A Lognormal Central Limit Theorem for Particle Approximations of Normalizing Constants

Abstract

This paper deals with the numerical approximation of normalizing constants produced by particle methods, in the general framework of Feynman-Kac sequences of measures. It is well-known that the corresponding estimates satisfy a central limit theorem for a fixed time horizon $n$ as the number of particles $N$ goes to infinity. Here, we study the situation where both $n$ and $N$ go to infinity in such a way that ${\lim }_{n\to \infty }$. In this context, Pitt et al. \cite{pitt2012} recently conjectured that a lognormal central limit theorem should hold. We formally establish this result here, under general regularity assumptions on the model. We also discuss special classes of models (time-homogeneous environment and ergodic random environment) for which more explicit descriptions of the limiting bias and variance can be obtained.

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