  We prove that the maximum of the sample importance weights in a
high-dimensional Gaussian particle filter converges to unity unless the
ensemble size grows exponentially in the system dimension. Our work is
motivated by and parallels the derivations of Bengtsson, Bickel and Li (2007);
however, we weaken their assumptions on the eigenvalues of the covariance
matrix of the prior distribution and establish rigorously their strong
conjecture on when weight collapse occurs. Specifically, we remove the
assumption that the nonzero eigenvalues are bounded away from zero, which,
although the dimension of the involved vectors grow to infinity, essentially
permits the effective system dimension to be bounded. Moreover, with some
restrictions on the rate of growth of the maximum eigenvalue, we relax their
assumption that the eigenvalues are bounded from above, allowing the system to
be dominated by a single mode.
