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 X0 is any random variable, {Zn} and {Yn} are independent sequences of i.i.d. random variables both independent of X0. 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.

Random Dynamical Systems on Ordered Topological Spaces

“…the case of a finite invariant measure, as treated in Section 6. There are numerous studies considering E = R + with its total order and random transformations preserving this order as, for instance, Alpuim/Athayde [1], Goldie [19], Helland/Nilsen [21], Lund et al [30], Yahav [40]. There is, moreover, a number of papers dealing also with only partially ordered state spaces as Bougerol/Picard [10], Diaconis/Freedman [12, Section 3], Glasserman/Yao [18], Jarner/Tweedie [23], Mairesse [31], Rachev/Samorodnitsky [34], Rachev/Todorovic [35], and in particular Brandt et al [11,Section 1.3] (where, however, a metric compatible with the order is postulated).…”

On the Extremal Behaviour of Some Stationary Markov Sequence

“…Let H n (x) denote the marginal distribution function of X n , n = 0,1, 2,... defined by (1). It is easy to obtain that H n (x) = F(x)\H n .…”

“…and X is a random variable with distribution function F then E(X + ) 1 < oo, 0 < 7 < a, where X+ = max(X,0).…”

“…Lei {X n ,n > 1} 6e defined by (1) where B\ is unbounded above. Let {X n , n > 1} /ias a siafe space 5 sucA i/iai sup S = 00 and /eZ If a n > 0 are constants such that limn-too n(l -F(a n x)) = x a , then for each k = 1,2,..., (8) lim P(Mf < a n r«i)…”

“…THEOREM 2. Let {X n ,n > 1} be defined by (1) where B\ is unbounded above. Suppose that 1 -F is regularly varying at oo with index -a < 0.…”