## The Annals of Statistics

### Strong invariance principles for sequential Bahadur–Kiefer and Vervaat error processes of long-range dependent sequences

#### Abstract

In this paper we study strong approximations (invariance principles) of the sequential uniform and general Bahadur–Kiefer processes of long-range dependent sequences. We also investigate the strong and weak asymptotic behavior of the sequential Vervaat process, that is, the integrated sequential Bahadur–Kiefer process, properly normalized, as well as that of its deviation from its limiting process, the so-called Vervaat error process. It is well known that the Bahadur–Kiefer and the Vervaat error processes cannot converge weakly in the i.i.d. case. In contrast to this, we conclude that the Bahadur–Kiefer and Vervaat error processes, as well as their sequential versions, do converge weakly to a Dehling–Taqqu type limit process for certain long-range dependent sequences.

#### Article information

Source
Ann. Statist., Volume 34, Number 2 (2006), 1013-1044.

Dates
First available in Project Euclid: 27 June 2006

https://projecteuclid.org/euclid.aos/1151418250

Digital Object Identifier
doi:10.1214/009053606000000164

Mathematical Reviews number (MathSciNet)
MR2283402

Zentralblatt MATH identifier
1113.60034

#### Citation

Csörgő, Miklós; Szyszkowicz, Barbara; Wang, Lihong. Strong invariance principles for sequential Bahadur–Kiefer and Vervaat error processes of long-range dependent sequences. Ann. Statist. 34 (2006), no. 2, 1013--1044. doi:10.1214/009053606000000164. https://projecteuclid.org/euclid.aos/1151418250

#### References

• Bahadur, R. R. (1966). A note on quantiles in large samples. Ann. Math. Statist. 37 577–580.
• Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press.
• Csáki, E. (1977). The law of iterated logarithm for normalized empirical distribution functions. Z. Wahrsch. Verw. Gebiete 38 147–167.
• Csáki, E., Csörgő, M., Földes, A., Shi, Z. and Zitikis, R. (2002). Pointwise and uniform asymptotics of the Vervaat error process. J. Theoret. Probab. 15 845–875.
• Csörgő, M. (1983). Quantile Processes with Statistical Applications. SIAM, Philadelphia.
• Csörgő, M. and Horváth, L. (1993). Weighted Approximations in Probability and Statistics. Wiley, Chichester.
• Csörgő, M. and Révész, P. (1978). Strong approximations of the quantile process. Ann. Statist. 6 882–894.
• Csörgő, M. and Révész, P. (1981). Strong Approximations in Probability and Statistics. Academic Press, New York.
• Csörgő, M. and Shi, Z. (2001). An $L^p$-view of a general version of the Bahadur–Kiefer process. J. Math. Sci. 105 2534–2540.
• Csörgő, M. and Shi, Z. (2005). An $L^p$-view of the Bahadur–Kiefer theorem. Period. Math. Hungar. 50 79–98.
• Csörgő, M. and Szyszkowicz, B. (1998). Sequential quantile and Bahadur–Kiefer processes. In Order Statistics: Theory and Methods (N. Balakrishnan and C. R. Rao, eds.) 631–688. North-Holland, Amsterdam.
• Csörgő, M. and Zitikis, R. (1999). On the Vervaat and Vervaat-error processes. Acta Appl. Math. 58 91–105.
• Csörgő, M. and Zitikis, R. (2001). The Vervaat process in $L_p$-spaces. J. Multivariate Anal. 78 103–138.
• Csörgő, M. and Zitikis, R. (2002). On the general Bahadur–Kiefer, quantile and Vervaat processes: Old and new. In Limit Theorems in Probability and Statistics: In Honour of the 65th Birthday of Professor Pál Révész (Balatonlelle, Hungary, June 28–July 2, 1999) (I. Berkes, E. Csáki and M. Csörgő, eds.) 1 389–426. János Bolyai Math. Soc., Budapest.
• Csörgő, S. and Mielniczuk, J. (1995). Density estimation under long-range dependence. Ann. Statist. 23 990–999.
• de Haan, L. (1970). On Regular Variation and Its Application to the Weak Convergence of Sample Extremes. Mathematical Centre Tracts 32. Mathematisch Centrum, Amsterdam.
• Dehling, H., Mikosch, T. and Sørensen, M., eds. (2002). Empirical Process Techniques for Dependent Data. Birkhäuser Boston.
• Dehling, H. and Philipp, W. (2002). Empirical process techniques for dependent data. In Empirical Process Techniques for Dependent Data (H. Dehling, T. Mikosch and M. Sørensen, eds.) 3–113. Birkhäuser, Boston.
• Dehling, H. and Taqqu, M. S. (1988). The functional law of the iterated logarithm for the empirical process of some long-range dependent sequences. Statist. Probab. Lett. 7 81–85.
• Dehling, H. and Taqqu, M. S. (1989). The empirical process of some long-range dependent sequences with an application to $U$-statistics. Ann. Statist. 17 1767–1783.
• Dobrushin, R. L. and Major, P. (1979). Non-central limit theorems for non-linear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 27–52.
• Giraitis, L. and Surgailis, D. (2002). The reduction principle for the empirical process of a long memory linear process. In Empirical Process Techniques for Dependent Data (H. Dehling, T. Mikosch and M. Sørensen, eds.) 241–255. Birkhäuser, Boston.
• Kiefer, J. (1967). On Bahadur's representation of sample quantiles. Ann. Math. Statist. 38 1323–1342.
• Kiefer, J. (1970). Deviations between the sample quantile process and the sample DF. In Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.) 299–319. Cambridge Univ. Press.
• Kiefer, J. (1972). Skorohod embedding of multivariate rv's and the sample df. Z. Wahrsch. Verw. Gebiete 24 1–35.
• Koul, H. L. and Surgailis, D. (2002). Asymptotic expansion of the empirical process of long memory moving averages. In Empirical Process Techniques for Dependent Data (H. Dehling, T. Mikosch and M. Sørensen, eds.) 213–239. Birkhäuser, Boston.
• Major, P. (1981). Multiple Wiener–Itô Integrals. Lecture Notes in Math. 849. Springer, Berlin.
• Mori, T. and Oodaira, H. (1987). The functional iterated logarithm law for stochastic processes represented by multiple Wiener integrals. Probab. Theory Related Fields 76 299–310.
• Müller, D. W. (1970). On Glivenko–Cantelli convergence. Z. Wahrsch. Verw. Gebiete 16 195–210.
• Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
• Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287–302.
• Taqqu, M. S. (1977). Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long range dependence. Z. Wahrsch. Verw. Gebiete 40 203–238.
• Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 53–83.
• Vervaat, W. (1972). Functional central limit theorems for processes with positive drift and their inverses. Z. Wahrsch. Verw. Gebiete 23 245–253.
• Vervaat, W. (1972). Success Epochs in Bernoulli Trials: With Applications to Number Theory. Mathematical Centre Tracts 42. Mathematisch Centrum, Amsterdam.
• Zitikis, R. (1998). The Vervaat process. In Asymptotic Methods in Probability and Statistics –- A Volume in Honour of Miklós Csörgő (B. Szyszkowicz, ed.) 667–694. North-Holland, Amsterdam.