Limiting distributions and almost sure limit theorems for the normalized maxima of complete and incomplete samples from Gaussian sequence
Zuoxiang Peng, Ping Wang, and Saralees Nadarajah
Source: Electron. J. Statist. Volume 3 (2009), 851-864.
Abstract
Let {Xk,k⩾1} be a stationary Gaussian sequence with partial maximum Mn=max {Xk,1⩽k⩽n} and sample mean X̅n=∑k=1nXk/n. Suppose that some of the random variables X1,X2,… can be observed and the others not. Denote by M̃n the maximum of the observed random variables from the set {X1,X2,…,Xn}. Under some mild conditions, we prove the joint limiting distribution and the almost sure limit theorem for (M̃n−X̅n,Mn−X̅n).
Primary Subjects: 62F15
Secondary Subjects: 60G70, 60F15
Keywords: Almost sure limit theorem; complete and incomplete samples; limiting distribution; maximum; stationary Gaussian sequence
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.ejs/1250880018
Digital Object Identifier: doi:10.1214/09-EJS443
References
[1] Berkes, I. and Csáki, E. (2001). A universal result in almost sure central limit theory. Stochastic Processes and Their Applications 94 105–134.
[2] Berman, S.M. (1964). Limit theorems for the maximum term in stationary sequences. Annals of Mathematical Statistics 35 502–516.
[3] Chen, S.Q. and Lin, Z.Y. (2006). Almost sure max-limits for nonstationary Gaussian sequence. Statistics and Probability Letters 76 1175–1184.
[4] Chen, S.Q. and Lin, Z.Y. (2007). Almost sure limit theorems for a stationary normal sequence. Applied Mathematics Letters 20 316–322.
[5] Cheng, S., Peng, L. and Qi, Y. (1998). Almost sure convergence in extreme value theory. Mathematische Nachrichten 190 43–50.
[6] Csáki, E. and Gonchigdanzan, K. (2002). Almost sure limit theorems for the maximum of stationary Gaussian sequences. Statistics and Probability Letters 58 195–203.
[7] Dudziński, M. (2003). An almost sure limit theorem for the maxima and sums of stationary Gaussian sequences. Probability and Mathematical Statistics 23 139–152.
[8] Dudziński, M. (2008). The almost sure central limit theorems in the joint version for the maxima and sums of certain stationary Gaussian sequences. Statistics and Probability Letters 78 347–357.
[9] Fahrner, I. and Stadtmüller, U. (1998). On almost sure max-limit theorems. Statistics and Probability Letters 37 229–236.
[10] Lin, F. (2009). Almost sure limit theorem for the maxima of strongly dependent Gaussian sequence. Electronic Communications in Probability 14 224–231.
[11] Leadbetter, M.R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
[12] McCormick, W.P. (1980). Weak convergence for the maxima of stationary Gaussian processes using random normalization. Annals of Probability 8 483–497.
[13] McCormick, W.P. and Mittal, Y. (1976). On weak convergence of the maximum. Technical Report No 81, Department of Statistics, Stanford University.
[14] Mittal, Y. and Ylvisaker, D. (1975). Limit distributions for the maxima of stationary Gaussian processes. Stochastic Processes and Their Applications 3 1–18.
[15] Mladenović, P. and Piterbarg, V. (2006). On asymptotic distribution of maxima of complete and incomplete samples from stationary sequences. Stochastic Processes and Their Applications 116 1977–1991.
[16] Peng, Z., Li, J. and Nadarajah, S. (2009). Almost sure convergence of extreme order statistics. Electronic Journal of Statistics 3 546–556.
[17] Peng, Z., Wang, L. and Nadarajah, S. (2009). Almost sure central limit theorem for partial sums and maxima. Mathematische Nachrichten 282 632–636.
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