Erratum to A New Class of Particle Filters for Random Dynamic Systems with Unknown Statistics
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Erratum to “A New Class of Particle Filters for Random Dynamic Systems with Unknown Statistics” ´ Joaqu´ın M´ıguez,1 Monica F. Bugallo,2 and Petar M. Djuri´c2 1 Departamento 2 Department
de Teor´ıa de la Se˜nal y las Comunicaciones, Universidad Carlos III de Madrid, 28911 Leganes, Spain of Electrical and Computer Engineering, Stony Brook University, Stony Brook, NY 11794, USA
Received 28 August 2005; Accepted 9 November 2005 Recommended for Publication by Marc Moonen We have found an error in the proof of Lemma 1 presented in our paper “A New Class of Particle Filters for Random Dynamic Systems with Unknown Statistics” (EURASIP Journal on Applied Signal Processing, 2004). In the sequel, we provide a restatement of the lemma and a corrected (and simpler) proof. We emphasize that the original result in the said paper still holds true. The only difference with the new statement is the relaxation of condition (3), which becomes less restrictive. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
Lemma 1 in [1] should be as follows.
then
Lemma 1. Let {xt(i) }M i=1 be a set of particles drawn at time t using the propagation pdf ptM (x), let y1:t be a fixed bounded sequence of observations, let ΔC(x | yt ) ≥ 0 be a continuous opt cost function, bounded in S{xt , ε}, with a minimum at x = opt (i) M xt , and let μt : A ⊆ {xt }i=1 → [0, ∞) be a set function defined as
M
μt A ⊆ xt(i)
i=1
=
μ ΔC x | yt .
(1)
x∈A
lim Pr 1 −
ptM (x)dx = γ > 0,
∀ε > 0,
(2)
(2) the supremum of the function μ(ΔC(· | ·)) for points opt outside S(xt , ε) is a finite constant, that is, Sout =
sup opt
μt xt(i)
lim
μ ΔC xt | yt
< ∞,
(3)
lim E
M i=1
= 0,
∀δ > 0,
(5)
i=1
opt
=1 (i) M
μt xt
(i.p.),
(6)
i=1
where i.p. stands for “in probability.”
lim Pr 1 −
M →∞
/M
opt
μt SM xt , ε
μt xt(i)
= lim Pr
μt
M →∞
= lim Pr
M ≥ δ i=1
μt xt(i)
≥δ
(7)
opt \ SM xt , ε ≥δ . (i) M i=1
μt xt
i=1
Using the second condition, we infer that lim Pr
(4)
M
μt xt(i)
opt \ SM xt , ε ≥δ (i) M
M
i=1
μt xt
= 0,
M opt xt(i) i=1 − μt SM xt , ε (i) M μt xt i=1
M →∞
M →∞
1
μt xt(i)
M ≥ δ
(3) the expected value of 1/μt ({xt(i) }M i=1 ) satisfies
M →∞
μt SM xt , ε
M →∞
xt ∈RLx \S(xt ,ε)
opt
where Pr[·] denotes probability, that is,
has a nonzero probability (1) Any ball with center at under the propagation density, that is, opt
Proof. The proof is based on Markov inequality. We write
opt xt
S{xt ,ε}
μt SM xt , ε
M →∞
If the following three conditions are met:
i=1
MSout ≤ lim Pr M ≥ δ . M →∞ μt xt(i) i=1
(8)
2
EURASIP Journal on Applied Signal Processing
Finally, we apply Markov inequality to the last expression on the right and obtain
lim Pr
μt xt(i)
M →∞
opt \ SM xt , ε ≥δ (i) M
M
i=1
μt xt
i=1
Sout 1 ≤ lim E M δ M →∞ μt xt(i) i=1 /M
(9)
.
Clearly, if
lim
M →∞
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