Linear Variance Bounds for Particle Approximations of Time-Homogeneous Feynman-Kac Formulae

Linear V ariance Bounds for P article Appro ximati ons of Time-Homo geneous F eynman-Kac F orm ul ae NICK WHITELEY, NIK OLAS KANT AS & AJA Y JASRA Dep artment of Mathematics, University of Bristol, Bristol, BS8 1TW, UK. Dep artment of Ele ctric al & Ele ctr onic Engine ering, Imp erial Col le ge L ondon, L ondon, SW7 2AZ, UK. Dep artment of Statistics & Applie d Pr ob abili t y, National University of Singap or e, Singap or e, 117546, Sg. Octob er 29, 2018 Abstract This article establishes sufficien t cond itions for a linear-in-t i me b o u nd on the non-asymptotic v ariance for particle appro x i mations of time-homogeneous F eynman-Kac form ulae. These form u l ae app ea r in a wide v ariet y of app l ications including option pricing in finance and risk sensitiv e con trol in engineering. In direct Mon te Carlo appro ximation of these formula e, th e non-asymptotic v ariance typicall y increases at an exp onen tial rate in the time parameter. It is sh own that a linear b o und holds when a non-n e gativ e kernel, defined by the logarithmic potential function and Marko v kernel which sp ecif y th e F eynman-Kac model, satisfies a type of multipli cativ e drift condition and other regularit y assumptions. Examples illustrate that these conditions are general and flex ible enough to ac commodate tw o rather extreme cas es, whic h can occur in the con text of a non-compact state space: 1) when the p oten tial function is b oun ded ab o ve, not boun ded b elo w and the Mark o v kernel is not ergodic; and 2) when th e p oten tial function is not b ounded ab o ve, but the Marko v kernel itself satisfies a multiplicati ve d ri ft condition. Keywo rds: F eynman-Kac F orm u lae; Non-Asymp to tic V ariance; Multiplicativ e Drift Condition. 1 In tro duction On a state space X endow ed with a σ -algebra B ( X ) let M be a Marko v k er nel and let U : X → R be a logar it hmic p oten tial function. Then for x ∈ X , consider the s e q u ence of measures { γ n,x ; n ≥ 1 } defined by γ n,x ( ϕ ) := E x " exp n − 1 X k =0 U ( X k ) ! ϕ ( X n ) # , (1.1) for a suitable test function ϕ and where E x denotes exp ectation with resp ect to the law o f a Ma rk ov chain { X n ; n ≥ 0 } with transition kernel M , initialised from X 0 = x . F eynman-Kac for m ulae as in (1.1) arise in a v a riet y o f application domains. In the case that U is non-p ositiv e, the quantit y γ n,x (1) can be in ter pr eted as the probability of surviv al up to time step n of a 1 Marko v ia n pa rticle exploring an absorbing medium [Del Mora l a n d Miclo, 2003, Del Mora l and Doucet, 2004]; the particle evolv es a ccording to M and at time step k it is k illed with pro babilit y 1 − exp ( U ( X k )) . Another applica tio n is the ca lculation o f e xpectations at a terminal time with r espect to jump-diffusion pro cesses which may or may not b e pa rtially obser v ed (e.g. Jasr a and Doucet [20 09 ]). In particular, for option pricing in fina nce , there are a v ar iet y of options, (e.g. asian, barrie r ) whic h can b e written in the form (1.1) wher e the p oten tial function arise s from the pay-off function/change of mea- sure and the Markov kernel sp ecifies finite dimensional marg inals of so me partially obs e rv ed Lévy pro - cess (e.g. Jasr a and Del Moral [20 11 ]). It is remarked that in this latter example, the finite dimensio nal marginals can induce a time-homog eneous Markov chain that i s no t necessar ily erg odic. F