A GENERAL THEORY OF PARTICLE FILTERS IN HIDDEN MARKOV MODELS AND SOME APPLICATIONS. By Hock Peng Chan National University of Singapore and

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1 Subitted to the Annals of Statistics A GENERAL THEORY OF PARTICLE FILTERS IN HIDDEN MARKOV MODELS AND SOME APPLICATIONS By Hock Peng Chan National University of Singaore and By Tze Leung Lai Stanford University By aking use of artingale reresentations, we derive the asytotic norality of article filters in hidden Markov odels and a relatively sile forula for their asytotic variances. Although reeated resalings result in colicated deendence aong the sale aths, the asytotic variance forula and artingale reresentations lead to consistent estiates of the standard errors of the article filter estiates of the hidden states. 1. Introduction. Let X = {X t,t 1} be a Markov chain and let Y 1,Y 2,...be conditionally indeendent given X, such that (1.1) X t t ( X t 1 ), Y t g t ( X t ), in which t and g t are density functions with resect to soe easures ν X and ν Y, and 1 ( X 0 ) denotes the initial density 1 ( ) ofx 1. Gordon, Salond and Sith [12] introduced article filters for Monte Carlo estiation of E[ψ(X T ) Y 1,...,Y T ]. More generally, letting X t =(X 1,...,X t ), Y t =(Y 1,...,Y t ), and ψ be a easurable real-valued function of X T,we consider estiation of ψ T := E[ψ(X T ) Y T ]. The density function of X T conditional on Y T in the above hidden Markov odel (HMM) is T (1.2) T (x T Y T ) [ t (x t x t 1 )g t (Y t x t )]. t=1 However, this conditional distribution is often difficult to sale fro and the noralizing constant is also difficult to coute for high-diensional Suorted by the National University of Singaore grant R Suorted by the National Science Foundation grant DMS AMS 2000 subject classifications: Priary 60F10, 65C05; secondary 60J22, 60K35 Keywords and hrases: article filter, iortance saling and resaling, standard error 1

2 2 CHAN AND LAI or colicated state saces. Particle filters that use sequential Monte Carlo (SMC) ethods involving iortance saling and resaling have been develoed to circuvent this difficulty. Asytotic norality of the article filter estiate of ψ T when the nuber of siulated trajectories becoes infinite has also been established [6, 8, 9, 13]. However, no exlicit forulas are available to estiate the standard error of ψ T consistently. In this aer, we rovide a corehensive theory of the SMC estiate ψ T, which includes asytotic norality and consistent standard error estiation. The ain results are stated in Theores 1 and 2 in Section 2 for the case in which bootstra resaling is used, and extensions are given in Section 4 to residual Bernoulli (instead of bootstra) resaling. The roof of Theore 1 for the case where bootstra resaling is used at every stage, given in Section 3.2, roceeds in two stes. First we assue that the noralizing constants (i.e. constants of roortionality in (1.2)) are easily coutable. We call this the basic rototye, for which we derive in Section 3.1 artingale reresentations of ( ψ T ψ T ) for novel SMC estiates ψ T that involve likelihood ratios and general resaling weights. We first encountered this rototye in connection with rare event siulation using SMC ethods in [5]. Although traditional article filters also use siilar sequential iortance saling rocedures and resaling weights that are roortional to the likelihood ratio statistics, the estiates of ψ T are coarser averages that do not have this artingale roerty. Section 3.2 gives an asytotic analysis of ( ψ T ψ T )as in this case, by aking use of the results in Section 3.1. In contrast to Gilks and Berzuini [11] who use article set sizes t which increase with t and aroach in a certain way to guarantee consistency of their standard error estiate which differs fro ours, we use the sae article set size for every t, where is the nuber of SMC trajectories. As noted in [11,.132], the ode of convergence of the theore is not directly relevant to the ractical context in which 1 = 2 = =. A ajor reason why we are able to overcoe this difficulty in develoing a consistent standard error estiate lies in the artingale aroxiation for our SMC estiate ψ T. We note in this connection that for the basic rototye, although both artingale reresentations of ( ψ T ψ T ) develoed in Section 3.1 can be used to rove the asytotic norality of ψ T as the nuber of SMC trajectories aroaches, only one of the leads to an estiable exression for the asytotic variance. Further discussion of the ain differences between the traditional and our article filters is given in Section 5.

3 PARTICLE FILTER 3 2. Standard error estiation for bootstra filters. Since Y 1,Y 2,... are the observed data, we will treat the as constants in the sequel. Let q t ( x t 1 ) be a conditional density function with resect to ν X such that q t (x t x t 1 ) > 0 whenever t (x t x t 1 ) > 0. In the case t =1,q t ( x t 1 ) = q 1 ( ) Bootstra resaling at every stage. Let (2.1) w t (x t )= t (x t x t 1 )g t (Y t x t )/q t (x t x t 1 ). To estiate ψ T, a article filter first generates conditionally indeendent rando variables X 1 t,..., X t at stage t, with X i t having density q t( X i t 1 ), to for X i t =(X i t 1, X i t) and then use noralized resaling weights (2.2) Wt i = w t ( X / i t) w t ( X j t) to draw sale aths X j t,1 j, fro { X i t, 1 i }. Note that Wt i are the iortance weights attached to X i t and that after resaling the X i t have equal weights. The following recursive algorith can be used to ileent article filters with bootstra resaling. It generates not only the X i T but also the ancestral origins Ai T 1 that are used to coute the standard error estiates. It also coutes H t i and Ht i recursively, where (2.3) H t i = w 1 w t h t ( X i t), Ht i = w 1 w t h t (X i t), w k = w k ( X / t j k /, ) h t (x t )=1 w k (x k ). k=1 Initialization: Let A i 0 = i and H i 0 = 1 for all 1 i. Iortance saling at stage t = 1,...,T: Generate conditionally indeendent X i t fro q t ( X i t 1), 1 i. Bootstra resaling at stage t = 1,...,T 1: Generate i.i.d. rando variables B 1 t,...,b t such that P (B 1 t = j)=w j t and let (X j t,a j t,h j t )=( X Bj t t,a Bj t t 1, H Bj t t )for1 j. The SMC estiate is defined by (2.4) ψ T =( w T ) 1 ψ( X i T)w T ( X i T ).

4 4 CHAN AND LAI Let E T denote exectation with resect to the robability easure under which X T has density (1.2), and let E q denote that under which X t X t 1 has the conditional density function q t for all 1 t T. In Section 3.2 we rove the following theore on the asytotic norality of ψ T and the consistency of its standard error estiate. Define [ t / t (2.5) η t = E q w k (X k ) ], h t (x t)=η t w k (x k ), ζ t = η t 1 /η t, Γ t (x t )= k=1 t k=1 [w k (x k )+w 2 k (x k)], k=1 with the convention η 0 = 1 and h 0 Γ 0 1. Let / T (2.6) L T (x T )= T (x T Y T ) q t (x t x t 1 ). Let f 0 = 0 and define for 1 t T, (2.7) f t (x t )[= f t,t (x t )] = E q {[ψ(x T ) ψ T ]L T (X T ) X t = x t }. Theore 1. Assue that σ 2 <, where σ 2 = 2T 1 k=1 σ2 k and (2.8) σ2t 1 2 = E q {[ft 2 (X t ) ft 1(X 2 t 1 )]h t 1(X t 1 )}, σ2t 2 = E q [ft 2 (X t )h t (X t )]. Then ( ψ T ψ T ) N(0,σ 2 ). Moreover, if E q Γ T (X T ) <, then σ 2 ( ψ T ) σ 2, where for any real nuber µ, (2.9) σ 2 (µ) = 1 ( w T ( Xi T ) [ψ( w X ) 2. i T) µ] i:a i T T 1 =j 2.2. Extension to occasional resaling. Due to the coutational cost of resaling and the variability introduced by the resaling ste, occasional resaling has been recoended, e.g., by Liu [14, Chater 3.4.4] who roose to resale at stage k only when cv 2 k, the coefficient of variation of the resaling weights at stage k, exceeds soe threshold. Let τ 1 <τ 2 < <τ r be the successive resaling ties and let τ(k) be the ost recent resaling tie before stage k, with τ(k) = 0 if no resaling has been carried out before tie k. Define k (2.10) v k (x k )= w t (x t ), Vk i = v k ( X / i k) ( v k ), t= τ(k)+1 t=1

