An extremal markovian sequence

Abstract

In this paper we consider an independent and identically distributed sequence {Yn} with common distribution function F(x) and a random variable X0, independent of the Yi's, and define a Markovian sequence {Xn} as Xi = X0, if i = 0, Xi = k max{Xi− 1, Yi}, if i ≧ 1, k ∈ R, 0 < k < 1. For this sequence we evaluate basic distributional formulas and give conditions on F(x) for the sequence to possess a stationary distribution. We prove that for any distribution function H(x) with left endpoint greater than or equal to zero for which log H(ex) is concave it is possible to construct such a stationary sequence with marginal distributions equal to it. We study the limit laws for extremes and kth order statistics.

Extremes of scale mixtures of multivariate time series

“…The pARMAX sequence Y (1) has lag-m upper tail dependence function Λ (Y 1 ,Y 1+m ) (x, y) = 0, for all m ≥ 1, and thus it is an upper tail independent process. Therefore, in this case, the lag-m, s upper tail dependence function Λ (X 1,1 ,X 1+m,1+s ) (x, y) is also null, for all m ≥ 1 and s = 1, .…”

“…Observe that sequence X generated from an ARMAX recursion Y (1) corresponds to a multivariate formulation of the RARMAX process introduced in Alpuim and Athayde ( [2], 1990), with applications within reliability and various natural phenomena.…”

“…. , d, with Y (1) = {Y n } n≥1 a stationary sequence having common Pareto-type d.f. with shape parameter α > 0, and…”

“…The case m ≥ 2 corresponding to perfect independence of Y (1) has been analyzed in (22). Observe that each marginal X n,j is a moving maxima process with a random coefficient Z n,j , j = 1, .…”

Finite sample behaviour of the mixed moment estimator in dependent frameworks

“…The Max-AutoRegressive (ARMAX) sequences here considered, introduced and studied in Alpuim (1989), are defined by…”