Asymptotics of a Class of Markov Processes Which Are Not in General Irreducible

Stochastic stability in monotone economies

“…On the splitting condition, see, e.g., Bhattacharya and Lee (1988) or Bhattacharya and Majumdar (2001).…”

“…For this reason, mixing properties tend to be related to contraction mapping arguments (because contractions are operators that map distinct points closer together). In fact, at least for the Euclidean case, the existence, uniqueness and stability results of Hopenhayn and Prescott (1992) can all be obtained simultaneously via Banach's contraction mapping theorem (see Bhattacharya and Lee 1988).…”

“…Studies of monotone Markov theory in the mathematical literature include Dubins and Freedman (1966), Yahav (1975), Bhattacharya and Lee (1988), Heikkila and Salonen (1996), Chueshov (2002), and Bhattacharya et al (2010).…”

“…As clarified below, our order reversing condition is weaker than this monotone mixing condition. The papers by Dubins and Freedman (1966), Bhattacharya and Lee (1988), Bhattacharya and Majumdar (2001), and Bhattacharya et al (2010) analyze stability in the monotone setting via a mixing condition called splitting. Our order reversing condition is also weaker than splitting.…”

Estimating the stationary distribution of a Markov chain

“…When the Markov Chain {X n } is Harris irreducible (see Orey [1971]), with respect to some nontrivial -finite measure, (this includes irreducible countable state space Markov Chains) the techniques of regeneration due to Athreya and Ney [1978] and Nummelin [1978] can be exploited to find such conditions (see the books by Nummelin [1984], and Meyn and Tweedie [1993]). However, there are many Markov Chains that are generated by iterations of independent identically distributed random maps (also known as Iterated Function Systems (IFS)) that are in general not irreducible (see Bhattacharya and Lee (1988), Athreya and Stenflo (2000)). This is especially true when the IFS consists of a finite or countable number of maps and the stationary distribution turns out to be a nonatomic one.…”