5 PARTICLE FILTER 5 where v k = 1 v k ( X j k ). Note that if bootstra resaling is carried out at stage k, then Vk i,1 i, are the resaling weights. Define the SMC estiator (2.11) ψ OR = VT i ψ( X i T). In Section 4.3, we show that the resaling ties, which resale at stage k when cv 2 k exceeds soe threshold, satisfy (2.12) r r and τ s τ s for 1 s r, as, for soe nonrando ositive integers r,τ 1,..., and τ r. In Section 4.2 we rove the following extension of Theore 1 to ψ OR and give an exlicit forula (4.6), which is assued to be finite, for the liiting variance σ 2 OR of ( ψ OR ψ T ). Theore 2. Under (2.12), ( ψ OR ψ T ) N(0,σOR 2 ) as. Moreover, if E q Γ T (X T ) <, then σ OR 2 ( ψ OR ) σor 2, where for any real nuber µ, σ OR 2 (µ) ( v T ( Xi = 1 T ) [ψ( v X ) 2. i T ) µ] i:a i τr =j T 2.3. A nuerical study. Yao [16] has derived exlicit forulas for ψ T = E(X T Y T ) in the noral ean shift odel {(X t,y t ) : t 1} with the unobserved states X t generated recursively by { Xt 1 with rob. 1 ρ, X t = Z t with rob. ρ, where 0 <ρ<1 is given and Z t N(0,ξ) is indeendent of X t 1. The observations are Y t = X t + ɛ t, in which the ɛ t are i.i.d. standard noral, 1 t T. Instead of X T, we siulate using the bootstra filter the changeoint indicators I T := (I 1,...,I T ), where I t = 1 {Xt X t 1 } for t 2 and I 1 = 1; this technique of siulating E(X T Y T ) via realizations of E(X T I T, Y T ) is known as Rao-Blackwellization. Let C t = ax{j t : I j =1} denote the ost recent change-oint u to tie t, λ t =(t C t +1+ξ 1 ) 1 and µ t = λ t ti=ct Y i. It can be shown that E(X T I T, Y T )=µ T and that for t 2, P {I t =1 I t 1, Y t } = a(ρ)/{a(ρ)+b(ρ)}, where a(ρ) =ρφ 0,1+ξ (Y t ) and b(ρ)=(1 ρ)φ µt 1,1+λ t 1 (Y t ), in which φ µ,v denotes the N(µ, V ) density function.

6 6 CHAN AND LAI Table 1 Siulated values of ψor ± σ OR( ψor)/. An italicized value denotes that ψ T is outside the interval but within two standard errors of ψor; a boldfaced value denotes that ψ T is ore than two standard errors away fro ψor. c T = 200 T = 400 T = 600 T = 800 T = ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±.0047 ψ T The results in Table 1 are based on a single realization of Y 1000 with ξ =1andρ =.01. We use article filters to generate {I i 1000 :1 i } for = 10,000 and coute ψ OR for T =200, 400, 600, 800, 1000, using bootstra resaling when cv 2 t c, for various thresholds c. The values of σ OR ( ψ OR ) in Table 1 suggest that variability of ψ OR is iniized when c = 2. Aong the 40 siulated values of ψ OR in Table 1, 24 fall within one estiated standard error (= σ OR ( ψ OR )/ ) and 39 within two estiated standard errors of ψ T. This agrees well with the corresonding nubers 27 and 39, resectively, given by the central liit theore in Theore 2. Table 2 reorts a larger siulation study involving 500 realizations of Y 1000, with an indeendent run of the article filter for each realization. The table shows confidence intervals for each T, with resaling threshold c = 2. The results for σ OR ( ψ OR ) agree well with the coverage robabilities of.683 and.954 given by the liiting standard noral distribution in Theore 2. Table 2 also shows that the Gilks-Berzuini estiator V T described in Section 5 is Table 2 Fraction of confidence intervals ψor ± σ/ containing the true ean ψ T. σ 2 = σ OR( 2 ψor) σ 2 = VT T 1se 2se 1se 2se

7 PARTICLE FILTER 7 very conservative. 3. Martingales and roof of Theore 1. In this section, we first consider in Section 3.1 the basic rototye entioned in the second aragrah of Section 1, for which ( ψ T ψ T ) can be exressed as a su of artingale differences for a new class of article filters. We cae u with this artingale reresentation, which led to a standard error estiate for α T, in [5] where we used a article filter α T to estiate the robability α of a rare event that X T belongs to Γ. Unlike the resent setting of HMMs, the X T is not a vector of hidden states in [5], where we showed that ( α T α) is a artingale, thereby roving the unbiasedness of α T and deriving its standard error. In Section 3.1 we extend the arguents for ( α T α) in[5] to ( ψ T ψ T ) under the assution that the constants of roortionality in (1.2) are easily coutable, which we have called the basic rototye in Section 1. We then aly the results of Section 3.1 to develo in Section 3.2 artingale aroxiations for traditional article filters and use the to rove Theore Martingale reresentations for the basic rototye. We consider here the basic rototye, which requires T to be fixed, in a ore general setting than HMM. Let (X T, Y T ) be a general rando vector, with only Y T observed, such that the conditional density T (X T Y T )ofx T given Y T is readily coutable. As in Section 2, since Y T reresents the observed data, we can treat it as constant and sily denote T (x T Y T )by T (x T ). The bootstra resaling schee in Section 2.1 can be alied to this general setting and we can use general weight functions w t (x t ) > 0, of which (2.1) is a secial case for the HMM (1.1). Because T (x T ) is readily coutable, so is the likelihood ratio L T (x T )= T (x T )/ T t=1 q t (x t x t 1 ). This likelihood ratio lays a fundaental role in the theory of the article filter estiate α T of the rare event robability α in [5]. An unbiased estiate of ψ T under the basic rototye is (3.1) ψ T = 1 L T ( X i T )ψ( X i T )Hi T 1, where Ht i is defined in (2.3). The unbiasedness follows fro the fact that ( ψ T ψ T ) can be exressed as a su of artingale differences. There are in fact two artingale reresentations of ( ψ T ψ T ). One is closely related to that of Del Moral [8, Chater 9]; see (3.11) below. The other is related to the variance estiate σ 2 ( ψ T ) in Theore 1 and is given in Lea 1. Recall the eaning of the notation E q introduced in the second aragrah

