  It has been widely realized that Monte Carlo methods (approximation via a
sample ensemble) may fail in large scale systems. This work offers some
theoretical insight into this phenomenon in the context of the particle filter.
We demonstrate that the maximum of the weights associated with the sample
ensemble converges to one as both the sample size and the system dimension
tends to infinity. Specifically, under fairly weak assumptions, if the ensemble
size grows sub-exponentially in the cube root of the system dimension, the
convergence holds for a single update step in state-space models with
independent and identically distributed kernels. Further, in an important
special case, more refined arguments show (and our simulations suggest) that
the convergence to unity occurs unless the ensemble grows super-exponentially
in the system dimension. The weight singularity is also established in models
with more general multivariate likelihoods, e.g. Gaussian and Cauchy. Although
presented in the context of atmospheric data assimilation for numerical weather
prediction, our results are generally valid for high-dimensional particle
filters.
