Mathematics – Statistics Theory
Scientific paper
2008-05-21
IMS Collections 2008, Vol. 3, 318-329
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.1214/074921708000000228 the IMS Collections (http://www.imstat.org/publications/imscollec
Scientific paper
10.1214/074921708000000228
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.
Bengtsson Thomas
Bickel Peter
Li Bo
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