8 8 CHAN AND LAI of Section 2.1. Let # i k denote the nuber of coies of X i k generated fro { X 1 k,..., X k } to for the articles in the kth generation. Then conditionally, (# 1 k,...,# k ) Multinoial(, W k 1,..., W k ). Lea 1. Let f 0 = ψ T and define for 1 t T, (3.2) f t (x t )=E q [ψ(x T )L T (X T ) X t = x t ]. Then f t (x t )=E q [ f T (X T ) X t = x t ] and (3.3) ( ψ T ψ T )= (ɛ j ɛj 2T 1 ), where (3.4) ɛ j 2t 1 = ɛ j 2t = [ f t ( X i t ) f t 1 (X i t 1 )]Hi t 1, i:a i t 1 =j (# i t Wt i )[ f t ( X i t) H t i f 0 ]. i:a i t 1 =j Moreover, for each fixed j, {ɛ j k, F k, 1 k 2T 1} is a artingale difference sequence, where (3.5) F 2t 1 = σ({ X 1 i :1 i } {(X i s, X i s+1,a i s):1 s<t,1 i }), F 2t = σ(f 2t 1 {(X i t,a i t):1 i }) are the σ-algebras generated by the rando variables associated with the articles just before and just after the tth resaling ste, resectively. Proof. Recalling that the first generation of the articles consists of X 1,..., 1 X 1 (before resaling), A i t = j if the first coonent of X i t is X j 1. Thus Ai t reresents the ancestral origin of the genealogical article X i t. It follows fro (2.2), (2.3) and sile algebra that for 1 i, (3.6) (3.7) Wt i = Ht 1/ i H t, i f t (X i t)h i t = # i f t t ( X i t) H t, i i:a i t =j i:a i t 1 =j

9 PARTICLE FILTER 9 noting that X i t has the sae first coonent as X i t 1. Multilying (3.6) by f t ( X i t) H t i and cobining with (3.7) yield (3.8) T t=1 i:a i t 1 =j [ f t ( X i t) f t 1 (X i t 1)]H i t 1 T 1 + t=1 i:a i t 1 =j (# i t W i t ) f t ( X i t) H i t = i:a i T 1 =j f T ( X i T )H i T 1 f 0, recalling that A i 0 = i and H i 0 = 1. Since f 0 = ψ T and f T (x T )=ψ(x T )L T (x T ) by (3.2), it follows fro (3.1) that (3.9) ( ψ T ψ T )= f T ( X i T )H i T 1 f 0. Since # i t = W i t = for 1 t T 1, cobining (3.9) with (3.8) yields (3.3). Let E denote exectation under the robability easure induced by the rando variables in the SMC algorith with articles. Then E (ɛ j 2 F 1)=E {(# j 1 W j 1 ) X 1 1,..., X 1 }[ f 1 ( X j 1 ) H j 1 f 0 ]=0 since E (# j 1 F 1)=W j 1. More generally, the conditional distribution of (X 1 t,...,x t ) given F 2t 1 is that of i.i.d. rando vectors which take the value X j t with robability W j t. Moreover, the conditional distribution of ( X t 1,..., X t ) given F 2(t 1) is that of indeendent rando variables such that X t i has density function q t ( X i t 1 ), and therefore E [ f t ( X i t) F 2(t 1) ]= f t 1 (X i t 1 ) by (3.2) and the tower roerty of conditional exectations. Hence in articular, E (ɛ j 3 F 2)= {E [ f 2 ( X i 2) X1] i f 1 (X1)}H i 1 i =0. i:a i 1 =j Proceeding inductively in this way shows that {ɛ j k, F k, 1 k 2T 1} is a artingale difference sequence for all j. Without tracing their ancestral origins as in (3.4), we can also use the successive generations of the articles to for artingale differences directly and thereby obtain another artingale reresentation of ( ψ T ψ T ). Secifically, the receding arguent also shows that {(Z 1 k,...,z k ), F k, 1

10 10 CHAN AND LAI k 2T 1} is a artingale difference sequence, where (3.10) Z2t 1 i = [ f t ( X i t) f t 1 (X i t 1)]Ht 1, i Z2t i = f t (X i t)ht i W j f t t ( X j t ) H j t. Moreover, Zk 1,...,Z k are conditionally indeendent given F k 1. It follows fro (3.1), (3.6), (3.10) and an arguent siilar to (3.8) that (3.11) ( ψ T ψ T )= 2T 1 k=1 (Z 1 k + + Z k ). This artingale reresentation yields the liiting noral distribution of ( ψ T ψ T ) for the basic rototye in the following. Corollary 1. Let σ 2 C = σ σ 2 2T 1, where { (3.12) σ k 2 Eq {[ f t 2 (X t ) f t 1 2 = (X t 1)]h t 1 (X t 1)} if k =2t 1, E q {[ f t (X t )h t (X t ) f 0 ] 2 /h t (X t )} if k =2t. Assue that σc 2 <. Then as, (3.13) ( ψ T ψ T ) N(0,σ 2 C). The roof of Corollary 1 is given in the Aendix. The ain result of this section is consistent estiation of σc 2 in the basic rototye. Define for every real nuber µ, σ 2 (µ) = 1 { (3.14) L T ( X i T )ψ( X i T )Hi T 1 i:a i T 1 =j and note that by (3.4) and (3.8), [ T 1 ] 2, 1+ (# i t W t µ} i) t=1 i:a i t 1 =j σ 2 (ψ T )= 1 (ɛ j ɛj 2T 1 )2. We next show that σ 2 (ψ T ) σc 2 by aking use of the following two leas which are roved in the Aendix. Since for every fixed T as, ψ T = ψ T +O (1/ ) by (3.13), it then follows that σ 2 ( ψ T ) is also consistent for σc 2.

11 PARTICLE FILTER 11 Lea 2. Let 1 t T and let G be a easurable real-valued function on the state sace. Define h t, η t and ζ t as in (2.5). (i) If E q [ G(X t ) /h t 1 (X t 1)] <, then as, (3.15) 1 G( X i t) E q [G(X t )/h t 1(X t 1 )]. (ii) If E q [ G(X t ) /h t (X t )] <, then as, (3.16) 1 G(X i t) E q [G(X t )/h t (X t )]. In articular, for the secial case G = w t, (3.15) yields w t ζ 1 t and hence (3.17) H i t h t ( X i t) = Hi t h t (X i t) = η 1 t w 1 w t 1 as. Moreover, if E q [ G(X t ) /h t 1(X t 1 )] <, then alying (3.15) to G( ) 1 { G( ) >M} and letting M yield (3.18) 1 G( X i t) 1 { G( X i t ) >ɛ } 0 as, for every ɛ>0. Lea 3. Let 1 t T and let G be a easurable nonnegative valued function on the state sace. (i) If E q [G(X t )/h t 1(X t 1 )] <, then as, (3.19) 1 ax G( X i t ) 0. 1 j i:a i t 1 =j (ii) If E q [G(X t )/h t (X t )] <, then as, (3.20) 1 ax G(X i t) 0. 1 j i:a i t =j Corollary 2. Suose σ 2 C <. Then σ2 (ψ T ) σ 2 C as. Proof. By (3.4) and (3.8), 2T 1 k=1 ɛ j k = f T ( X i T )Hi T 1 i:a i T 1 =j [ T 1 1+ t=1 i:a (i) t 1 =j (#i t W i t ) ] f 0.

