Numerically stable online estimation of variance in particle filters

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by  Jimmy Olsson

Numerically stable online estimation of
variance in particle filters
Jimmy Olsson1,∗ and Randal Douc2
7
1 1Department of Mathematics
0 KTH Royal Institute of Technology
2 SE-100 44 Stockholm, Sweden
e-mail: [email protected]
n
a 2D´epartement CITI
J TELECOM SudParis

4 9 rue Charles Fourier, 91000 EVRY
e-mail: [email protected]

]
E Abstract: This paper discusses variance estimation in sequential Monte
Carlomethods,alternativelytermedparticlefilters.Thevarianceestimator
M
thatweproposeisanaturalmodificationofthatsuggestedbyH.P.Chan
. andT.L.Lai[AgeneraltheoryofparticlefiltersinhiddenMarkovmodels
t andsomeapplications.Ann. Statist.,41(6):2877–2904, 2013],whichallows
a
the variance to be estimated in a single run of the particle filter by trac-
t
s ingthe genealogical historyof the particles.However, due particlelineage
[ degeneracy, theestimatorofthementionedworkbecomesnumericallyun-

stable as the number of sequential particle updates increases. Thus, by
1
tracingonlyapartoftheparticles’genealogyratherthanthefullone,our
v
estimator gains long-term numerical stability at the cost of a bias. The
1
scope of the genealogical tracing is regulated by a lag, and under mild,
0
easily checked model assumptions, we prove that the bias tends to zero
0 geometrically fast as the lag increases. As confirmed by our numerical re-
1 sults,thisallowsthebiastobetightlycontrolledalsoformoderateparticle
0 samplesizes.
.
1
MSC 2010 subject classifications:Primary62M09;secondary62F12.
0 Keywords and phrases: Asymptotic variance, Feynman-Kac models,
7 hidden Markov models, particle filters, sequential Monte Carlo methods,
1 state-space models,varianceestimation.
:
v
i
X 1. Introduction
r
a
Sincethebootstrap particle filter wasintroducedin[14],sequential Monte Carlo
(SMC) methods, alternatively termed particle filters, have been successfully ap-
plied within a wide rangeof applications,including computer vision,automatic
control,signalprocessing,optimisation,robotics,econometrics,andfinance;see,
e.g., [13, 23] for introductions to the topic. SMC methods approximate a given
sequence of distributions by a sequence of possibly weighted empirical mea-
suresassociatedwithasampleofparticles evolvingrecursivelyandrandomlyin
time. Each iteration of the SMC algorithm comprises two operations: a muta-
tion step,whichmovestheparticlesrandomlyinthestatespace,andaselection
∗J.OlssonissupportedbytheSwedishResearchCouncil,Grant2011-5577.
1
J. Olsson and R. Douc/Estimation of variance inparticle filters 2
step, which duplicates/eliminates, through resampling, particles with high/low
importance weights, respectively.
In parallelwith algorithmicdevelopments,the theoreticalproperties ofSMC
have been studied extensively during the last twenty years, and there is cur-
rently a number of available results describing the convergence, as the number
of particles tends to infinity, of Monte Carlo estimates produced by the algo-
rithm;see,e.g.,the monographs[5, 6]and[1, Chapter9].The firstcentral limit
theorem (CLT)forSMCmethodswasestablishedin[7],andthisresultwaslater
refinedinthe seriesofpapers[3, 18,10].Inthe mentionedCLT,the asymptotic
variance of the weak Gaussian limit is expressed through a recursive formula
involving high-dimensional integrals over generally complicated integrands and
is hence intractable in general.
Duetoitscomplexity,onlyaveryfewrecentworkshavetreatedtheimportant—
althoughchallenging—topicofvarianceestimationinSMCalgorithms.Abreak-
throughwasmadebyH.P.ChanandT.L.Lai,whoproposed,in[2]andwithin
the frameworkofgeneralstate-space models (or,hidden Markov models), anes-
timator, from now on referred to as the Chan & Lai estimator (CLE), that
allows the sequence of asymptotic variances to be estimated on the basis of a
single realisation of the algorithm, without the need of additional simulations.
