Introduction 4
1 Gaussian maxima and Cezaro convergence of covariances 5
2 Relations between types of convergence for Gaussian maxima 9
3 Asymptotic independence of the values of the support function of a Gaussian process 12
4 The speed of convergence of Gaussian convex hulls 15
5 Maxima of Gaussian sequences with nonidentical variances 19
Conclusion 23
6 References 25
In this work we consider various questions in the held of maxima of Gaussian processes and Gaussian convex hulls.
Section 1 is devoted to the behaviour of maxima of centered stationary Gaussian processes in terms of covariance functions or spectral measures. Section 2 contains some results on the eqivalence of convergence in Lp-mean, convergence in probability and almost sure convergence for Gaussian maxima. In Section 3 we provide a central limit theorem (which can be viewed as an asymptotic independence result) for the values of the support function of the convex hull of a stationary Gaussian process. Section 4 is devoted to the question of speed of convergence of the normalized convex hulls of iid copies of a centered Gaussian process to their limit shape. Section 5 describes the case of maxima of centered independent Gaussian variables with nonequal variances. Finally, in the conclusion we present some open questions related to Gaussian maxima.
All the Gaussian processes and variables in this work are assumed to be centered.
Below we formulate some open questions and suggest some directions of research on the topics discussed above.
1) In connection with the obtained results in the future it makes sense to investigate more deeply the question of the behaviour of the quantities
1- • rmax(X1 ,-,Xn)
m(X) = liminf . —
n!+i ^2log(n)
and
M (X) = limsup™**1"-*-)
n!+1 V2 log(n)
for spectral measures vX, which are singular with the Lebesgue measure.
It seems reasonable to start from Cantor type measures, which are the distributions of random series (for example, as above, we consider as i'. the distribution of the variable J/ уёу) and try to estimate m(X) and M(X) k=i
in terms of the density of the set Л = {a1,a2,...} (the set of "frequencies") in the set of positive integers, taking into account that such a method was used to construct our example with M(X) = m(X) = 0 almost surely: it is obtained from a lacunary series. If, on the contrary, Ж>0Л is finite, then the spectral measure turns out to be uniform on some finite union of segments, in partical, it is absolutely continuous, which implies m(X) = M(X) = 1. Therefore, it is natural to assume that the denser is Л in Z>0, the closer (in some sense) are m(X) and M(X) to 1.
2) One may also try to generalize our central limit theorem/asymptotic independence result for the support functions of Gaussian polytopes to the case of stationary sequences of centered weakly correlated Gaussian processes.
3) Instead of the support functions one may try to prove central limit theorems for the volumes and surface areas of convex hulls of centered stationary Gaussian processes.
4) In our counterexample to Theorem 6 we have
p( w„,w) = o(-X;
V2ln n J Win nJ
but in [2] it is only proven that p ( <-1— Wn, W) = o (1) in general without 2 1П —
the assumption of continuity of X.
Hence, it is interesting to check whether the o(1) can be improved or one can build examples with p (x Jl|in Wn, W j = Q(f (n)) for arbitrary f such that f (n) = o(1) as n ! +1.
