On the Convergence of the Number of Exceedances of Nonstationary Normal Sequences

It is known that the number of exceedances of normal sequences is asymptotically a Poisson random variable, under certain restrictions. We analyze the rate of convergence to the Poisson limit and extend the result known in the stationary case to nonstationary normal sequences by using the Stein-Chen method. In addition, we consider the cases of exceedances of a constant level as well as of a particular nonconstant level.


1, Introduction and Result
The extreme value theory of Gaussian sequences has interested many authors, for instance Refs.[1,4,7,8], dealing with the limit distribution of the suitably normalized extreme value.
Let {Ai,/^!} be a standardized normal sequence with correlations E(X^i)=r",iJ ^ 1, and <P(-} the distribution function of X,.
For practical use of the asymptotic theory, it is rather important to know the rate of convergence or at least some upper bound for this rate.
For the stationary case, results on the rate of convergence have been obtained for instance by Refs.[2,3,9,10], The aim of this paper is to give an upper bound for the total variation distance dn-between N" and P{\.,), in the nonstationary case, extending the results of these mentioned papers.
Suppose that for some sequence p,,: Ki^py-ji for i^j, and that the two conditions P"<1 for all n^\ (1) pt^Allogk,k ^2, for some constant A (2) are satisfied.Define p as p = max(0, ni, jVy)<l.
In addition, we assume that the boundary values tend uniformly to = : The exceedances of a constant boundary u"i=u", l^j^n, are considered first, where only the tools given in Ref. [3] are used.We show in the second result that the method of Ref. [3] can be used also for nonconstant boundaries {u"}.But these boundaries are restricted such that the condition holds.If we want to extend (he results lo a more general class of boundaries such that only holds, we need to combine the method developed by Ref. [4] with that of Ref. [3] to get satisfactory results (see Ref. [6]).Our first result for boundaries which are constant for fixed n, shows that the given upper bound of the rate of convergence depends mainly on the largest positive correlation value p. Theorem 1: Let {A",,/ ^1} be a standardized nonstationary normal sequence with correlations {nj, ij^\}.Suppose that y,j\^pf^,\ for i^j, such that Eqs. ( 1) and ( 2

) hold. Let the boundary values {u"i-u",\<^i^n} and A,, he real values with
A,, -n(\ -*(u«)).Suppose that A,, ^S C < oc, for some constant C. Then as n-*«!rf",(N,"#'{\")) = o(/i-T^ ■ (log «)- This extends the result of Ref. [3] showing that for a constant boundary their upper bound of the rate of convergence in the stationary case holds also in the nonstalionary ca.sc.
The first term of Eq. ( 6) dominates the rate of convergence in cases with itsi pi < » and p >0.
Then the rate of convergence depends only on the lowest value «".m:n of the particular boundary u," and also on the largest positive correlation p.It extends naturally the results of the stationary case with boundary values which are constant for fixed n.This rate is only good if u"^"," is not a uniquely low value, which is supposed for reasonable boundaries.For the case u".min is uniquely low, the rate can be improved.

Proof
The proof of Theorem 1 is an adaption of that used by Ref. [3] in the stationary case.We use the following lemma which is a straightforward extended version of Lemma 3.4 of Ref. [3].