Blanchet, J., and Chen, X., Rates of Convergence to Stationarity for Multidimensional RBM. To appear in Mathematics of Operations Research.

Summary:

This is the first paper that studied rates of convergence to stationarity as the dimension increases for multidimensional reflected Brownian motion. The result was subsequently (and substantially!) improved, building on the methodology introduced in this paper, by Banarjee and Budhiraja (2019). The strategy is to use an asynchronous coupling with a stationary version of the process. A key observation involves noting that every time a “tour” is completed, the distance to stationarity contracts by a fixed factor. A tour is completed when we can guarantee that each coordinate has hit zero at least once. We then use an upper bound process which has orthogonal reflections to estimate the times to complete tours and a Lyapunov bound applied to the upper bound process. It is right at this point where we incur in a rather suboptimal estimate, which Banarjee and Budhiraja (2019) fix by constructing a substantially improved Lyapunov function.

Bibtex:

@Article{BC_RBMrc_2019,

author = { J. Blanchet and Chen, X.},

title = {Rates of convergence to stationarity for multidimensional RBM.},

journal = {Mathematics of Operations Research},

year = {2020},

volume = {to appear},

% pages = {271-276}

}