  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 $\lim_{n\rightarrow\infty}% n/N=\alpha>0$. 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.
