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 | References | Similar Articles | Additional Information

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.

**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 -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