Remarks on a monotone Markov chain

P. Todorovic

International Journal of Stochastic Analysis, 1987, vol. 1, 1-18

Abstract:

In applications, considerations on stochastic models often involve a Markov chain { ζ n } 0 ∞ with state space in R + , and a transition probability Q . For each x R + the support of Q ( x , . ) is [ 0 , x ] . This implies that ζ 0 ≥ ζ 1 ≥ … . Under certain regularity assumptions on Q we show that Q n ( x , B u ) → 1 as n → ∞ for all u > 0 and that 1 − Q n ( x , B u ) ≤ [ 1 − Q ( x , B u ) ] n where B u = [ 0 , u ) . Set τ 0 = max { k ; ζ k = ζ 0 } , τ n = max { k ; ζ k = ζ τ n − 1 + 1 } and write X n = ζ τ n − 1 + 1 , T n = τ n − τ n − 1 . We investigate some properties of the imbedded Markov chain { X n } 0 ∞ and of { T n } 0 ∞ . We determine all the marginal distributions of { T n } 0 ∞ and show that it is asymptotically stationary and that it possesses a monotonicity property. We also prove that under some mild regularity assumptions on β ( x ) = 1 − Q ( x , B x ) , ∑ 1 n ( T i − a ) / b n → d Z ∼ N ( 0 , 1 ) .

Date: 1987
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:802803

DOI: 10.1155/S1048953388000103

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