## Variance of partial sums of stationary sequences.(English)Zbl 1291.60068

The authors give necessary and sufficient conditions for the variance of partial sums of centered sequences of weakly stationary random variables to be regularly varying of index $$\gamma$$ at infinity. The authors derive lower and upper bounds for the variance of partial sums and use them to prove the main result of this paper.

### MSC:

 60G10 Stationary stochastic processes 42A24 Summability and absolute summability of Fourier and trigonometric series
Full Text:

### References:

 [1] Aaronson, J. and Weiss, B. (2000). Remarks on the tightness of cocycles. Colloq. Math. 84/85 363-376. · Zbl 0980.28010 [2] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27 . Cambridge Univ. Press, Cambridge. · Zbl 0617.26001 [3] Bolthausen, E. (1989). A central limit theorem for two-dimensional random walks in random sceneries. Ann. Probab. 17 108-115. · Zbl 0679.60028 [4] Bradley, R. C. (2007). Introduction to Strong Mixing Conditions. Vol. 1. Kendrick Press, Heber City, UT. · Zbl 1134.60004 [5] Bryc, W. and Dembo, A. (1995). On large deviations of empirical measures for stationary Gaussian processes. Stochastic Process. Appl. 58 23-34. · Zbl 0833.60027 [6] Doob, J. L. (1990). Stochastic Processes . Wiley, New York. · Zbl 0696.60003 [7] Gel’fand, I. M. and Shilov, G. E. (1964). Generalized Functions. Vol. I : Properties and Operations . Academic Press, New York. · Zbl 0115.33101 [8] Giraitis, L., Taqqu, M. S. and Terrin, N. (1998). Limit theorems for bivariate Appell polynomials. II. Non-central limit theorems. Probab. Theory Related Fields 110 333-367. · Zbl 0927.60031 [9] Hardy, G. H. and Littlewood, J. E. (1924). Solution of the Cesàro summability problem for power-series and Fourier series. Math. Z. 19 67-96. · JFM 49.0232.01 [10] Ibragimov, I. A. (1962). Some limit theorems for stationary processes. Theory Probab. Appl. 7 349-382. · Zbl 0119.14204 [11] Kesten, H. and Spitzer, F. (1979). A limit theorem related to a new class of self-similar processes. Z. Wahrsch. Verw. Gebiete 50 5-25. · Zbl 0396.60037 [12] Merlevède, F., Peligrad, M. and Utev, S. (2006). Recent advances in invariance principles for stationary sequences. Probab. Surv. 3 1-36. · Zbl 1189.60078 [13] Peligrad, M. and Utev, S. (2005). A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33 798-815. · Zbl 1070.60025 [14] Robinson, E. A. (1960). Sums of stationary random variables. Proc. Amer. Math. Soc. 11 77-79. · Zbl 0099.12602 [15] Rosenblatt, M. (1979). Some limit theorems for partial sums of quadratic forms in stationary Gaussian variables. Z. Wahrsch. Verw. Gebiete 49 125-132. · Zbl 0388.60048 [16] Samorodnitsky, G. (2006). Long range dependence. Found. Trends Stoch. Syst. 1 163-257. · Zbl 1242.60033 [17] Surgailis, D. (2000). Long-range dependence and Appell rank. Ann. Probab. 28 478-497. · Zbl 1130.60306 [18] Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287-302. · Zbl 0303.60033 [19] Zygmund, A. (2002). Trigonometric Series. Vol. I , II , 3rd ed. Cambridge Univ. Press, Cambridge. · Zbl 1084.42003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.