12 12 CHAN AND LAI We ake use of Leas 2 and 3, (3.12) and atheatical induction to show that for all 1 k 2T 1, (3.21) 1 (ɛ j ɛj k )2 k σ l 2. By the weak law of large nubers, (3.21) holds for k = 1. Next assue that (3.21) holds for k =2t and consider the exansion l=1 (3.22) 1 [(ɛ j ɛj k+1 )2 (ɛ j ɛj k )2 ] = 1 (ɛ j k+1 ) (ɛ j ɛj k )ɛj k+1. Let C j t = {(i, l) :i l and Ai t = A l t = j}. Suressing the subscrit t, let U i =[ f t+1 ( X i t+1) f t (X i t)]h i t. Then ɛ j 2t+1 = i:a i t =j U i and (3.23) 1 (ɛ j 2t+1 )2 = 1 Ui By (3.12), (3.17) and Lea 2(i), (3.24) 1 Ui 2 σ 2 2t+1. (i,l) C j t U i U l. Since U 1,...,U are indeendent ean zero rando variables conditioned on F 2t, it follows fro (3.17), Leas 2(ii) and 3(ii) that (3.25) ( 2 ) F2t Var U i U l (i,l) C j t = 2 (i,l) C j t [ 1 ][ E (Ui 2 F 2t) By (3.23) (3.25), E (U 2 i U 2 l F 2t ) 2 1 ax 1 j (3.26) 1 (ɛ j 2t+1 )2 σ 2t+1. 2 { E (Ui 2 F 2t ) i:a i t =j ] E (U 2 i F 2t) 0. i:a i t =j } 2

13 We next show that PARTICLE FILTER 13 (3.27) 1 (ɛ j ɛj 2t )ɛj 2t+1 0. Since ɛ j 1,...,ɛj 2t are easurable with resect to F 2t for 1 j and ɛ 1 2t+1,...,ɛ 2t+1 indeendent conditioned on F 2t, by the induction hyothesis and by (3.17) and Lea 3(ii), it follows that 2 Var ((ɛ j ɛj 2t )ɛj 2t+1 F 2t) = 2 {(ɛ j } ɛj 2t )2 E (Ui 2 F 2t ) i:a i t =j [ 1 (ɛ j ɛj 2t )2][ 1 ] ax E (U 2 i F 2t ) 0, 1 j i:a i t =j and therefore (3.27) holds. By (3.22), (3.26), (3.27) and the induction hyothesis, (3.21) holds for k =2t +1. Next assue (3.21) holds for k =2t 1. Suressing the subscrit t, let S i =(# i t Wt i )[ f t ( X i t) H t i f 0 ]. Then (3.28) 1 (ɛ j 2t )2 = 1 Si S i S l. (i,l) C j t 1 Fro Lea 2(i), (3.12) and (3.17), it follows that 1 E (Si 2 F 2t 1 )= 1 (3.29) Var (# i t F 2t 1 )[ f t ( X i t) H t i f 0 ] 2 = 1 w t ( X i t ) ( 1 w t( X i t ) ) [ w t w f t ( X i t) H t i f 0 ] 2 σ 2t. 2 t Technical arguents given in the Aendix show that ( 2 ) (3.30) Var F 2t 1 (3.31) (3.32) Si 2 [( 2 E ) 2 ] F2t 1 S i S l (i,l) C j t 1 [{ 2 E } 2 ] F2t 1 (ɛ j ɛj 2t 1 )ɛj 2t 0, 0, 0

14 14 CHAN AND LAI under (3.21) for k =2t 1. By (3.22) and (3.28) (3.32), (3.21) holds for k =2t. The induction roof is colete and Corollary 2 holds. Let Ψ(x T )=ψ(x T ) ψ T and define Ψ T as in (3.1) with Ψ in lace of ψ. Then relacing ψ T by Ψ T in Corollaries 1 and 2 shows that as, (3.33) Ψ T N(0,σ 2 ), σ 2 (0) σ 2, where σ 2 (µ) is defined by (3.14) with ψ relaced by Ψ, noting that relacing ψ by Ψ in the definition of σc 2 in Corollary 1 gives σ2 defined in (2.8) as f 0 = Aroxiation of ψ T ψ T by Ψ T and roof of Theore 1. We now return to the setting of Section 2.1, in which we consider the HMM (1.1) and use weight functions of the for (2.1) and the article filter (2.4) in lieu of (3.1). We ake use of the following lea to extend (3.33) to ψ T ψ T. Lea 4. Let w t be of the for (2.1) for 1 t T and define η T and ζ T by (2.5) and Ψ T by (3.1) with ψ relaced by Ψ=ψ ψ T. Then (3.34) (3.35) (3.36) T (x T ) [= T T (x T Y T )] = ηt 1 [ t (x t x t 1 )g t (Y t x t )], t=1 L T (x T ) = T ηt 1 w t (x t ), t=1 ψ T ψ T = ( w T ) 1 Ψ( X i T )w T ( X i T)=( w 1 w T ) 1 η Ψ T T. Proof. By (2.1) and (2.5), T η T = [w t (x t )q t (x t x t 1 )]dν X (x T )= t=1 T t=1 [ t (x t x t 1 )g t (Y t x t )]dν X (x T ), and (3.34) follows fro (1.2). Moreover, (3.35) follows fro (2.1), (2.6) and (3.34). The first equality in (3.36) follows fro (2.3) and (2.4). By (2.3) and (3.35), (3.37) L T ( X ( i T )HT i w1 w ) T wt ( 1 = X i T ). η T w T Hence the second equality in (3.36) follows fro (3.1). The following lea, roved in the Aendix, is used to rove Theoere 1. Although it resebles Lea 3, its conclusions are about the square

15 PARTICLE FILTER 15 of the sus in Lea 3, and accordingly it assues finiteness of second (instead of first) oents. Lea 5. Let 1 t T, G be a easurable function on the state sace, Γ t (x t ) be defined as in (2.5). (i) If E q [G 2 (X t )Γ t 1 (X t 1 )] <, then 1 [ (ii) If E q [G 2 (X t )Γ t (X t )] <, then 1 G( X ] 2 i t) = O (1) as. i:a i t 1 =j [ ] 2 G(X i t) = O (1) as. i:a i t =j Proof of Theore 1. By (3.17), (3.33) and (3.36), (3.38) ( ψ T ψ T )=(1+o (1)) Ψ T N(0,σ 2 ). Since σ 2(0) = 1 { i:a i T 1 =j L T ( X i T )Hi T 1 Ψ( X i T )}2 σ 2 by (3.14) and (3.33), and since σ 2(0) = (1 + o (1)) σ 2 (ψ T ) in view of (2.9), (3.17) and (3.37), (3.39) σ 2 (ψ T ) σ 2. Letting c j = w T ( Xi T ) i:a i T 1 =j w T [ψ( X i T ) ψ T ], b j = i:a i T 1 =j w T ( X i T )/ w T and a = ψ T ψ T, it follows fro (2.9) that (3.40) σ 2 ( ψ T )= 1 (c j + ab j ) 2 = 1 c 2 j + 2a b j c j + a2 b 2 j. Since a = O ( 1/2 ) by (3.38), 1 c 2 j = σ2 (ψ T ) σ 2 by (3.39), and since 1 b 2 j = O (1) by Lea 5 (with G = w t ), 2a b j c j a ( 1 Hence by (3.40), σ 2 ( ψ T ) σ 2. b 2 j + 1 c 2 j) 0.