Remarkably, the CLE can be shown to be consistent (see [2, Theorem 2]), i.e.,
to converge in probability to the true asymptotic variance as the number of
particles tends to infinity. At a given time step, the CLE estimates the asymp-
totic variance by tracing genealogically the time-zero ancestors of the particles
at the time step in question. The variance estimators proposed recently in [19]
are based on the same principle, and may be viewed as refinements of the CLE
within a more general framework of Feynman-Kac models and particle algo-
rithms with time varying particle population sizes. Moreover, [19] provides an
elegant,deepened(asymptoticaswellasnon-asymptotic)theoreticalanalysisof
the technique,andthese resultsareessentialforthe developmentofthe present
paper.
Appealingly,thesetoftime-zeroancestorsmaybeupdatedrecursivelyinthe
SMC algorithm by adding just a single line to the code. This allows variance
estimates to be computed online with essentially the same computational com-
plexity and memory demands as the original algorithm. Nevertheless, since the
SMC algorithm performs repeatedly selection, it is well established that all the
particles in the sample will, eventually, share the same time-zero ancestor (see,
e.g.,[16]foratheoreticalanalysisofthisparticlepathdegeneracyphenomenon).
Unfortunately, this implies that the CLE collapses eventually to zero as time
increases. Increasing the particle sample size or resampling less frequently the
particles will postpone somewhat, but not avoid, this collapse. This makes the
CLE impractical in the long run. Thus, although the SMC methodology has
become a standard tool in statistics and engineering, a numerically stable esti-
matoroftheSMCvariancehas,surprisingly,hithertobeenlacking,andtheaim
of the present paper is to—at least partially—fill this gap.
The natural solution that we propose in the present paper is to estimate
the SMC asymptotic variance by tracing only a part of the particles’genealogy
J. Olsson and R. Douc/Estimation of variance inparticle filters 3
rather than the full one (i.e., back to the time-zero ancestors). By tracing ge-
nealogicallyonlythelastgenerations,depletedancestorsetsareavoidedaslong
astheparticlesamplesizeisatleastmoderatelylarge.Still,thismeasureleadsto
abias,whosesizedeterminescompletelythe successoftheapproach.Neverthe-
less,in [12],whichstudies the stochasticstability ofthe sequenceof asymptotic
variances within the framework of general hidden Markov models, or, viewed
differently, randomly perturbed Feynman-Kac models (see [12, Remark 1]), it
is established that the variance, which at time n may be expressed in the form
of a sum of n+1 terms, is, in the case where the perturbations form a station-
arysequence,uniformly stochasticallyboundedintime,or,tight.Moreover,the
analysisprovidedin [12],which is drivenby mixing assumptions,indicates that
the size of the mth term in the variance at time n decreases geometrically fast
with the difference n m. Consequently, we may expect the last terms of the
−
sumtorepresentthemajorpartofthevariance,andaslongasthe numberλof
traced generations, the lag, and the particle sample size are not too small, the
biasshouldbenegligible.Thisargumentisconfirmedbyourmainresult,Theo-
rem6,whichatanytimepointnprovidesanorderρλ boundontheasymptotic
(as the number of particles tends to infinity) bias, where ρ (0,1) is a mixing
∈
rate. Consequently, as long as the number of particles is large enough, the bias
stays numerically stable in the long run and may, as it decreases geometrically
fast with the lag λ, be controlled efficiently. Methodologically, the estimator
that we propose has similarities with the fixed-lag smoothing approach studied
in [17, 21, 22], and we here face the same bias-variance tradeoff, in the sense
that a too greedy/generouslag design leads to high bias/variance, respectively.
The developments of the present paper are cast into the framework of ran-
domly perturbed Feynman-Kac models, and the theoretical analysis is driven
by the assumptions of [11, 12] (going back to [9]), which are easily checked
for many models used in practice. In particular, by replacing the now classical
strong mixing assumption on the transition kernel of the underlying Markov
chain—astandardassumptionin the literature that typically requiresthe state
space of the Markov chain to be a compact set—by a local Doeblin condition,
we are able to verify the assumptions for a wide class of models with possibly
non-compact state space.
