Explicit Rates of Convergence of Stochastically Ordered Markov Chains*

Abstract

Let $\Phi = \left\{ {{\Phi _n}} \right\}$be a Markov chain on a half-line [0, ∞) that is stochastically ordered in its initial state. We find conditions under which there are explicit bounds on the rate of convergence of the chain to a stationary limit π: specifically, for suitable rate functions r which may be geometric or subgeometric and “moments” f ≥ 1, we find conditions under which

$$ r\left( n \right)\mathop {\sup }\limits_{|g| \leqslant f} |{E_x}\left[ {g\left( {{\Phi _n}} \right)} \right] - \pi \left( g \right)| \leqslant M\left( x \right) $$

for all n and all x. We find bounds on r ( n ) and M(x) both in terms of geometric and subgeometric “drift functions”, and in terms of behaviour of the hitting times on {0} and on compact sets [0 c ] for c > 0. The results are illustrated for random walks and for a multiplicative time series model.

Article Footnote

Work supported in part by NSF Grants DMS-9205687 and DMS-9504561