A stationary, rst order Markov process is generated from p(xtxt 1)

given by xt where N(xt xt 1 v) with probability N(xt0s)

otherwise,

<1 s=v (1 2)andfor some probability ; in the appli cation of interest, is relatively large, such as 0.9.

(a) What is the distribution of (xtxt 1)?

(b) What is the conditional mean E(xtxt 1) of this state transition dis tribution?

(c) Show that the conditional variance V (xtxt 1) depends quadratically on xt 1 Interpret this.

(d) The process is stationary. What is the marginal distribution p(xt) for all t?

(e) Comment on the forms of trajectories generated by such a model, and speculate on possible applications. Use simulation examples to help generate insights. Discuss how it di ers from the basic, normal AR(1)
model (the special case = 1).

(f) Is the process reversible? Either prove or disprove.