As a numerical illustration, we apply our estimator in the context of SMC-
basedpredictorflowapproximationingeneralstate-spacemodels,includingthe
widelyusedstochasticvolatility modelproposedin[15].Weareabletoreportan
efficient, numerically stable performance of our variance estimator, with tight
control of the bias at low variance.
Thepaperisstructuredasfollows.Section2introducessomenotation,defines
the framework of perturbed Feynman-Kac models, and provides some back-
ground to SMC. In Section 3, focus is set on asymptotic variance estimation
andafteraprefatorydiscussionontheCLEweintroducetheproposedfixed-lag
varianceestimator.Wealsodescribehowourestimatorcanbestraightforwardly
extendedto so-calledupdated Feynman-Kacdistribution flows.Theoreticaland
numerical results are found in Section 4 and Section 5, respectively, and Ap-
pendix A contains all proofs.Our theoreticalanalysis is divided into two parts:
J. Olsson and R. Douc/Estimation of variance inparticle filters 4
first, the identification of the limiting bias and, second, the construction of a
tight upper bound on the same. The first part relies on the theoretical machin-
ery developed in [19], whose key elements are recalled briefly in the beginning
of Section A.1. Finally, Section 6 concludes the paper.
2. Preliminaries
2.1. Some notation and conventions
We assume that all random variables are defined on a common probability
space (Ω, ,P). The set of natural numbers is denoted by N = 0,1,2,... ,
and we leFt N∗ = N 0 be the positive ones. For all (m,n) {N2, we se}t
\ { } ∈
Jm,nK:= m,m+1,...,n .The setof nonnegativerealnumbers is denotedby
R . For an{y quantities a}m , vectors are denoted by am :=(a ,...,a ).
+ { ℓ}ℓ=1 ℓ ℓ m
We introduce some measure and kernel notation. Given some state space
(E, ),wedenotebyF( )andM( )thespacesofboundedmeasurablefunctions
andEprobability measuEres on (E,E ), respectively. For any functions (h,h′)
F( )2 we define the product funcEtion h(cid:15)h′ : E2 (x,x′) h(x)h′(x′). Th∈e
ideEntity functionx xis denotedbyid.Letµ be a∋measure7→on(E, );then for
7→ E
any µ-integrable function h, we denote by
µh:= h(x)µ(dx)
Z
theLebesgueintegralofhw.r.t.µ.Inaddition,let(E′, ′)besomeothermeasur-
ablespaceandKsomepossiblyunnormalisedtransitioEnkernelK:E ′ R .
+
×E →
The kernel K induces two integral operators, one acting on functions and the
otheronmeasures.Morespecifically,givenameasureν on(E, )andameasur-
able function h on (E′, ′), we define the measure E
E
νK: ′ A K(x,A)ν(dx)
E ∋ 7→
Z
and the function
Kh:E x h(y)K(x,dy),
∋ 7→
Z
whenever these quantities are well defined.
2.2. Randomly perturbed Feynman-Kac models
Let (X, ) and (Z, ) be a pair of general measurable spaces. Moreover, let
K and χXbe a MarkZov transition kernel and a probability measure on (X, ),
respectively, and g z : z Z a family of real-valued, positive, and mXea-
surable potential f{unhctiions o∈n (X}, ). For all vectors zm Zm−k+1, we define
X k ∈
unnormalised transition kernels
m
Q zm :X (x ,A) (x ) g z (x )M(x ,dx ),
h k i ×X ∋ k 7→ ··· 1A m+1 h ℓi ℓ ℓ ℓ+1
Z Z ℓ=k
Y
J. Olsson and R. Douc/Estimation of variance inparticle filters 5
withtheconventionQ zm (x,A)=δ (A) ifm<k (whereδ denotesthe Dirac
h k i x x
mass located at x), and probability measures
χQ zm
η zm : A h k i1A. (1)
h k i X ∋ 7→ χQ zm X
h k i1
Using these definitions we may,givena sequence zn n∈N ofperturbations inZ,
{ }
express the Feynman-Kac distribution flow {ηhz0ni}n∈N recursively as
η zn−1 Q z
η zn = h 0 i h ni , n N (2)
h 0i ηhz0n−1iQhzni1X ∈
(where, by the previous convention, η z−1 = χ). Even though the previous
h 0 i
model may be applied in a non-temporal context, we will often refer to the
index n as “time”.
