Limit theorems for mixing sequences without rate assumptions

Limit theorems for mixing sequences without rate assumptions

István Berkes and Walter Philipp

Source: Ann. Probab. Volume 26, Number 2 (1998), 805-831.

Abstract

We extend Lévy’s classical criterion for a sequence of independent identically distributed random variables to belong to the domain of partial attraction of a nondegenerate Gaussian law to stationary $\phi$-mixing sequences. We also extend some results of Kesten and of Kuelbs and Zinn on the LIL behavior of independent identically distributed random variables to stationary $\phi$-mixing sequences. No assumptions on the rate of decay for the mixing coefficient are made.

Primary Subjects: 60F05, 60F12, 60F17

Keywords: Domain of attraction; mixing; central limit theorem; law of the iterated logarithm

Full-text: Open access

PDF File (164 KB)

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1022855651
Mathematical Reviews number (MathSciNet): MR1626531
Digital Object Identifier: doi:10.1214/aop/1022855651
Zentralblatt MATH identifier: 0943.60020

back to Table of Contents

References

Berkes, I. and Philipp, W. (1979). Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 29-54.

Mathematical Reviews (MathSciNet): MR80k:60008

Zentralblatt MATH: 0392.60024

Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.

Mathematical Reviews (MathSciNet): MR38:1718

Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York. Bradley, R. C. (1980a). On the -mixing condition for stationary random sequences. Duke Math. J. 47 421-433. Bradley, R. C. (1980b). A remark on the central limit theorem for dependent random variables. J. Appl. Probab. 17 94-101.

Mathematical Reviews (MathSciNet): MR95k:60001

Bradley, R. C. (1981). A sufficient condition for the linear growth of variances in a stationary sequence. Proc. Amer. Math. Soc. 83 586-589.

Mathematical Reviews (MathSciNet): MR82i:60058

Bradley, R. C. (1986). Basic properties of strong mixing conditions. In Dependence in Probability and Statistics (E. Eberlein and M. S. Taqqu, eds.) 165-192, Birkh¨auser, Boston.

Mathematical Reviews (MathSciNet): MR88g:60039

Bradley, R. C. (1988). A central limit theorem for stationary -mixing sequences with infinite variance. Ann. Probab. 16 313-332.

Mathematical Reviews (MathSciNet): MR89a:60053

Dehling, H., Denker, M. and Philipp, W. (1986). A central limit theorem for mixing sequences of random variables under minimal conditions. Ann. Probab. 14 1359-1370.

Mathematical Reviews (MathSciNet): MR88d:60065

Zentralblatt MATH: 0605.60027

Doukhan, P. (1994). Mixing Properties and Examples. Springer, New York.

Mathematical Reviews (MathSciNet): MR96b:60090

Zentralblatt MATH: 0790.60037

Gnedenko, B. W. and Kolmogorov, A. N. (1954). Limit distributions for sums of independent random variables. Addison-Wesley, Cambridge, MA.

Mathematical Reviews (MathSciNet): MR16,52d

Herrndorff, N. (1983). Stationary strongly mixing sequences not satisfying the central limit theorem. Ann. Probab. 11 809-813.

Heyde, C. C. (1969). A note concerning behaviour of iterated logarithm type. Proc. Amer. Math. Soc. 23 85-90.

Mathematical Reviews (MathSciNet): MR40:4999

Ibragimov, I. A. (1962). Some limit theorems for stationary processes. Theory Probab. Appl. 7 349-382.

Zentralblatt MATH: 0119.14204

Ibragimov, I. A. (1975). A note on the central limit theorem for dependent random variables. Theory Probab. Appl. 20 135-140.

Ibragimov, I. A. and Linnik, Y. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.

Iosifescu, M. and Theodorescu, R. (1969). Random Processes and Learning. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR45:2781

Kesten, H. (1972). Sums of independent random variables without moment conditions. Ann. Math. Statist. 43 701-732.

Mathematical Reviews (MathSciNet): MR46:941

Kuelbs, J. and Zinn, J. (1983). Some results on LIL behavior. Ann. Probab. 11 506-557.

Mathematical Reviews (MathSciNet): MR85e:60007

Lo´eve, M. (1963). Probability Theory, 3rd ed. Van Nostrand, Princeton.

Mathematical Reviews (MathSciNet): MR34:3596

Peligrad, M. (1982). Invariance principles for mixing sequences of random variables. Ann. Probab. 10 968-981.

Mathematical Reviews (MathSciNet): MR84c:60054

Zentralblatt MATH: 0503.60044

Peligrad, M. (1985). An invariance principle for -mixing sequences. Ann. Probab. 13 1304-1313.

Mathematical Reviews (MathSciNet): MR87b:60056

Peligrad, M. (1990). On Ibragimov-Iosifescu conjecture for -mixing sequences. Stochastic Process. Appl. 35 293-308.

Mathematical Reviews (MathSciNet): MR91i:60067

Peligrad, M. (1993). Asymptotic results for -mixing sequences. Contemp. Math. 149 163-169.

Mathematical Reviews (MathSciNet): MR94h:60030

Petrov, V. V. (1995). Limit Theorems of Probability Theory. Clarendon Press, Oxford.

Rogozin, B. A. (1968). On the existence of exact upper sequences. Theory Probab. Appl. 13 667- 672.

Mathematical Reviews (MathSciNet): MR39:6393

previous :: next