|
Abstract:
|
The subject of this doctoral dissertation is related to the problems of
extreme values in strictly stationary random sequences. It belongs to the topical
area of probability and statistics, broadly applicable to real life situations and in
many scienti c elds. It relies on large number of seminal articles and monographs.
The main aim of the dissertation is to determine the asymptotic behavior of
maxima of some incomplete samples from the rst-order auto-regressive processes
with uniform marginal distributions.
The dissertation consists of three chapters.
New results (the theoretical ones and the results of computer simulations) are
presented in the third chapter. Two types of the uniform ARp1q process pXnqnPN
are considered: positively correlated and negatively correlated process, with the
lag one correlation p1q : CorrpXn 1;Xnq equal to
1
r
and
1
r
, respectively, where
r ¥ 2 is the parameter of the underlying process. Let pcnqnPN be a non-random
0 1 sequence, such that lim
nÑ8
1
n
n ¸j 1
cj p P r0; 1s. This sequence of degenerate
random variables is introduced with the purpose to correspond to the sequence
pXnq in the following sense: r.v. Xj is observed if cj 1, otherwise r.v. Xj is not
observed (missing observation). Let us use the notation: the r.v. Mn : max
1¤j¤n
Xj is
maximum of the complete (size n) sample from the random sequence pXnq, and
the r.v. Mn is what is called partial maximum, i.e. the maximal element of
incomplete sample tXj : cj 1; 1 ¤ j ¤ nu. Based on di erent, speci c
deterministic sequences pcnq it is proved that the limiting distribution, as n Ñ 8,
of the two-dimensional random vector Mn;Mn , is not uniquely determined by
the limit value p. This appears as a consequence of the fact that for the uniform
ARp1q process one of the weak dependence conditions does not apply. Namely, the
uniform ARp1q process does not satisfy the local condition under which clustering
of extremes is restricted. As a consequence of this property, some interesting
conclusions about asymptotic joint distributions of random variables Mn and Mn
are reached. In the cases when the partial maximum Mn is determined by an
arbitrary point process there are presented results obtained by simulations.
The rst two chapters are rather informative. Having in mind interest in studying
the asymptotic behavior of linearly standardized two-dimensional component-wise
maxima the role of the rst chapter is to anticipate the concept of multivariate
extreme values. In the second chapter the basic terms in the time series analysis
are formulated, with the accent on the linear stationary models, especially on
rst-order auto-regressive models. The special attention is dedicated to the
uniform ARp1q processes, their properties and existing results concerning their
extremal behavior.
Still open questions are mentioned in the conclusion, in the very end of the third
chapter. |