Example 1 (partiallydominatedstate-spacemodels). Let (X, ) be a measur-
able space, M : X [0,1] a Markov transition kernel, andXχ a probability
measure on (X, )×(Xthe→latter being referred to as the initial distribution). In
addition, let (Y,X) be another measurable space and g :X Y R a Markov
+
transition densityYwith respect tosome reference measureν×on (→Y, ). By agen-
Y
eral state-space model we mean the canonical version of the bivariate Markov
chain (Xn,Yn) n∈N having transition kernel
{ }
X Y (cid:15) ((x,y),A) (x′,y′)g(x′,y′)ν(dy′)M(x,dx′)
A
× ×X Y ∋ 7→ 1
ZZ
and initial distribution
(cid:15) A (x,y)g(x,y)ν(dy)χ(dx).
A
X Y ∋ 7→ 1
ZZ
Herethemarginal process Xn n∈N,referredtoas the stateprocess,is onlypar-
{ }
tiallyobservedthroughthe observationprocess Yn n∈N.Forthemodel (Xn,Yn) n∈N
{ } { }
defined in this way,
(i) the state process is a Markov chain with transition kernel M and initial
distribution χ,
(ii) the observations are, given the states, conditionally independent and such
that the marginal conditional distribution of each Y depends on X only
n n
and has density g(X , )
n
·
(we refer to [1, Section 2.2] for details). When operating on a well-specified
state-space model, a key ingredient is typically the computation of the flow of
predictor distributions, where the predictor η yn−1 at time n N is defined
as the conditional distribution of the state Xh 0givein the record∈yn−1 Yn of
n 0 ∈
realised historical observations uptotimen 1.UsingBayes’ formula (see,e.g.,
−
[1,Section3.2.2]fordetails),itisstraightforwardlyshownthatthepredictorflow
satisfies a perturbed Feynman-Kac recursion (2) with (X, ), M, and χ given
X
J. Olsson and R. Douc/Estimation of variance inparticle filters 6
above, the observations Yn n∈N playing the role of perturbations (i.e., Z Y
and ), and the l{ocal}likelihood functions g(,y) : y Y playing←the
role oZf p←oteYntial functions g y : y Y . We wil{l re·turn to t∈his }framework in
{ h i ∈ }
Section 5.
2.3. Sequential Monte Carlo methods
SMC methods approximate online the Feynman-Kac flow generated by (2) and
agivensequence zn n∈N ofperturbationsbypropagatingrecursivelyarandom
sample ξi N o{f X-}valued particles. More specifically, given a particle sample
ξi N {tanr}gie=t1ing η zn−1 in the sense that for all h F( ), η zn−1 h ⋍
η{ znn}−i=11h as N tendhs t0o inifinity, where ∈ X Nh 0 i
h 0 i
N
1
η zn−1 : A (ξi)
Nh 0 i X ∋ 7→ N 1A n
i=1
X
denotes the empirical measure associated with the particles, an updated parti-
cle sample ξi N approximating η zn is, as the perturbation z becomes
{ n+1}i=1 h 0i n
accessible, formed by Algorithm 1.