16 16 CHAN AND LAI 4. Extensions and roof of Theore 2. In this section we first extend the results of Section 2.1 to the case where residual Bernoulli resaling is used in lieu of bootstra resaling at every stage. We then consider occasional bootstra resaling and rove Theore 2, which we also extend to the case of occasional residual Bernoulli resaling Residual Bernoulli resaling. The residual resaling schee introduced in [2, 3] often leads to saller asytotic variance than that of bootstra resaling; see [6, 15]. We consider here the residual Bernoulli schee given in [7], which has been shown in [9, Section 2.4] to yield a consistent and asytotically noral article filtering estiate of ψ T. To ileent residual Bernoulli resaling, we odify bootstra resaling at stage t as follows: Let M(1) = and let ξt 1,...,ξ M(t) t be indeendent Bernoulli rando variables conditioned on (M(t),Wt i ) satisfying P {ξt i = 1 F 2t 1 } = M(t)Wt i M(t)Wt i. For each 1 i M(t) and t T 1, let H t i = Ht 1/(M(t)W i t i ) and ake # i t := M(t)Wt i +ξt i coies of ( X i t,a i t 1, H t). i These coies constitute an augented sale {(X j t,a j t,ht j ): 1 j M(t +1)}, where M(t +1)= M(t) #i t. Estiate ψ T by M(T ) ψ T,R := [M(T ) w T ] 1 ψ( X i T)w T ( X i T ). Let Ψ T,R be (3.1) with Ψ = ψ ψ T in lace of ψ and M(T ) in lace of. Since Ht 1 i is still given by (2.3) and Lea 2 still holds for residual Bernoulli resaling, it follows fro (3.17) and (3.37) that (4.1) = M(T )( ψ M(T ) T,R ψ T )= w T 1 Ψ( X i T )w T ( X i T) ( η ) M(T ) T Ψ( w 1 w X i T )L T ( X i T )h t 1 (X i t 1) =(1+o (1))M(T ) Ψ T,R. T Because relacing ψ by Ψ in (3.2) odifies f t to f t as defined in (2.7), we have for residual Bernoulli resaling the following analog of (3.11): (4.2) M(T ) Ψ T,R = 2T 1 (Zk,R 1 k=1 + + ZM( k/2 ) k,r ), where Z i 2t 1,R =[f t( X i t ) f t 1(X i t 1 )]Hi t 1, Zi 2t,R =(#i t M(t)W i t )f t( X i t ) H i t.

17 Let σ 2 R = 2T 1 k=1 σ2 k,r, with PARTICLE FILTER 17 σ 2 2t 1,R = σ 2 2t 1, σ 2 2t,R = E q [γ(w t (X t ))f 2 t (X t )h t (X t )], in which γ(x) =(x x )(1 x + x )/x, wt (x t )=h t 1(x t 1 )/h t (x t )= ζ t w t (x t ) and σ2t 1 2 is defined in Theore 1. Since Var(ξ) =γ(x)x when ξ is a Bernoulli(x x ) rando variable, Var ( M(t) ) M(t) Z2t,R i F 2t 1 = and therefore it follows fro Lea 2 that (4.3) 1 Var ( M( k/2 ) Zk,R i γ(m(t)w i t )M(t)W i t f 2 t ( X i t )( H i t )2, F k 1 ) σ 2 k,r, 1 k 2T 1. By (4.1) (4.3) and a siilar odification of (3.3) for residual Bernoulli resaling, Theore 1 still holds with ψ T relaced by ψ T,R ; secifically, (4.4) ( ψ T,R ψ T ) N(0,σ 2 R), σ 2 ( ψ T,R ) σ 2 R, where σ 2 (µ) is defined in (2.9). The following exale shows the advantage of residual resaling over bootstra resaling. Exale. Consider the bearings-only tracking roble in [12], in which X t =(X t1,...,x t4 ) and (X t1,x t3 ) reresents the coordinates of a shi and (X t2,x t4 ) the velocity at tie t. A erson standing at the origin observes that the shi is at angle Y t relative to hi. Taking into account easureent errors and rando disturbances in velocity of the shi, and letting Φ be a 4 4 atrix with Φ 12 =Φ 34 =1=Φ ii for 1 i 4 and all other entries 0,Γbea4 2 atrix with Γ 11 =Γ 32 =0.5, Γ 21 =Γ 42 = 1 and all other entries 0, the dynaics can be described by state-sace odel X t =ΦX t 1 +Γz t, Y t = tan 1 (X t3 /X t1 )+u t, where X 11 N(0,.5 2 ), X 12 N(0, ), X 13 N(.4,.3 2 ), X 14 N(.05,.01 2 ), z t+1 N(0, I 2 ) and u t N(0, ) are indeendent rando variables for t 1. The quantities of interest are E(X T 1 Y T ) and E(X T 3 Y T ). To ileent the article filters, we let q t = t for t 2. Unlike [12] in which q 1 = 1 as well, we use an iortance density q 1 that involves 1 and Y 1 to generate the initial location (X 11,X 13 ). Let ξ and ζ be indeendent standard noral rando variables, and r = tan(y 1 +

18 18 CHAN AND LAI Table 3 Siulated values of ψt ± σ( ψt )/ (for bootstra resaling) and ψ T,R ± σ( ψt,r)/ (for residual Bernoulli resaling). E(X T 1 Y T ) E(X T 3 Y T ) T boot(p ) boot resid boot(p ) boot resid ±.026 ±.009 ±.007 ±.007 ±.002 ± ±.043 ±.012 ±.011 ±.008 ±.003 ± ±.042 ±.014 ±.010 ±.027 ±.009 ± ±.044 ±.015 ±.010 ±.046 ±.016 ± ±.045 ±.016 ±.011 ±.066 ±.023 ± ±.047 ±.016 ±.012 ±.088 ±.029 ± ξ). Since u 1 has sall variance, we can estiate X 13 /X 11 well by r. This suggests choosing q 1 to be degenerate bivariate noral with suort y = rx, with (y, x) denoting (X 13,X 11 ), so that its density function on this line is roortional to ex{ (x µ) 2 /(2τ)}, where µ =.4r/(.36 + r 2 ), τ =.09/(.36 + r 2 ). Thus, q 1 generates X 11 = µ + τζ and X 13 = rx 11, but still follows 1 to generate (X 12,X 14 ). By (2.1), w 1 (x 1 )= x 11 { τ(1 + r 2 ) ex x2 11 2(.5) 2 (x 13.4) 2 2(.3) 2 + ζ2 }, 2 noting that.005 x 11 τ(1 + r 2 ) is the Jacobian of the transforation of (ξ, ζ) to(x 11,x 13 ). For t 2, (2.1) yields the resaling weights w t (x t )= ex{ [Y t tan 1 (x t3 /x t1 )] 2 /[2(.005) 2 ]}. Weuse = 10,000 articles to estiate E((X T 1,X T 3 ) Y T ) by article filters, using bootstra (boot) or residual (resid) Bernoulli resaling at every stage, for different values of T. Table 3, which reorts results based on a single realization of Y T, shows that residual Bernoulli resaling has saller standard errors than bootstra resaling. Table 3 also considers bootstra resaling that uses q t = t for t 1, as in [12] and denoted by boot(p ), again with =10,000 articles. Although boot and boot(p ) differ only in the choice of the iortance densities at t = 1, the standard error of boot is substantially saller than that of boot(p ) over the entire range 4 T 24. Unlike Section 2.3, the standard error estiates in Table 3 use a sale slitting refineent that is described in the first aragrah of Section 5.

