Abstract
Abstract In this paper, we study the problem of a variety of nonlinear time series model X n+1= F(X n , e n+1(Z n+1)) in which {Z n+1} is a Markov chain with finite state space, and for every state i of the Markov chain, {e n (i)} is a sequence of independent and identically distributed random variables. Also, the existence of the stationary distribution of the sequence {X n } defined by the above model is investigated. Some new novel results on the underlying models are presented. 2010 Mathematics Subject Classification: 60J10
Highlights
1 Introduction It is known that stochastic difference equations provide models that represent a broad class of discrete-time stochastic systems, and a unified representation leads to the following general model: Xn+1 = F(Xn, en+1), n ≥ 0, (1:1)
Let (, F, P) be a probability space, and (Rq, Bq) be a measurable space, where Rq is a q-dimensional real space, and Bq is the s-algebra consisting of all Boreal subsets of Rq. μq denotes Lebesgue measure on (Rq, Bq)
Let {Zn, n ≥ 1} be an irreducible, aperiodic, and time homogeneous Markov chain, which values on state space (E, F) and its probability space is (, F, P)
Summary
Where F : Rq × Rq ↦ Rq is a Boreal measurable mapping, {en} is a sequence of independent and identically distributed q-dimensional random vectors on a probability space ( , F , P). {en(1)}, ..., {en(m)} are i.i.d random vector sequences, which value on state space (Rq, Bq) and are defined on ( , F , P) They are mutually independent and ∀i Î E, {Zn} is m independent of {en(i)}. (ii) If X0 ~ π, and π is the invariant distribution of model (1.2), the sequence {Xn} which is generated iteratively by (1.2) and started from the initial value X0, is called a stationary solution of the model (1.2). We call a probability measure π defined on F is a stationary distribution for {Xn}, if the following equality holds: to any A ∈ F , π (A) = π (dx)P(x, A).
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