Data: ξi N , z
{ n}i=1 n
Result: ξi N
{ n+1}i=1
1 set Ω 0;
n
←
2 for i=1 N do
→
3 set ωi g z (ξi);
n ← h ni n
4 set Ω Ω +ωi;
n ← n n
5 for i=1 N do
6 draw I→i Cat( ωℓ/Ω N );
n+1 ∼ { n n}ℓ=1
7 draw ξni+1 ∼M(ξnIni+1,·);
Algorithm 1: SMC particle update
(In the algorithm above, Cat( ωℓ/Ω N ) denotes the categorical distribu-
{ n n}ℓ=1
tion induced by the normalised particle weights ωℓ/Ω N .) Algorithm 1 is
initialised at time n = 0 by drawing ξi N {χ(cid:15)nN. nF}oℓr=1all n N and all
h F( ), the convergence, as N tend{s 0to}i=in1fin∼ity, of η zn−1 h∈to η zn−1 h
∈ X Nh 0 i h 0 i
can be established in several probabilistic senses. In particular, the first CLT
for SMC methods was provided by [7], establishing that
√N η zn−1 h η zn−1 h D σ zn−1 (h)Z, (3)
Nh 0 i − h 0 i −→ h 0 i
where Z is standard(cid:0)normally distributed an(cid:1)d the asymptotic variance is given
by σ zn−1 :=σ zn−1 with
h 0 i 0h 0 i
n η zm−1 Q zn−1 (h η zn−1 h) 2
σ2 zn−1 :F( ) h h 0 i{ h m i − h 0 i } (4)
ℓh 0 i X ∋ 7→mX=ℓ (ηhz0m−1iQhzmn−1i1X)2
J. Olsson and R. Douc/Estimation of variance inparticle filters 7
(see also [3, 18, 10] for similar results). The fact that we in (4) define a trun-
catedversionof the variancewith only n ℓ+1 terms will be clear later on. In
−
the coming section we propose a lag-based, numerically stable estimator of the
{seσq2uhezn0nc−e1i{}σn2∈hNz0no−n1liin}en,∈Nasonf ainsycmrepastoest,icuvnadreiranccoenss.tTanhteceosmtimpuattaotrioanpaplrcooxmimpaletxes-
ity and memory requirements. Importantly, the estimator is obtained as a by-
productoftheparticlefilteroutputanddoesnotrequireadditionalsimulations.
The numerical stability is obtained at the price of a small bias, which may be
controlled under weak assumptions on the mixing properties of the model.
3. A lag-based variance estimator
3.1. The variance estimator proposed in [2]
Since Algorithm 1 resamples the particles at each time step, the particle cloud
may be associated with a tree describing the genealogical lineages of the par-
ticles. The estimators proposed in [2] and [19] are based on the particles’ Eve
indices Ei N (theterminologyisadoptedfrom[19]),whichare,foralln N,
{ n}i=1 ∈
defined as the indices of the time-zero ancestors of the particles ξi N . More
{ n}i=1
specifically, the Eve indices may, for all i J1,NK, be computed recursively in
∈
Algorithm 1 (just after Line 6) by letting
i for n=0,
Ei :=
n (EIni for n N∗.
n−1 ∈
Using the Eve indices, H. P. Chan and T. L. Lai proposed, in [2], for all n N,
σ2 zn−1 (h), with ∈
Nh 0 i
2
N
1
σ2 zn−1 :F( ) h h(ξj) η zn−1 h , (5)
Nh 0 i X ∋ 7→ N  n − Nh 0 i 
Xi=1 j:XEnj=i(cid:8) (cid:9)
 
as an estimator of σ2 zn−1 (h) for all h F( ). As mentioned in the introduc-
h 0 i ∈ X
tion,wewillrefertothisestimatorastheCLE.(Moreprecisely,in[2],focuswas
setontheupdated distributionflowsdiscussedinSection3.3below;theadapta-
tion is however straightforward.) In [19], a generalisation of the CLE, allowing
the particle population size N to vary between SMC iterations, is presented.
As the main result of [2], the consistency, as N tends to infinity, of the CLE is
established; see also [19, Theorem 1 and Corollary 1] for a generalisation.