19 PARTICLE FILTER Proof of Theore 2. First consider the following odification of Ψ T : (4.5) Ψ OR := 1 L T ( X i T )Ψ( X i T)H ĩ τ(t ) = η 1 T v τ 1 v τr+1 ( ψ OR ψ T ). Analogous to (3.11), Ψ OR = 2r+1 k=1 (Z1 k + + Z k ), where Z2s 1 i = [f τs ( X i τ s ) f τs 1 (X i τ s 1 )]Hτ i s 1, Z2s i = f τs ( X i τ s ) H τ i s Vτ j s f( X j τ s ) H τ j s. Let τ0 = 0. Since v τ s (2.5) and (4.5) that τ s t=τ s 1 +1 ζ t by (2.12) and Lea 2, it follows fro ( ψ OR ψ T )=(1+o (1)) Ψ OR N(0,σ 2 OR), siilarly to (3.38), where (4.6) σ 2 OR = r +1 s=1 + E q {[f 2 τ s (X τ s ) f 2 τ s 1 (X τ s 1 )]h τ s 1 (X τ s 1 )} r s=1 E q [f 2 τ s (X τ s )h τ s (X τ s )]. Moreover, analogous to (3.3), Ψ OR = (ɛ j ɛj 2r+1 ), where ɛ j 2s 1 = i:a i τ s 1 =j [f τ s ( X i τ s ) f τs 1 (X i τ s 1 )]Hτ i s 1, ɛ j 2s = i:a i τ s 1 =j (#i τ s Vτ i s )f τs ( X i τ s ) H τ i s, therefore the roof of σ 2 OR ( ψ OR ) σ 2 OR is siilar to that of Theore Occasional residual resaling and assution (2.12). In the case of occasional residual resaling, Theore 2 still holds with ψ OR relaced by ψ ORR and with σor 2 relaced by σ 2 ORR = r +1 s=1 + E q {[f 2 τ s (X τ s ) f 2 τ s 1 (X τ s 1 )]h τ s 1 (X τ s 1 )} r s=1 ( τs E q [γ ] wt (X t) )fτ 2 s (X τ s )h τ s (X τ s ). t=τs 1 +1

20 20 CHAN AND LAI In articular, ( ψ ORR ψ T ) N(0,σ 2 ORR ) and σ2 OR ( ψ ORR ) σ 2 ORR. Assution (2.12) is often satisfied in ractice. In articular, if one follows Liu [14] and resales at stage t whenever (4.7) cv 2 t := 1 (Vt i ) 2 1= [v t ( X i t) v t ] 2 v 2 t c, where c>0 is a re-secified threshold for the coefficient of variation, then (2.12) can be shown to hold by aking use of the following lea. Lea 6. Let resaling be carried out at stage t whenever cv 2 t c. Then (2.12) holds with (4.8) { τs = inf t>τs 1 : ([ t k=τ E s 1 +1 w k (X k)] 2 ) } q h 1 c, for 1 s r, τs 1(X τ s 1 ) where τ 0 =0and r = ax{s : τ s <T}, rovided that (4.9) E q ([ τ s k=τ s 1 +1 w k (X k)] 2 h τ s 1(X τ s 1 ) ) 1 c for 1 s r. Proof. Let τs 1 +1 l τs and G(x l )= l k=τ s 1 +1 wk (x k). Aly Lea 2(i) to G and G 2, with t 1 in the subscrit relaced by the ost recent resaling tie τs 1. It then follows fro (4.7) that (4.10) cv 2 l E q [G 2 (X l )/h τ s 1 (X τ s 1 )] 1on{τ s 1 = τ s 1 }. In view of (4.8) and (4.9), it follows fro (4.10) with l = τ s that P {τ s > τ s,τ s 1 = τ s 1 } 0, for 1 s r, and fro (4.10) with τ s 1 +1 l<τ s that P {τ s = l, τ s 1 = τ s 1 } Discussion and concluding rearks. The central liit theore for article filters in this aer and in [6, 8, 9, 13] considers the case of fixed T as the nuber of articles becoes infinite. In addition to the standard error estiates (2.9), one can use the following odification that substitutes ψ T in σ 2 (ψ T ) by out-of-sale estiates of ψ T. Instead of generating additional articles to accolish this, we aly a sale-slitting technique to the articles, which is siilar to k-fold cross-validation. The standard

21 PARTICLE FILTER 21 error estiates in the exale in Section 4.1 use this technique with k =2. Divide the articles into k grous of equal size r = /k excet for the last one that ay have a larger sale size. For definiteness consider bootstra resaling at every stage; the case of residual resaling or occasional resaling can be treated siilarly. Denote the articles at the tth generation, before resaling, by { :1 i r, 1 j k}, and let Xt 1j,...,X rj 1j rj t be saled with relaceent fro { X t,..., X t }, with weights W ij ij t = w t ( X t )/( wj t ), where wj t = r 1 r ij w t ( X t ). With these stratified resaling weights, we estiate ψ T by X ij t (5.1) ψ T = k 1 k ψ j T, where ψ j T =(r wj T ) 1 r ij ψ( X T )w ij T ( X T ). Siilarly, letting ψ j T =(k 1) 1 l:l j ψ l T, we estiate σ2 by (5.2) σ 2 SP = 1 k r { l:a lj T 1 =i lj w T ( X w j T T ) [ψ( } lj j 2. X T ) ψ T ] Choin [6, Sect. 2.1] suarizes a general fraework of traditional article filter estiates of ψ T, in which the resaling (also called selection ) weights w t (X t ) are chosen to convert a weighted aroxiation of the osterior density of X t given Y t, with likelihood ratio weights associated with iortance saling (called utation in [6]), to an unweighted aroxiation (that has equal weights 1), so that the usual average ψ T := 1 ψ(x i T ) converges a.s. to ψ T as. He uses induction on t T to rove the central liit theore for ( ψ T ψ T ): conditional on ast iterations, each ste generates indeendent (but not identically distributed) articles, which follow soe (conditional) central liit theore [6,.2392] and oints out that the article filter of Gilks and Berzuini [11] is a variant of this fraework. Let V t be the variance of the liiting noral distribution of t ( ψ T ψ T ) in the Gilks-Berzuini article filter [11, ] that assues (5.3) 1, then 2,, and then T, which eans that 1 and t t 1 for 1 <t T. The estiates of V t roosed in [11] uses the idea of ancestor, instead of the ancestral origin we use. For s<t, article X i s is called an ancestor of X k t if X k t descends fro X i s. Thus, the ancestral origin is the secial case corresonding to s =1

22 22 CHAN AND LAI (the first generation of the articles). Let (5.4) (5.5) t Nt k,l s = 1 { Xi s is an ancestor of Xk t and of Xl t }, s=1 V t = 1 t t 2 Nt k,l {ψ( X k t ) ψ T }{ψ( X l t) ψ T }. t k=1 l=1 Theore 3 of Gilks and Berzuini [11, Aendix A] shows that V t /V t 1 under (5.3). The basic idea is to use the law of large nubers for each generation conditioned on the receding one and an induction arguent which relies on the assution (5.3) that is not directly relevant to the ractical context. In contrast, our Theores 1 or 2 is based on a uch ore recise artingale aroxiation of ( ψ T ψ T )or( ψ OR ψ T ). While we have focused on iortance saling in this aer, another aroach to choosing the roosal distribution is to use Markov chain Monte Carlo iterations for the utation ste, as in [11] and iortant advances in this aroach have been develoed recently in [1, 4]. APPENDIX A: LEMMAS 2, 3, 5, COROLLARY 1, AND (3.30) (3.32) Proof of Lea 2. We use induction on t and show that if (ii) holds for t 1 then (i) holds for t, and that if (i) holds for t then (ii) also holds for t. Since G = G + G, we can assue without loss of generality that G is non-negative. By the law of large nubers, (i) holds for t = 1, noting that h 0 1. Let t>1 and assue that (ii) holds for t 1. To show that (i) holds for t, assue µ t = E q [G(X t )/h t 1(X t 1 )] to be finite and suose that, contrary to (3.15), there exist 0 <ɛ<1 and 1 < 2 < such that (A.1) { 1 P G( X } i t,) µ t > 2ɛ >ɛ(2 + µ t ) for all M, in which M = { 1, 2,...} and we write X i t,(= X i t) to highlight the deendence on. Let δ = ɛ 3. Since x =(x y)+(x y) +, we can write G( X i t,) =U i t, + V i t, + S i t,, where (A.2) Ut, i = G( X i t,) (δ) E [G( X i t,) (δ) F 2t 2 ], Vt, i = E [G( X i t,) (δ) F 2t 2 ], St, i =[G( X i t,) δ] +. Since E (U i t, F 2t 2 ) = 0 and Cov (U i t,,u l t, F 2t 2 ) = 0 for i l, it

