Statistics – Computation
Scientific paper
2012-02-29
Statistics
Computation
Scientific paper
Perron-Frobenius theory treats the existence of a positive eigen-vector associated with the principal eigen-value \lambda_{\star} of a non-negative matrix, say Q . A simple method for approximating this eigen-vector involves computing the iterate \lambda_{\star}^{-n}Q^{(n)}, for large n. In the more general case that Q is a non-negative integral kernel, an extended Perron-Frobenius theory applies, but it is typical that neither the principal eigen-function nor the iterate \lambda_{\star}^{-n}Q^{(n)} can be computed exactly. In this setting we propose and study an interacting particle algorithm which yields a numerical approximation of the principal eigen-function and the associated twisted Markov kernel. We study a collection of random integral operators underlying the algorithm, address some of their mean and path-wise properties, and obtain L_{r} error estimates. Examples are provided in the context of a classical neutron model studied by Harris, a Bellman optimality equation and a rare event estimation problem. For the rare event problem we show how the proposed algorithm allows unbiased approximation of a Markov importance sampling method by conditional simulation.
Kantas Nikolas
Whiteley Nick
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