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Long strange segments of a stochastic process. (English) Zbl 1052.60025

Long strange intervals in a linear stationary stochastic process with regularly varying tails are studied. Let X n =μ+ - ϕ n-j Z j , n=1,2,, where ,Z -1 ,Z 0 ,Z 1 , is a sequence of zero mean iid random variables and μ is a constant. It is assumed that the tails of the distribution of Z=Z 1 satisfy the regular variation condition P(|Z|>λ)=L(λ)λ -α , as λ, for some α>1, and L be a slowly varying function at infinity, and balance conditions

lim λ P(Z>λ) P(|Z|>λ)=p,lim λ P(Z<-λ) P(|Z|>λ)=q,

0<p=1-q1. The behavior of the statistic

R n (θ)=supj-i:0i<jn,X i+1 ++X j j-i>θ

is studied. R n (θ) can be considered as the greatest length of time interval when the system X n runs under the nominal value. The main result of the paper is the limit theorem for the distribution of the vector (R n (θ),M n ), where M n =max(X 1 ,...,X n ). It is assumed that the sum of the modules of the coefficients ϕ i is finite.

60G10Stationary processes
60G70Extreme value theory; extremal processes (probability theory)