urthermore, functionals as in (1.1 ) a rise in certain s t o c hastic control pr oblems, where one co nsiders the biv aria t e pro cess { X n = ( Y n , A n ); n ≥ 0 } with Y n being a controlled Markov chain and { A n ; n ≥ 0 } a control input pro cess. In some case s the transition kernel M can b e expressed a s M 1 ( y n , da n ) M 2 ( y n , a n , dy n +1 ) with M 1 corresp onding to the control law or p o licy and M 2 to the controlled pro cess dynamics. In a risk-sensitive optimal co n trol framework 1 n log γ n,x (1) aris es as a cost function one aims to minimise with resp ect to a n appropr iate class of p o licies; see [Whittle , 1990, Di Mas i and Stettner, 199 9 ] for de- tails. In such problems it is common to choos e U ( y , a ) to b e unbounded from ab o ve, e.g. U is usually chosen to be a quadra tic for linear and Gaussian state space mo dels [Whittle, 199 0 ]. More generally (1.1) arises as a spec i al case o f a time-inhomogeneous F eynman-Kac form ulae studied b y Del Mora l [2004]. The non-negative kernel Q ( x, dy ) := exp ( U ( x )) M ( x, dy ) , defines a linear op erator on functions Q ( ϕ ) ( x ) := R Q ( x, dy ) ϕ ( y ) and (1.1) can b e rewritten as γ n,x ( ϕ ) = Q n ( ϕ ) ( x ) , where Q n denotes the n -fold iterate of Q . In the applications des cribed a bov e, the F eynman-Ka c for m ulae (1.1) typically cannot b e e v aluated analy tica lly . Ho wev er, they may be approximated using a system of interacting particles [Del Mor al , 200 4 ]. These par ticle sy stems, a lso known as sequent ia l Monte Carlo metho ds in the computational statistics literature (e.g. Doucet et a l. [200 1]), hav e themselves b ecome an ob ject of in tensive s tudy , see amongst others [Crisan and Bain, 200 8 , Del Moral e t al., 2009, v an Handel, 2009, Chopin et al., 201 1 , Del Mo r al et al., 20 11 ] and r eferences therein for recent developmen ts in a v ar iet y of settings. The present work is concerned with second moment prop erties o f err ors asso ciated with the particle approximations of { γ n,x } . In or d er to obtain b ounds o n the relative v a riance, we control certain tenso r- pro duct functionals of these pa r ticle appr oximations, rece ntly addressed b y Cérou et al. [2011], using stability prop e r ties of the o perators { Q n ; n ≥ 1 } . These sta bility prop erties are themselv e s derived from the multiplicativ e erg odic and sp ectral theories of linea r opera t ors o n weighted ∞ -norm spaces due to Kon toyiannis and Meyn [2 003 , 2005]; this is one of the main nov elties of the pap er. By doing so we obtain a linea r-in- n relative v ariance b ound under assumptions on Q w hich are weak er than those relied up on in the literature to date and which rea dily hold o n non- c ompact spa ces. F ur th ermore, to the knowledge of the author s, these are the firs t res ult s which establish • that a linear-in- n bo und holds under conditions which can accommo date Q defined in terms o f a non-ergo dic Mar k ov kernel M , • that any form o f non-asymptotic sta bility result for par t icle approximations of F eynman K a c formulae holds under conditions which can accommo date U not bo un ded ab o ve. 2 1.1 In teracting Particle Systems Let N ∈ N be a p opulation size para meter . F or n ∈ N , let ζ ( N ) n := n ζ ( N ,i ) n ; 1 ≤ i ≤ N o be the n -th generation of the pa rticle sy stem , where ea ch particle, ζ ( N ,i ) n , is a random v ariable v alued in X . Denote η N n := 1 N P N i =1 δ ζ ( N,i ) n . The genera ti ons of the particle system n ζ ( N ) n ; n ≥ 0 o form