On the stationary distribution of some extremal Markovian sequences

Abstract

This paper is concerned with the Markovian sequence Xn = Zn max{Xn– 1, Yn },n ≧ 1, where X 0 is any random variable, {Zn } and {Yn } are independent sequences of i.i.d. random variables both independent of X 0. We consider the problem of characterizing the class of stationary distributions arising in such a model and give criteria for a d.f. to belong to it. We develop further results when the Zn 's are random variables concentrated on the interval [0, 1], namely having a beta distribution.

Extremes of scale mixtures of multivariate time series

“…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.…”

“…Factor models have been used in the modeling of data within hydrology (Nadarajah [26,27] 2006/2009, Nadarajah and Masoom [28] 2008), storm insurance (Lescourret and Robert, [22] 2006), soil erosion in crops (Todorovic and Gani [36] 1987, Alpuim and Athayde [2] 1990), reliability (Alpuim and Athayde [2] 1990, Kotz et al [19] 2000), economy (Arnold,[3] 1983) and finance (Ferreira and Canto e Castro, [13] 2010). Let X n = (X n1 , .…”

Stationary max-stable processes with the Markov property

“…Related works by Tavares [16], Alpuim [4], Alpuim et al [2] characterized stationary max-AR(1) processes, and Alpuim et al [3] study maxautoregressive processes and the Markov property in extreme value theory. Extremes of Markov chains have been considered by Perfekt [11] and Smith [14], while Smith et al [15] consider Markov chain models for threshold exceedances (see the monograph by Beirlant et al section 10.4 for further discussion on extremes and Markov chains).…”

“…Theorem 1. Any stationary simple max-stable process η = (η(t)) t∈Z satisfying the Markov property is equal in distribution to a max-AR(1) process (1) or to a time-reversed max-AR(1) process (2).…”

Minification processes with discrete marginals

“…It is interesting to note from (3), that if FE [F0 then F belongs to [Foj for every positive integer j. Littlejohn (1992), Adke and Balakrishna (1992), Alpuim (1989), Alpuim and Athayde (1990) and Chernick et al (1988) have also worked on related topics.…”