Electronic Journal of Probability

A Hölderian FCLT for some multiparameter summation process of independent non-identically distributed random variables

Vaidotas Zemlys

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Abstract

We introduce a new construction of a summation process based on the collection of rectangular subsets of unit d-dimensional cube for a triangular array of independent non-identically distributed variables with d-dimensional index, using the non-uniform grid adapted to the variances of the variables. We investigate its convergence in distribution in some Holder spaces. It turns out that for dimensions greater than 2, the limiting process is not necessarily the standard Brownian sheet. This contrasts with a classical result of Prokhorov for the one-dimensional case.

Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 75, 2259-2282.

Dates
Accepted: 21 December 2008
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819148

Digital Object Identifier
doi:10.1214/EJP.v13-590

Mathematical Reviews number (MathSciNet)
MR2469611

Zentralblatt MATH identifier
1190.60026

Subjects
Primary: 60F17: Functional limit theorems; invariance principles

Keywords
Brownian sheet functional central limit theorem H"older space invariance principle triangular array summation process

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Zemlys, Vaidotas. A Hölderian FCLT for some multiparameter summation process of independent non-identically distributed random variables. Electron. J. Probab. 13 (2008), paper no. 75, 2259--2282. doi:10.1214/EJP.v13-590. https://projecteuclid.org/euclid.ejp/1464819148


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References

  • Bickel, P. J., Wichura, M. J. Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42 (1971), 1656–1670.
  • Goldie, Charles M., Greenwood, Priscilla E. Characterisations of set-indexed Brownian motion and associated conditions for finite-dimensional convergence. Ann. Probab. 14 (1986), no. 3, 802–816. (88e:60038a)
  • Goldie, Charles M., Greenwood, Priscilla E. Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes. Ann. Probab. 14 (1986), no. 3, 817–839. (88e:60038b)
  • Khoshnevisan, Davar. Multiparameter processes. An introduction to random fields. Springer Monographs in Mathematics. Springer-Verlag, New York, 2002. xx+584 pp. ISBN: 0-387-95459-7
  • Prohorov, Yu. V. Convergence of random processes and limit theorems in probability theory. (Russian) Teor. Veroyatnost. i Primenen. 1 (1956), 177–238. (18,943b) (18,943b)
  • Rackauskas, Alfredas, Suquet, Charles. Holder versions of Banach space valued random fields. Dedicated to Professor Nicholas Vakhania on the occasion of his 70th birthday. Georgian Math. J. 8 (2001), no. 2, 347–362.
  • Rackauskas, A., Suquet, Ch. Principe d'invariance holderien pour des tableaux triangulaires de variables aleatoires. [Holderian invariance principle for triangular arrays of random variables] (French) Liet. Mat. Rink. 43 (2003), no. 4, 513–532; translation in Lithuanian Math. J. 43 (2003), no. 4, 423–438
  • Rackauskas, A., Suquet, C. Central limit theorems in Holder topologies for Banach space valued random fields. Teor. Veroyatn. Primen. 49 (2004), no. 1, 109–125; translation in Theory Probab. Appl. 49 (2005), no. 1, 77–92
  • Rackauskas, Alfredas, Suquet, Charles. Holder norm test statistics for epidemic change. J. Statist. Plann. Inference 126 (2004), no. 2, 495–520.
  • Rackauskas, Alfredas, Suquet, Charles. Testing epidemic changes of infinite dimensional parameters. Stat. Inference Stoch. Process. 9 (2006), no. 2, 111–134.
  • Rackauskas, Alfredas, Suquet, Charles, Zemlys, Vaidotas. A Holderian functional central limit theorem for a multi-indexed summation process. Stochastic Process. Appl. 117 (2007), no. 8, 1137–1164.
  • Rackauskas, A., Zemlys, V. Functional central limit theorem for a double-indexed summation process. Liet. Mat. Rink. 45 (2005), no. 3, 401–412; translation in Lithuanian Math. J. 45 (2005), no. 3, 324–333
  • van der Vaart, Aad W., Wellner, Jon A. Weak convergence and empirical processes. With applications to statistics. Springer Series in Statistics. Springer-Verlag, New York, 1996. xvi+508 pp. ISBN: 0-387-94640-3