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The joint distribution of the maximum and minimum of an AR(1) process. (English) Zbl 1328.60129

Summary: Consider a sequence of \(n\) observations from an autoregressive process of order \(1\) with maximum \(M_n\) and minimum \(m_n\). We give their joint cumulative distribution function first in terms of \(n\) repeated integrals and then, for the case where the marginal distribution of the observations is absolutely continuous, as a weighted sum of \(n\)th powers of eigenvalues of a certain Fredholm kernel. This enables good approximations for the joint distribution when \(n\) repeated integrals are not a practical solution.

MSC:

60G70 Extreme value theory; extremal stochastic processes
62E17 Approximations to statistical distributions (nonasymptotic)

Software:

R; FitAR
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References:

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