# On extreme values in stationary sequences

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## Summary

In this paper, extreme value theory is considered for stationary sequences

*ζ*_{n}satisfying dependence restrictions significantly weaker than strong mixing. The aims of the paper are:- (i)
To prove the basic theorem of Gnedenko concerning the existence of three possible non-degenerate asymptotic forms for the distribution of the maximum

*M*_{n}= max(*ξ*_{1}...*ξ*_{n}), for such sequences. - (ii)To obtain limiting laws of the formwhere$$\mathop {\lim }\limits_{n \to \infty } \Pr \{ M_n^{(r)} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } u\} = e^{ - \tau } \sum\limits_{s = 0}^{r - 1} {\tau ^s /S!} $$
*M*_{n}^{(r)}is the*r*-th largest of*ξ*_{1}...*ξ*_{n}, and Pr*ξ*_{1}>*u*_{n}∼*Τ/n*. Poisson properties (akin to those known for the upcrossings of a high level by a stationary*normal*process) are developed and used to obtain these results. - (iii)
As a consequence of (ii), to show that the asymptotic distribution of

*M*_{n}^{(r)}(normalized) is the same as if the {*ξ*_{n}} were i.i.d. - (iv)
To show that the assumptions used are satisfied, in particular by stationary

*normal*sequences, under mild covariance conditions.

## Keywords

Covariance Stochastic Process Probability Theory Mathematical Biology Normal Process
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## References

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