## Strong approximation of quantile processes by iterated Kiefer processes.(English)Zbl 1044.60011

Summary: The notion of a $$k$$th iterated Kiefer process $${\mathcal K} (\nu,t;k)$$ for $$k\in\mathbb{N}$$ and $$\nu,t\in\mathbb{R}$$ is introduced. We show that the uniform quantile process $$\beta_n(t)$$ may be approximated on $$[0,1]$$ by $$n^{-1/2} {\mathcal K} (n,t;k)$$, at an optimal uniform almost sure rate of $$O(n^{-1/2+1/2^{k+1} +o(1)})$$ for each $$k\in\mathbb{N}$$. Our arguments are based in part on a new functional limit law, of independent interest, for the increments of the empirical process. Applications include an extended version of the uniform Bahadur-Kiefer representation, together with strong limit theorems for nonparametric functional estimators.

### MSC:

 60F05 Central limit and other weak theorems 60F17 Functional limit theorems; invariance principles 62G20 Asymptotic properties of nonparametric inference 60G15 Gaussian processes
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### References:

 [1] Bahadur, R. (1996). A note on quantiles in large samples. Ann. Math. Statist. 37 577-580. · Zbl 0147.18805 [2] Berkes, I. and Philipp, W. (1979). Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 29-54. · Zbl 0392.60024 [3] Berthet, P. (1997). On the rate of clustering to the Strassen set for increments of the uniform empirical process. J. Theoret. Probab. 10 557-579. · Zbl 0884.60029 [4] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201 [5] Bosq, D. and Lecoutre, J. P. (1987). Théorie de l’Estimation Fonctionnelle. Economica, Paris. [6] Burdzy, K. (1993). Some path properties of iterated Brownian motion. In Seminar on Stochastic Processes (E. Çinlar, K. L. Chung and M. Sharpe, eds.) 67-87. Birkhäuser, Boston. · Zbl 0789.60060 [7] Castelle, N. and Laurent-Bonvalot, F. (1998). Strong approximation of bivariate uniform empirical processes. Ann. Inst. H. Poincaré 34 425-480. · Zbl 0915.60048 [8] Chung, K. L. (1949). An estimate concerning the Kolmogorov limit distribution. Trans. Amer. Math. Soc. 64 205-233. JSTOR: · Zbl 0034.22602 [9] Csáki, E., Cs örg o, M., F öldes, A. and Révész, P. (1989). Brownian local time approximated by a Wiener sheet. Ann. Probab. 17 516-537. · Zbl 0674.60072 [10] Csáki, E., Cs örg o, M., F öldes, A. and Révész, P. (1995). Global Strassen-type theorems for iterated Brownian motions. Stochastic Process. Appl. 59 321-341. · Zbl 0843.60072 [11] Csáki, E., F öldes, A. and Révész, P. (1997). Strassen theorems for a class of iterated processes. Trans. Amer. Math. Soc. 349 1153-1167. JSTOR: · Zbl 0867.60051 [12] Cs örg o, M. (1983). Quantile Processes with Statistical Applications. SIAM Philadelphia. · Zbl 0518.62043 [13] Cs örg o, M., Cs örg o, S., Horváth, L. and Mason, D. M. (1986). Weighted empirical and quantile processes. Ann. Probab. 14 86-118. · Zbl 0589.60029 [14] Cs örg o, M., Cs örg o, S., Horváth, L. and Révész, P. (1984). On weak and strong approximations of the quantile processes. In Proceedings of the Seventh Conference on Probability Theory (M. Iosifescu, ed.) 81-95, Editura Academiei, Bucarest. · Zbl 0591.60029 [15] Cs örg o, M. and Horváth, L. (1986). Approximations of weighted empirical and quantile processes. Statist. Probab. Lett. 4 275-280. · Zbl 0676.60042 [16] Cs örg o, M. and Horváth, L. (1993). Weighted Approximations in Probability and Statistics. Wiley, New York. · Zbl 0770.60038 [17] Cs örg o, M. and Révész, P. (1975). Some notes on the empirical distribution function and the quantile process. In Limit Theorems of Probability Theory. (P. Révész, ed.) 11 59-71. North-Holland, Amsterdam. · Zbl 0315.62013 [18] Cs örg o, M. and Révész, P. (1978). Strong approximations of the quantile process. Ann. Statist. 