Extreme values of linear processes with heavy-tailed innovations and missing observations
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Extreme values of linear processes with heavy-tailed innovations and missing observations Lenka Glavaˇs1 · Pavle Mladenovi´c1 Received: 13 March 2020 / Revised: 20 July 2020 / Accepted: 3 August 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We investigate maxima in incomplete samples from strictly stationary random sequences defined as linear models of i.i.d. random variables with heavy-tailed innovations that satisfy the tail balance condition. Using the point process approach we obtain limit theorems for the sequence of random vectors whose components are properly normalized maxima in complete and incomplete samples. Keywords Extreme values · Heavy-tailed innovations · Linear processes · Missing observations · Partial maxima · Point processes AMS 2000 Subject Classifications Primary 60G70 · Secondary 60G10
1 Introduction In many statistical investigations the asymptotic behavior of the maximum in certain growing collection of random variables appears to be important. In some cases not all random variables of interest are known, and the problem of comparing maxima in complete and incomplete samples arises naturally. In this paper we shall always consider a stationary sequence X = (Xn )n1 , and a deterministic sequence ci ∈ {0, 1}, i = 1, 2, . . . defining the registration of random variables from the sequence X as follows: the observation Xi is registered if ci = 1, and not registered otherwise.
Research supported by the Ministry of Education, Science and Technological Development, Republic of Serbia, Grant No. 174012. Pavle Mladenovi´c
[email protected] Lenka Glavaˇs [email protected] 1
Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000, Belgrade, Serbia
L. Glavaˇs, P. Mladenovi´c
The maxima we are interested in are the following: Mn = max Xi , 1in
n = M
max
Xi ,
1in: ci =1
n = M
max
Xi ,
1in: ci =0
(1.1)
i.e. Mn is the maximum of the complete sample consisting of the first n terms of n and M n are respectively maxima of observed and nonthe sequence X, while M n for every n ∈ N. observed random variables among them. It is obvious that Mn M We will always assume the existence of an asymptotic sampling frequency 1 ci ∈ (0, 1] . n→∞ n n
p1 = lim
(1.2)
i=1
Note that in many areas (for example in finance, hydrology, meteorology,. . . ), different frequency of observations may be of interest, and the problem of studying n and relations between them becomes important in such situamaxima Mn and M tions. Some data can also be lost by different reasons in a very irregular way. Probably the first paper that considers the question of asymptotic behavior of maxima in incomplete samples is Mittal (1978), where the Gaussian sequence was studied. Several more recent papers on this topic appeared. Scotto (2005), Hall and H¨usler (2006) and Hall and Scotto (2008) considered the cases where the registered observations appeared in a periodic manner. Under very general conditions it was shown in Mladenovi´c and Piterbarg (2006) that
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