The CLE is indeed remarkable, as it allows the variance to be estimated on-
lineinasinglerunoftheparticlefilterwithnofurthersimulation.Nevertheless,
asexplainedintheintroduction,thepreviousestimatorhasaseriousflawwhich
is related to the well-known particle path depletion phenomenon of SMC algo-
rithms. More specifically, resampling the particles systematically at each time
leadswithoutexceptiontoarandomtimepointbeforewhichallthegenealogical
J. Olsson and R. Douc/Estimation of variance inparticle filters 8
tracescoincide;wereferagainto[16],whichprovidesatimeuniform (NlogN)
O
bound on the expected number of generations back in time to this most recent
common ancestor. Thus, as n increases, the sets j J1,NK : Ej = i will
{ ∈ n }
eventually be empty for all indices i J1,NK except one, say, i , for which
0
j J1,NK:Ej =i =J1,NK.Asaco∈nsequence,eventually,σ2 zn−1 (h)=0
{for∈allh F( n),wh0ic}hmakestheestimatorimpractical.InthenNehxt0secition,we
∈ X
propose a simple modification of the CLE that stabilises numerically the same
at the cost of a negligible, controllable bias.
3.2. Our estimator
The estimator that we propose is based on the simple idea of stabilising nu-
merically the CLE by tracing, backwards in time, only a few generations of
the particle genealogy, rather than tracing the history all the way back to the
time-zeroancestors.In ourapproach,the Eveindices will be replacedby Enoch
indices1 defined, for all i J1,NK and m N, recursively as
∈ ∈
i for n=m,
Ei := (6)
m,n Ii
(E n for n>m.
m,n−1
In other words, for all n N, m J1,nK, and i J1,NK, ξEmi ,n is the ancestor
m
of ξi at time m. Now, let∈λ N∈be some fixed n∈umber, referred to as the lag,
andndefine n(λ):=(n λ) ∈0; then, we propose σ2 zn−1 (h), with
− ∨ N,λh 0 i
2
N
1
σ2 zn−1 :F( ) h h(ξj) η zn−1 h , (7)
N,λh 0 i X ∋ 7→ N  n − Nh 0 i 
Xi=1j:EnjX(λ),n=i(cid:8) (cid:9)
 
as an estimator of the variance σ2 zn−1 (h) for all n N, zn−1 Zn, and
h F( ). Online computation of thhe0Enioch indices E∈i N0 re∈quires the
∈ X { n(λ),n}i=1
propagation of a window Ei ,...,Ei N of indices; see Algorithm 2 for
{ n(λ),n n,n}i=1
a pseudo-code. As the length of the window is bounded by λ+1, the memory
demandoftheestimatoris (λN)independentlyofn.Moreover,sincegenealog-
O
ical tracing has a linear complexity in N, the total complexity of the estimator
is (λN), again independently of n.
O
1Two figures named Enoch appear in the 2nd as well as the 6th generations of the Ge-
nealogiesofGenesis,asthesonofCainandtheson-son-son-son-sonofSeth,respectively.
J. Olsson and R. Douc/Estimation of variance inparticle filters 9
Data: ξi N , Ei ,...,Ei N , z
{ n}i=1 { n(λ),n n,n}i=1 n
Result: ξi N , Ei ,...,Ei N
{ n+1}i=1 { (n+1)(λ),n+1 n+1,n+1}i=1
1 set Ω 0;
n
←
2 for i=1 N do
→
3 set ωi g z (ξi);
n ← h ni n
4 set Ω Ω +ωi;
n ← n n
5 for i=1 N do
6 draw I→i Cat( ωℓ/Ω N );
n+1 ∼ { n n}ℓ=1
7 draw ξni+1 ∼M(ξnIni+1,·);
8 for m=(n+1)(λ) n do
→
9 set Emi ,n+1 ←EmIni,+n1;
10 set Ei i;
n+1,n+1 ←
Algorithm 2: SMC particle and Enoch-index update
For n =0, Algorithm 2 is initialised by drawing ξi N χ(cid:15)N and setting
{ 0}i=1 ∼
Ei iforalli J1,NK.Attheendofthealgorithm,afterthesecondfor-loop,
0,0 ← ∈
an estimate
2
N
1
σ2 zn (h)= h(ξj ) η zn h
N,λh 0i N  { n+1 − Nh 0i }
Xi=1j:E(jn+1X)(λ),n+1=i 
 
of σ2 zn (h) may be formed for all h F( ).