23 PARTICLE FILTER 23 follows fro Chebyshev s inequality, (U i t,) 2 δg( X i t,) and δ = ɛ 3 that { } ( 1 P Ut, i >ɛ F 2t 2 (ɛ) 2 ) Var Ut, i (A.3) F 2t 2 = (ɛ) 2 E ( Ut, i 2 F 2t 2 ) ɛ 1 E [G( X i t,) F 2t 2 ] ɛµ t, as, by (3.16) alied to G (X t 1 ) = E q [G(X t ) X t 1 ], noting that E q [G (X t 1 )/h t 1(X t 1 )] = µ t. Alication of (3.16) to G (X t 1 ) also yields (A.4) 1 V i t, µ t. Fro (A.3), it follows that (A.5) { } 1 P Ut, i >ɛ ɛ(1 + µ t ) for all large. Since µ t <, we can choose n k such that E q [G(X t )1 {G(Xt)>n k }/h t 1(X t 1 )] <k 1. Hence by alying (3.16) to G k (X t 1 )=E q [G(X t )1 {G(Xt)>n k } X t 1 ], we can choose a further subsequence k that satisfies (A.1), k n k /δ and { P 1 E [G( X i t,)1 {G( X i t, )>n k} F 2t 2] 2k 1} k 1 for = k. Then for = k, with robability at least 1 k 1, P {St, i 0 F 2t 2 } = P {G( X i t,) >δ F 2t 2 } (δ) 1 E [G( X i t,)1 {G( X i )>δ} F 2t 2] 2δ 1 k 1. t, Therefore, P {St, i 0 for soe 1 i } 0as = k. Cobining this with (A.4) and (A.5), we have a contradiction to (A.1). Hence we conclude (3.15) holds for t. The roof that (ii) holds for t whenever (i) holds for t is siilar. In articular, since (i) holds for t = 1, so does (ii).

24 24 CHAN AND LAI Proof of Lea 3. We again use induction and show that if (ii) holds for t 1 then (i) holds for t, and that if (i) holds for t then so does (ii). First (i) holds for t = 1 because A i 0 = i, h 0 1 and E q G(X 1 ) < ilies that for any ɛ>0, P { 1 ax G( X 1 i) ɛ} P q{g(x 1 ) ɛ} 0as. 1 i Let t>1 and assue that (ii) holds for t 1. To show that (i) holds for t, suose µ t (= E q [G(X t )/h t 1(X t 1 )]) < and that, contrary to (3.19), there exist 0 <ɛ<1 and M = { 1, 2,...} with 1 < 2 < such that (A.6) P { 1 ax G( X } i t,) > 2ɛ >ɛ(2 + µ t ) for all M. 1 j i:a i t 1 =j As in (A.2), let δ = ɛ 3 and G( X i t, )=U t, i + V t, i + Si t,.for1 j, { 1 P U i t, i:a i t 1 =j } >ɛ F 2t 2 ɛ 1 i:a i t 1 =j E [G( X i t,) F 2t 2 ], by an arguent siilar to (A.3). Suing over j and then taking exectations, it then follows fro Lea 2 that (A.7) { } 1 P U i t,k >ɛ ɛ(1 + µ t ) for all large. i:a i t 1 =j By (3.20) alied to the function G (X t 1 )=E q [G(X t ) X t 1 ], (A.8) 1 ax 0as. 1 j i:a i t 1 =j V i t, As in the roof of Lea 2, select a further subsequence k such that (A.6) holds and P { St, i =0} 1as = k. Cobining this with (A.7) and (A.8) yields a contradiction to (A.6). Hence (3.19) holds for t. Next let t 1 and assue that (i) holds for t. To show that (ii) holds for t, assue µ t = E q [G(X t )/h t (X t )] to be finite and note that i:a i t =j G(Xi t,k )= i:a i t 1 =j #i t,k G( X i t,k ). Suose that, contrary to (3.20) there exist 0 <ɛ<1 and M = { 1, 2,...} with 1 < 2 < such that (A.9) P { 1 ax # i t,kg( X } i t,k) > 2ɛ >ɛ(2 + µ t ) for all M. 1 j i:a i t 1 =j

25 PARTICLE FILTER 25 Let δ = ɛ 3 and write # i t,g( X i t,) =Ũ i t, + Ṽ i t, +# i t,s i t,, where Ũ i t, =(#i t, W i t, )[G( X i t, )} (δ)], Ṽ i t, = W i t, [G( X i t, ) (δ)], and St, i is defined in (A.2). Since E (# i t, F 2t 1 )=Wt,, i Var (# i t, F 2t 1 ) Wt, i = w t( X i t, )/ w t, and since # 1 t,,...,# t, are airwise negatively correlated conditioned on F 2t 1, we obtain by Chebyshev s inequality that for 1 j, P { 1 i:a i t 1 =j Ũ t, i >ɛ F 2t 1 } is bounded above by 1 (ɛ) 2 Var (# i t, F 2t 1 )[G( X i t,) (δ)] 2 ɛ G ( w X i t,), i:a i t, t 1 =j i:a i t 1 =j where G ( ) =w t ( )G( ). Hence, alying Lea 2 to G and noting that by (2.5), E q [G (X t )/h t 1 (X t 1)] = ζt 1 µ t and w t, = 1 w ( X i t,) ζt 1 by Lea 2, we obtain { } 1 (A.10) P >ɛ ɛ(1 + µ t ) for all large. i:a i t 1 =j Ũ i t, By (3.19) alied to G and Lea 2, (A.11) 1 [ ax Ṽ t, i w t, 1 1 j i:a i t 1 =j 1 ax 1 j G ( X ] i t,) 0. i:a i t 1 =j Finally, by Lea 2 alied to G (X t )=w t (X t )G(X t )1 {G(Xt)>δ}, (A.12) P {# i t,st, i 0 F 2t 1 } = P {# i t, > 0 F 2t 1 }1 {S i t, 0} 1 E (# i δ t, F 2t 1 ) G ( X i t,) w t ( X i t, ) = 1 [ 1 G ( δ w t, X i t,)] 0 as = k. Since (A.10) (A.12) contradicts (A.9), (3.20) follows. Proof of Lea 5. The roof of Lea 5 is siilar to that of Lea 3, and in fact is siler because the finiteness assution on the second oents of G disenses with truncation arguents of the tye in (A.2). First (i) holds for t = 1 because A i 0 = i, h 0 = 1 and { P 1 G 2 ( X } 1 i) K K 1 E q G 2 (X 1 ) 0asK.

