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Maximum of partial sums
and an invariance principle
for a class of weak dependent random variables


Author: Magda Peligrad
Journal: Proc. Amer. Math. Soc. 126 (1998), 1181-1189
MSC (1991): Primary 60F15, 60E15, 60G10
DOI: https://doi.org/10.1090/S0002-9939-98-04177-X
MathSciNet review: 1425136
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Abstract: The aim of this paper is to investigate the properties of the maximum of partial sums for a class of weakly dependent random variables which includes the instantaneous filters of a Gaussian sequence having a positive continuous spectral density. The results are used to obtain an invariance principle and the convergence of the moments in the central limit theorem.


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  • 1. P. Billingsley, Convergence of Probability Measures. New York: Wiley (1968). MR 38:1718
  • 2. R.C. Bradley, On the spectral density and asymptotic normality of weakly dependent random fields, J. Theor. Probab. 5 (1992), 355-373. MR 93e:60094
  • 3. R.C. Bradley, Equivalent mixing conditions for random fields, Ann. Probab. 21, 4 (1993), 1921-1926. MR 94j:60103
  • 4. R.C. Bradley, On regularity conditions for random fields, Proc. Amer. Math. Soc. 121 (1994), 593-598. MR 94h:60074
  • 5. R.C. Bradley, Utev, S. On second order properties of mixing random sequences and random fields, Prob. Theory and Math. Stat., pp. 99-120, B. Grigelionis et al. (Eds.) VSP/TEV (1994).
  • 6. W. Bryc, W. Smolenski, Moment conditions for almost sure convergence of weakly correlated random variables, Proc. A.M.S.119, 2 (1993), 629-635. MR 93k:60071
  • 7. P. Doukhan, Mixing Properties and Examples, Lecture Notes in Statistics 85, Springer-Verlag (1994). MR 96b:60090
  • 8. I.A. Ibragimov, Y.A. Rozanov, Gaussian Random Processes, Berlin: Springer (1978). MR 80f:60038
  • 9. A.N. Kolmogorov, Y.A. Rozanov, On a strong mixing condition for stationary Gaussian processes, Theory Probab. Appl. 5 (1960), 204-208. MR 24:A3009 (Russian original)
  • 10. C. Miller, Three theorems on $\rho^*$-mixing random fields, J. of Theoretical Probability 7 , 4 (1994), 867-882. MR 95i:60023
  • 11. M. Peligrad, Invariance principles for mixing sequences of random variables, The Ann. of Probab.10, 4 (1982), 968-981. MR 84c:60054
  • 12. M. Peligrad, Recent advances in the central limit theorem and its weak invariance principle for mixing sequences of random variables, Progress in Prob. and Stat., Dependence in Prob. and Stat., 11, pp. 193-223, E. Eberlein, M. Taqqu (eds.). Birkhauser (1986). MR 88j:60053
  • 13. M. Peligrad, Invariance principles under weak dependence J. of Multiv. Anal. 19, 2 (1986), 299-310. MR 87m:60077
  • 14. M. Peligrad, On the asymptotic normality of sequences of weak dependent random variables, J. of Theoretical Probabilities 9 (1996), 703-715. MR 97e:60046
  • 15. M. Rosenblatt, Stationary Sequences and Random Fields. Boston: Birkhauser, (1985). MR 88c:60077

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Additional Information

Magda Peligrad
Affiliation: Department of Mathematical Sciences, University of Cincinnati, P.O. Box 210025, Cincinnati, Ohio 45221-0025
Email: peligrm@math.uc.edu

DOI: https://doi.org/10.1090/S0002-9939-98-04177-X
Keywords: Maximal inequalities, functional central limit theorem, weak dependent random variables
Received by editor(s): June 3, 1996
Received by editor(s) in revised form: October 7, 1996
Additional Notes: The author was supported in part by an NSF grant and cost sharing at the University of Cincinnati and a Tuft travel grant
Communicated by: Stanley Sawyer
Article copyright: © Copyright 1998 American Mathematical Society

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