×

Some asymptotic properties of the hybrids of empirical and partial-sum processes. (English) Zbl 1143.60326

Summary: The motivation of this paper is to study some properties of the local times (when it exists) of the hybrids of empirical and partial-sum processes defined by \[ \bar{A}_n(t)=\sum _{1\leq i \leq n} H(X_i)1_{\{X_i\leq t\}} \epsilon_i, - \infty < t < \infty , \] namely by using knowing results on empirical process and Brownian local times.

MSC:

60J55 Local time and additive functionals
60F15 Strong limit theorems
60F17 Functional limit theorems; invariance principles
62G30 Order statistics; empirical distribution functions
PDFBibTeX XMLCite
Full Text: DOI Euclid EuDML

References:

[1] Bass, R. and Khoshnevisan, D.: Laws of the iterated logarithm for local times of the empirical process. Ann. Probab. 23 (1995), no. 1, 388-399. · Zbl 0845.60079 · doi:10.1214/aop/1176988391
[2] Bass, R. and Khoshnevisan, D.: Strongs approximations to Brownian local time. In Seminar in Stochastic Processes 1992 (Seattle, WA, 1992) , 43-65. Prog. Probab. 33 . Birkh\Hauser Boston, Boston, MA, 1993. · Zbl 0789.60062
[3] Csáki, E. and Földes, A.: A note on the stability of the local time of a Wiener process. Stochastic Process. Appl. 25 (1987), no. 2, 203-213. · Zbl 0631.60072 · doi:10.1016/0304-4149(87)90198-0
[4] Csörgő, M., Shi, Z. and Yor, M.: Some asymptotics properties of the local time of the uniform empirical processes. Bernoulli 5 (1999), no. 6, 1035-1058. · Zbl 0960.60023 · doi:10.2307/3318559
[5] Csörgő, M. and Horváth, L.: Weighted Approximations in Probability and Statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley, Chichester, 1993. · Zbl 0770.60038
[6] Diebolt, J., Laib, N. and Gatchou Wandji, J.: Limiting distributions of weighted processes of residuals for parametric autoregressive models of time series. Rapport Technique 1, IMAG, 1997.
[7] Diebolt, J.: A non-parametric test for the regression function: Asymptotic theory. J. Statist. Plann. Inference 44 (1995), no. 1, 1-17. · Zbl 0812.62051 · doi:10.1016/0378-3758(94)00045-W
[8] Diebolt, J. and Laib, N.: Un principe d’invariance faible pour l’etude d’un test nonparametrique relatif à la fonction de regression. C. R. Acad. Sci. Paris, Ser. I Math. 312 (1991), no. 11, 887-891. · Zbl 0737.62077
[9] Heusler, E. and Mason, D.: Weighted approximations to continuous time martingales with applications. Scand. J. Statist. 26 (1999), no. 2, 281-295. · Zbl 0946.60032 · doi:10.1111/1467-9469.00150
[10] Horváth, L.: Approximations for hybrids of empirical and partial sums process. J. Statist. Plann. Inference 88 (2000), no. 1, 1-18. · Zbl 0976.60044 · doi:10.1016/S0378-3758(99)00207-4
[11] Horváth, L., Kokoszka, P. and Steinebach, J.: Approximations for weighted bootstrap processes with an application. Statist. Probab. Lett. 48 (2000), no. 1, 59-70. · Zbl 0982.60019 · doi:10.1016/S0167-7152(99)00190-X
[12] Kesten, H.: An iterated logarithm law for local time. Duke Math. J. 32 (1965), 447-456. · Zbl 0132.12701 · doi:10.1215/S0012-7094-65-03245-X
[13] Khoshnevisan, D.: Exact rates of convergence to Brownian local time. Ann. Probab. 22 (1994), no. 3, 1295-1330. · Zbl 0819.60067 · doi:10.1214/aop/1176988604
[14] Khoshnevisan, D.: An embedding of compensated compound Poisson processes with applications to local times. Ann. Probab. 21 (1993), no. 1, 340-361. · Zbl 0787.60097 · doi:10.1214/aop/1176989408
[15] Khoshnevisan, D.: Level crossings of the empirical process. Stochastic Process. Appl. 43 (1992), no. 2, 331-343. · Zbl 0762.60067 · doi:10.1016/0304-4149(92)90066-Y
[16] Maumy, M.: Etude du processus empirique composé. PhD. Thesis. University of Paris VI, 2002.
[17] Révész, P.: Random Walk in Random and Non-random Environments . World Scientific Publishing Co., Hackensack, NJ, 1998.
[18] Revuz, D. and Yor, M.: Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften 293 . Springer-Verlag, Berlin, 1999. · Zbl 0917.60006
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.