) and under the condition that $a_N=O(N^{\delta})$ as $N\to\infty$ for all $\delta\in(0,1)$. Note that the sequence $S_N$ is monotone in this case.

We also consider the cases where $a_N\sim C\log N$ or $a_N=O(\log N)$.

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

  • 1. Josef Steinebach, On the increments of partial sum processes with multidimensional indices, Z. Wahrsch. Verw. Gebiete 63 (1983), no. 1, 59–70. MR 699786,
  • 2. D. Plachky and J. Steinebach, A theorem about probabilities of large deviations with an application to queuing theory, Period. Math. Hungar. 6 (1975), no. 4, 343–345. MR 0410870,
  • 3. M. Csörgő and P. Révész, Strong approximations in probability and statistics, Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 666546
  • 4. K. Prachar, Primzahlverteilung, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957. MR 19:393b
  • 5. O. I. Klesov, The Hájek-Rényi inequality for random fields and the strong law of large numbers, Teor. Veroyatnost. i Mat. Statist. 22 (1980), 58–66, 163 (Russian, with English summary). MR 568238
  • 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. Vol. II, Translated from the Russian by Leo F. Boron, Frederick Ungar Publishing Co., New York, 1961. MR 0148805
  • 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

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