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We prove that the maximum of the sample importance weights in ahigh-dimensional Gaussian particle filter converges to unity unless theensemble size grows exponentially in the system dimension. Our work ismotivated by and parallels the derivations of Bengtsson, Bickel and Li (2007);however, we weaken their assumptions on the eigenvalues of the covariancematrix of the prior distribution and establish rigorously their strongconjecture on when weight collapse occurs. Specifically, we remove theassumption that the nonzero eigenvalues are bounded away from zero, which,although the dimension of the involved vectors grow to infinity, essentiallypermits the effective system dimension to be bounded. Moreover, with somerestrictions on the rate of growth of the maximum eigenvalue, we relax theirassumption that the eigenvalues are bounded from above, allowing the system tobe dominated by a single mode.

Sharp failure rates for the bootstrap particle filter in high dimensions