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A note on speed of convergence to the quasi-stationary distribution

Polak M., Rolski T.

Demonstratio Mathematica

2012

Vol. 45, nr 2

385-397

In this note we show that for Z being a birth and death process on Z or Brownian motion with drift and (…), the speed of convergence to the quasi-stationary distribution is of order 1/t. The corresponding version that X is the number of calls in M/M/1 queue or the reflected Brownian motion is also considered. The result is obtained by asymptotic expansions of some transition functions. For this we use some new asymptotic expansion of the Bessel function.

Markov processes conditioned to never exit a subspace of the state space

Palmowski Z., Rolski T.

Probability and Mathematical Statistics

2004

Vol. 24, Fasc. 2

339--353

In this paper we study Markov processes never exiting (NE) a subspace A of the state space E or, in other words, Markov processes conditioned to stay in the subspace A. We show how the knowledge of the exact asymptotics of the tail distribution of the exit time helps to find the suitable exponential martingale, which, in turn, serves for the change of measure. Under the new probability measure the process is the sought for never exiting one the subspace A. We also find its extended generator and study relationships between the invariant measure (INE) and the quasi-stationary (QS) distribution. We analyze in detail the PDMP processes.