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Paper No. 05-5

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SD Jacka, Z Lazic and J Warren

Conditioning an Additive Functional of a Markov Chain to stay Non-Negative II: Hitting a High Level

Date: May 2005

Abstract: Let X be a continuous-time irreducible Markov chain on a finite statespace E, and let ϕ be an additive functional of X. We consider the cases where the process ϕ is oscillating and where ϕ has a negative drift. In each of these cases we condition the process X on the event that ϕ hits level y before hitting zero, and prove weak convergence of the conditioned process as y tends to ∞. In addition, we show the relation between conditioning the process ϕ with a negative drift to oscillate and conditioning it to stay non-negative until large time, and the relation between conditioning a ϕ with negative drift to drift to ∞ and conditioning it to hit large levels before hitting zero.