26 26 CHAN AND LAI As in the roof of Lea 3, let t>1 and assue that (ii) holds for t 1. To show that (i) holds for t, suose µ t := E q [G 2 (X t )Γ t 1 (X t 1 )] <. By exressing G = G + G, we ay assue without loss of generality G is nonnegative. Write G( X i t )=U t i + V t i, where V t i = E [G( X i t ) Xi t 1 ]. By the induction hyothesis, 1 ( i:a i t 1 =j V t i ) 2 = O (1). To show that 1 ( i:a i t 1 =j U t i ) 2 = O (1), note that E (Ut i Ut l F 2t 2 )=0fori l. Hence it follows by an arguent siilar to (A.3) that { P 1 ( ) 2 } F2t 2 K (K) 1 E [(Ut i)2 F 2t 2 ] i:a i t 1 =j U i t K 1 E q { G(X t ) E q [G(X t ) X t 1 ] 2 /h t 1(X t 1 )}. Next assue that (i) holds for t and show that (ii) holds for t if µ t := E q [G 2 (X t )Γ t (X t )] is finite. Note that i:a i t =j G(Xi t)= i:a i t 1 =j #i tg( X i t). Write # i tg( X i t)=ũ t i + Ṽ t i, where Ṽ t i = Wt i G( X i t)=w t ( X i t)g( X i t)/ w t. Since w t ζ 1 t and since E q {[w t (X t )G(X t )] 2 Γ t 1 (X t 1 )} µ t <, it follows fro the induction hyothesis that 1 ( i:a i t 1 =j Ṽ t i)2 = O (1). Moreover, since Var (# i t F 2t 1 ) Wt i and Cov (# i t, # l t F 2t 1 ) 0 for i l, it follows fro an arguent siilar to (A.3) that { P 1 ( Ũ t i i:a i t 1 =j ) 2 K F2t 1 } (K) 1 Wt i G 2 ( X i t) K 1 E q [G 2 (X t )/h t (X t )], noting that E q [G 2 (X t )/h t (X t )] η 1 t µ t <. Hence 1 ( i:a i t 1 =j Ũ i t ) 2 = O (1). This shows that (ii) holds for t, coleting the induction roof. Proof of Corollary 1. Recall that {(Zk 1,...,Z k ), F k, 1 k 2T 1} is a artingale difference sequence and that Zk 1,...,Z k are conditionally indeendent given F k 1, where F 0 is the trival σ-algebra; oreover, (A.13) ( ψ T ψ T )= 2T 1 k=1 ( Zk/ i ). Fix k. Conditioned on F k 1, Y k i := Zi k / is a triangular array of row-wise indeendent (i.i.d. when k =2t) rando variables. Noting that E [ f t ( X i t ) X i t 1] = f t 1 (X i t 1), we obtain fro (3.10) that E [(Z i 2t 1 )2 F 2t 2 ] = {E [ f 2 t ( X i t ) Xi t 1 ] f 2 t 1 (Xi t 1 )}(Hi t 1 )2,

27 PARTICLE FILTER 27 E [(Z i 2t) 2 F 2t 1 ] = W j f t t 2 ( X j t)( H [ t j ) 2 W j f t t ( X j t) H ] 2. t j Let ɛ>0. Since W j t = w t ( X j t)/( w t ) and w t ζ 1 t, E [(Yi) k 2 F k 1 ] σ k, 2 E [(Yi) k 2 1 { Y k i >ɛ} F k 1] 0, by (2.5) and Lea 2. Hence by Lindeberg s central liit theore for triangular arrays of indeendent rando variables, the conditional distribution of Zk i / given F k 1 converges to N(0, σ k 2 )as. This ilies that for any real u, (A.14) E (e iu Zj k / Fk 1 ) e u2 σ 2 k /2, 1 k 2T 1, where i denotes the iaginary nuber. Let S t = t k=1 ( Z j k / ). Since { Z j k /, F k, 1 k 2T 1} is a artingale difference sequence, it follows fro (A.13), (A.14) and atheatical induction that E (e iu ( ψ T ψ T ) )=E [e ius 2T 2 E (e iu Zj 2T 1 / F2T 2 )] = e u2 σ 2 2T 1 /2 E (e ius 2T 2 )+o(1) = = ex( u 2 σ 2 C/2) + o(1). Hence (3.13) holds. Proof of (3.30) (3.32). For distinct i, j, k, l, define c ij =(Wt i)(w j t ), c ijkl =(Wt i )(Wt j )(Wt k )(Wt l ), c iijk =(Wt i ) 2 (Wt j )(Wt k ) and define c ijk, c iij, c ijj siilarly. Since # i t = h=1 1 {B h t =i}, where Bh t are i.i.d. conditioned on F 2t 1 such that P {Bt h = i F 2t 1 } = Wt i, cobinatorial arguents show that (A.15) E [(# i t Wt i )(# j t Wt j )(# k t Wt k )(# l t Wt l ) F 2t 1 ]= 4 [( 1)( 2)( 3) 4 2 ( 1)( 2) ( 1) ]c ijkl =( )c ijkl, (A.16) E [(# i t W t i)2 (# j t W j t )(#k t W t k) F 2t 1] = ( )c iijk +( )c ijk, (A.17) E [(# i t Wt i ) 2 (# j t Wt j ) 2 F 2t 1 ]= ( )c iijj +( )(c iij + c ijj )+(1 1 )c ij, (A.18) E [(# i t Wt i ) 4 F 2t 1 ]=[Wt i (1 Wt i ) 4 +(1 Wt i )(Wt i ) 4 ] +3( 1)[(Wt i ) 2 (1 Wt i ) 4 +2(Wt i ) 3 (1 Wt i ) 3 +(Wt i ) 4 (1 Wt i ) 2 ], (A.19) E {(# i t W t i)(#j t W j t ) F 2t 1} = 1 c ij.

28 28 CHAN AND LAI To rove (3.30), let T i = Wt i [ f t ( X i t) H t i f 0 ] 2, T = 1 T i and Tt = 1 ax 1 j i:a i t 1 =j T i. We can use (A.17) and (A.18) to show that 2 E [( Si 2)2 F 2t 1 ] is equal to 2 E (S 4 i F 2t 1 )+ 2 i j E (S 2 i S 2 j F 2t 1 )= T 2 + o P (1) as. By Lea 2(i), T 2 σ 2t. 4 Fro this and (3.29), (3.30) follows. To rove (3.31), let D i = Wt i [ f t ( X i t) H t i f 0 ], D = 1 D i and Dt = 1 ax 1 j i:a i t 1 =j D i. By (A.15) (A.17), the left-hand side of (3.31) is equal to 1 ( ){ 3 (i,l) C j t 1 4 ( ) { 2 3( DD t ) 2 + TT t for 2. } 2 1 D i D l + (1 2 1 ) i:a i t 1 =j T i ( l i:a l t 1 =j D l ) 2 } (i,l) C j t 1 T i T l By Leas 2 and 3, the uer bound in the above inequality converges to 0 in robability as, roving (3.31). By (A.19), E (S i S j F 2t 1 )= 1 D i D j and siilarly E (Si 2 F 2t 1) = T i 1 Di 2. Therefore by (3.4), the left-hand side of (3.32) is equal to 1 (ɛ j ɛj 2t 1 )2{ 1 T i ( 1 i:a i t 1 =j i:a i t 1 =j D i ) 2 } 0, since Tt +(D t )2 0 by Lea 3 and since (3.21) holds for k =2t 1by the induction hyothesis. REFERENCES [1] Andrieu, C., Doucet, A. and Holenstein, R. (2010). Particle Markov chain Monte Carlo ethods (with discussion). J. Roy. Statist. Soc. Ser. B [2] Baker, J.E. (1985). Adative selection ethods for genetic algoriths. In Proc. International Conference on Genetic Algoriths and Their Alications (J. Grefenstette, ed.) Erlbau, Mahwah, NJ. [3] Baker, J.E. (1987). Reducing bias and inefficiency in the selection algorith. In Genetic Algoriths and Their Alications (J. Grefenstette, ed.) Erlbau, Mahwah, NJ. [4] Brockwell, A., Del Moral, P. and Doucet, A. (2010). Sequentially interacting Markov chain Monte Carlo ethods. Ann. Statist

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