Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1403.6585

Moment Conditions for Convergence of Particle Filters with Unbounded Importance Weights

Isambi S. Mbalawata·Simo Särkkä
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we derive moment conditions for particle filter importance weights, which ensure that the particle filter estimates of the expectations of bounded Borel functions converge in mean square and $L^4$ sense, and that the empirical measure of the particle filter converges weakly to the true filtering measure. The result extends the previously derived conditions by not requiring the boundedness of the importance weights, but only boundedness of second or fourth order moments. We show that the boundedness of the second order moments of the weights implies the convergence of the estimates bounded functions in the mean square sense, and the $L^4$ convergence as well as the empirical measure convergence are assured by the boundedness of the fourth order moments of the weights. We also present an example class of models and importance distributions where the moment conditions hold, but the boundedness does not. The unboundedness in these models is caused by point-singularities in the weights which still leave the weight moments bounded. We show by using simulated data that the particle filter for this kind of model also performs well in practice.

Explore related subjects

math.DSmath.STstat.MEstat.TH

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Isambi S. Mbalawata, Simo Särkkä. 2014-08-18. Moment Conditions for Convergence of Particle Filters with Unbounded Importance Weights. https://arxiv.org/abs/1403.6585

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗
arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗
arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