λh 0i ∈ X
3.3. Variance estimators for flows of updated distributions
Some applications involve approximation of the updated measures
χQ zm−1 (g z )
η¯ zm : A h k i h mi1A , (8)
h k i X ∋ 7→ χQhzkm−1i(ghzmi1X)
for zm Zm−k+1, rather than the measures defined by (1).
k ∈
Example 2 (partially dominated state-spacemodels, revisited). In the case of
the partially dominated state-space models discussed in Example 1, the updated
mciseealsyu,riens t{hη¯ihsyc0noin}tne∈xNt,dfeofirnaeldl nthrouNg,hη¯(y8)narisetthheecfiolntedritdioisntarlibduistitorinbsu;tmionoreofptrhee-
state X given the realised obser∈vationhs y0ni Yn+1 up to time n (i.e., including
n 0 ∈
the last observation y ).
n
Since for all h F( ), by normalisation,
∈ X
η zm−1 (g z h)
η¯ zm h= h k i h mi ,
h k i η zm−1 g z
h k i h mi
J. Olsson and R. Douc/Estimation of variance inparticle filters 10
the flow {η¯hz0ni}n∈N of updated distributions is naturally approximated by the
flow of weighted empirical measures
η zm−1 (g z ) N ωi
η¯ zn :A Nh k i h mi1A = n (ξi), (9)
Nh 0i ∋X 7→ ηNhzkm−1ighzmi Xi=1 Ωn1A n
for some given sequence {zn}n∈N of perturbations, where the weights {ωni}Ni=1
andthe weightsumΩ arecomputedinAlgorithm1.Bythe normality(3)and
n
theconsistency(42)oneobtains,usingSlutsky’stheorem,forallzn Zn+1,the
0 ∈
central limit theorem
√N(η¯ zn h η¯ zn h) D σ¯ zn (h)Z, (10)
h 0i − h 0i −→ h 0i
asN tendstoinfinity,whereZ isstandardnormallydistributedandtheasymp-
totic variance is given by σ¯2 zn (h)=σ¯2 zn (h) with
h 0i 0h 0i
σ2 zn−1 (g z h η¯ zn h )
σ¯2 zn :F( ) h ℓh 0 i h ni{ − h 0i } (11)
ℓh 0i X ∋ 7→ (η zn−1 g z )2
h 0 i h ni
(where σ zn−1 is definedin (4) for the originalFeynman-Kacparticle model).
ℓh 0 i
In the case ℓ = 0, the expression (11) is found also in [12, Eqn. (17)]. In the
light of (11), casting our fixed-lag approach into the framework of updated
Feynman-Kac models yields the estimator
σ2 zn−1 (g z h η¯ zn h )
σ¯2 zn :F( ) h N,λh 0 i h ni{ − Nh 0i }
N,λh 0i X ∋ 7→ (η zn−1 g z )2
Nh 0 i h ni
2
N ωi
=N n h(ξj) η¯ zn h (12)
 Ω n − Nh 0i 
n
Xi=1j:EnjX(λ),n=i (cid:8) (cid:9)
 
for some suitable lag λ N (where the equality stems from the fact that
η zn−1 g z (h η¯ zn∈h) =0).
Nh 0 i{ h ni − Nh 0i }
4. Theoretical results
4.1. Main assumptions
Inthe following weassessthe theoreticalpropertiesofourestimator.All proofs
arepresentedinAppendixA.Weprefaceourmainassumptionsbythedefinition
of an r-local Doeblin set.
Definition 3. Let r N∗. A set C is r-local Doeblin with respect to M
and g if there exist p∈ositive function∈s Xε− : Zr R+ and ε+ : Zr R+, a
family µ z¯ ;z¯ Zr of probability meaCsures, a→nd a family Cϕ z¯ ;z¯→ Zr of
C C
positive{funhctiions∈such} that for all z¯ Zr, µ z¯ (C)=1 and{for halliA ∈ }and
C
∈ h i ∈X
x C,
∈
ε− z¯ ϕ z¯ (x)µ z¯ (A) Q z¯ (x,A C) ε+ z¯ ϕ z¯ (x)µ z¯ (A).
Ch i Ch i Ch i ≤ h i ∩ ≤ Ch i Ch i Ch i