## Institute of Mathematical Statistics Collections

- Collections
- Volume 3, 2008, 318-329

### Sharp failure rates for the bootstrap particle filter in high dimensions

Peter Bickel, Bo Li, and Thomas Bengtsson

#### Abstract

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.

#### Chapter information

**Source***Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2008)

**Dates**

First available in Project Euclid: 28 April 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.imsc/1209398477

**Digital Object Identifier**

doi:10.1214/074921708000000228

**Mathematical Reviews number (MathSciNet)**

MR2459233

**Subjects**

Primary: 93E11: Filtering [See also 60G35] 62L12: Sequential estimation 86A22: Inverse problems [See also 35R30] 60G50: Sums of independent random variables; random walks 86A32: Geostatistics 86A10: Meteorology and atmospheric physics [See also 76Bxx, 76E20, 76N15, 76Q05, 76Rxx, 76U05]

**Keywords**

Bayesian filter curse of dimensionality ensemble forecast ensemble methods importance sampling large deviations Monte Carlo numerical weather prediction sample size requirements state-space model

**Rights**

Copyright © 2008, Institute of Mathematical Statistics

#### Citation

Bickel, Peter; Li, Bo; Bengtsson, Thomas. Sharp failure rates for the bootstrap particle filter in high dimensions. Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, 318--329, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/074921708000000228. https://projecteuclid.org/euclid.imsc/1209398477

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