6 882-894. · Zbl 0378.62050 [19] Cs örg o, M. and Révész, P. (1981). Strong Approximations in Probability and Statistics. Academic Press, New York. · Zbl 0539.60029 [20] Cs örg o, M. and Révész, P. (1984). Two approaches to constructing simultaneous confidence bounds for quantiles. Probab. Math. Statist. 4 221-236. · Zbl 0591.62039 [21] Deheuvels, P. (1997). Strong laws for local quantile processes. Ann. Probab. 25 2007-2054. · Zbl 0902.60027 [22] Deheuvels, P. (1998). On the approximation of quantile processes by Kiefer processes. J. Theoret. Probab. 11 997-1018. · Zbl 0930.62053 [23] Deheuvels, P. and Mason, D. M. (1990). Bahadur-Kiefer-type processes. Ann. Probab. 18 669-697. Deheuvels, P. and Mason, D. M. (1992a). A functional LIL approach to pointwise Bahadur- Kiefer theorems. In Probability in Banach Spaces 8 (R. M. Dudley, M. G. Hahn and J. Kuelbs, eds.) 255-266. Birkhäuser, Boston. Deheuvels, P. and Mason, D. M. (1992b). Functional laws of the iterated logarithm for the increments of empirical and quantile processes. Ann. Probab. 20 1248-1287. · Zbl 0712.60028 [24] Deuschel, J. D. and Stroock, D. W. (1989). Large Deviations. Academic Press, New York. · Zbl 0675.60086 [25] Devroye, L. and Gy örfi, L. (1985). Nonparametric Density Estimation: The L1 View. Wiley, New York. · Zbl 0546.62015 [26] Donsker, M. (1952). Justification and extension of Doob’s heuristic approach to the Kolmogorov- Smirnov theorems. Ann. Math. Statist. 23 277-281. · Zbl 0046.35103 [27] Doob, J. L. (1949). Heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Statist. 20 393-403. · Zbl 0035.08901 [28] Finkelstein, H. (1971). The law of the iterated logarithm for empirical distributions. Ann. Math. Statist. 42 607-615. · Zbl 0227.62012 [29] Hu, Y., Pierre-Loti-Viaud, D. and Shi,(1995). Laws of the iterated logarithm for iterated Wiener processes. J. Theoret. Probab. 9 303-319. · Zbl 0816.60027 [30] Khoshnevisan, D. and Lewis, T. M. (1996). Chung’s law of the iterated logarithm for iterated Brownian motion. Ann. Inst. H. Poincaré. 32 349-359. · Zbl 0859.60025 [31] Kiefer, J. (1970). Deviations between the sample quantile process and the sample d.f. In Nonparametric Techniques in Statistical Inference (M. Puri, ed.) 299-319. Cambridge Univ. Press. [32] Kiefer, J. (1972). Skorohod embedding of multivariate rv’s and the sample df.Wahrsch. Verw. Gebiete 24 1-35. · Zbl 0267.60034 [33] Koml ós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent r.v.’s and the sample df. I.Wahrsch. Verw. Gebiete 32 111-131. · Zbl 0308.60029 [34] Koml ós, J., Major, P. and Tusnády, G. (1976). An approximation of partial sums of independent r.v.’s and the sample df. II.Wahrsch. Verw. Gebiete 34 33-58. · Zbl 0307.60045 [35] Schilder, M. (1966). Asymptotic formulas for Wiener integrals. Trans. Amer. Math. Soc. 125 63-85. JSTOR: · Zbl 0156.37602 [36] Scott, D. W. (1992). Multivariate Density Estimation. Wiley, New York. · Zbl 0850.62006 [37] Shorack, G. R. (1982). Kiefer’s theorem via the Hungarian construction.Wahrsch. Verw. Gebiete 61 369-373. · Zbl 0501.60038 [38] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York. · Zbl 1170.62365 [39] Strassen, V. (1964). An invariance principle for the law of the iterated logarithm.Wahrsch. Verw. Gebiete 3 211-226. · Zbl 0132.12903 [40] Stute, W. (1982). The oscillation behaviour of empirical processes. Ann. Probab. 10 86-107. · Zbl 0489.60038
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