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

MathSciNet review:
1425136

Full-text PDF Free Access

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.**Patrick Billingsley,*Convergence of probability measures*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0233396****2.**Richard C. Bradley,*On the spectral density and asymptotic normality of weakly dependent random fields*, J. Theoret. Probab.**5**(1992), no. 2, 355–373. MR**1157990**, 10.1007/BF01046741**3.**Richard C. Bradley,*Equivalent mixing conditions for random fields*, Ann. Probab.**21**(1993), no. 4, 1921–1926. MR**1245294****4.**Richard C. Bradley,*On regularity conditions for random fields*, Proc. Amer. Math. Soc.**121**(1994), no. 2, 593–598. MR**1219721**, 10.1090/S0002-9939-1994-1219721-3**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 and W. Smoleński,*Moment conditions for almost sure convergence of weakly correlated random variables*, Proc. Amer. Math. Soc.**119**(1993), no. 2, 629–635. MR**1149969**, 10.1090/S0002-9939-1993-1149969-7**7.**Paul Doukhan,*Mixing*, Lecture Notes in Statistics, vol. 85, Springer-Verlag, New York, 1994. Properties and examples. MR**1312160****8.**Il′dar Abdulovich Ibragimov and Y. A. Rozanov,*Gaussian random processes*, Applications of Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Translated from the Russian by A. B. Aries. MR**543837****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.**Curtis Miller,*Three theorems on 𝜌*-mixing random fields*, J. Theoret. Probab.**7**(1994), no. 4, 867–882. MR**1295544**, 10.1007/BF02214377**11.**Magda Peligrad,*Invariance principles for mixing sequences of random variables*, Ann. Probab.**10**(1982), no. 4, 968–981. MR**672297****12.**Magda Peligrad,*Recent advances in the central limit theorem and its weak invariance principle for mixing sequences of random variables (a survey)*, Dependence in probability and statistics (Oberwolfach, 1985) Progr. Probab. Statist., vol. 11, Birkhäuser Boston, Boston, MA, 1986, pp. 193–223. MR**899991****13.**Magda Peligrad,*Invariance principles under weak dependence*, J. Multivariate Anal.**19**(1986), no. 2, 299–310. MR**853060**, 10.1016/0047-259X(86)90034-5**14.**Magda Peligrad,*On the asymptotic normality of sequences of weak dependent random variables*, J. Theoret. Probab.**9**(1996), no. 3, 703–715. MR**1400595**, 10.1007/BF02214083**15.**Murray Rosenblatt,*Stationary sequences and random fields*, Birkhäuser Boston, Inc., Boston, MA, 1985. MR**885090**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
60F15,
60E15,
60G10

Retrieve articles in all journals with MSC (1991): 60F15, 60E15, 60G10

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