a X N -v alued Mar k ov chain: for x ∈ X , the law of this chain is denoted by P N x and has tr a nsitions given in integral form by: P N x  ζ ( N ) 0 ∈ dy  = N Y i =1 δ x  dy i  , P N x  ζ ( N ) n ∈ dy    ζ ( N ) n − 1  = N Y i =1  η N n − 1 Q ( dy i ) η N n − 1 Q (1)  , n ≥ 1 , (1.2) where dy = d  y 1 , . . . y N  , 1 is the unit function and for some tes t function ϕ , η N n ( ϕ ) := 1 N P N i =1 ϕ  ζ ( N ,i ) n  (here the dep endence of η N n on x is suppressed from the notation). These tr ansition proba b ilities cor - resp ond to a simple selection- m utation ope ration: at each time step N pa rticles a re selected with replacement fro m the po pulation, on the basis of “fitness ” defined in terms o f e U , follo wed by each particle mut ating in a conditionally- ind ependent ma n ner accor ding to M . The empirical measur es  γ N n,x ; n ≥ 0  , defined by γ N n,x ( ϕ ) := n − 1 Y k =0 η N k  e U  η N n ( ϕ ) , n ≥ 1 , and γ N 0 ,x := δ x , a re taken a s approximations of { γ n,x } . It is well known [Del Mor al , 200 4 , Chapter 9] that E N x  γ N n,x ( ϕ )  = γ n,x ( ϕ ) , where E N x denotes exp ectation with res p ect to the law of the N -particle system. 1.2 Standard Regularit y Assumptions for Stability Recent work on a nalysis of tensor pro duct functionals ass o ciated with  γ N n,x ; n ≥ 0  , [Del Moral et al., 2009], has lead to imp ortan t results reg arding higher moments of the error asso ciated with these particle approximations; in a p ossibly time-inho m ogeneous context Cérou et al. [20 11] hav e prov ed a remark able linear-in- n b ound on the relative v ar iance of γ N n,x (1) . In the cont ext of time-homogeneo us F eynman-Kac mo dels, the a s sumpt io ns of Cérou et a l. [2011] are that sup x ∈ X U ( x ) < ∞ (1.3) and that for some m 0 ≥ 1 , there exists a finite constant c such that Q m 0 ( x, dy ) ≤ cQ m 0 ( x ′ , dy ) , ∀ ( x, x ′ ) ∈ X 2 . (1.4) 3 The result of Cérou et a l. [2011] is then of the form: N > c ( n + 1) = ⇒ E N x   γ N n,x (1) γ n,x (1) − 1 ! 2   ≤ c 4 N ( n + 1) , ∀ x ∈ X . (1.5) where c is as in (1.4). The efficiency of the par ticle approximation is therefore quite remark able: a natural alternative scheme for estimation o f γ n,x (1) is to s im ulate N independent copies o f the Markov chain with transition M and a ppro ximate the expec ta t ion in (1.1) by simple av era ging, but the relative v ariance in that case t y p ically explo des exp onential ly in n . The restrictio n is that (1.4) ra rely ho lds on non- compact spaces. The present w or k is concer ned with pr o ving a r esult of the same form as (1.5) under assumptions whic h are mor e r eadily verifiable when X is non-compact. The main result is summarized after the following discussion of (1 .3)-(1.4) and how they relate to the ass um ptions w e consider. The condition of (1.4) and its v aria n ts ar e very common in the literature on exp onen tial s ta b ilit y of nonlinear filters and their par t icle approximations, s e e for example [Del Mor al and Guionnet, 2 001 , Le Gland and Oudjane, 2004] and r eferences therein. It can b e int e r preted a s implying a uniform bo und on the r e la tiv e o scillations of the tota l mass of Q m 0 , i.e., Q m 0 (1) ( x ) Q m 0 (1) ( x ′ ) ≤ c, ∀ ( x, x ′ ) ∈ X 2 , (1.6) and this is very useful when controlling v ar ious functionals whic h a rise w hen analysing the relative v ariance a s in (1.5), (see Cérou et a l . 2 011 , Proof of Theorem 5.1). How e ver one may take the inter- pretation of (1.4) in another direction: it implies immediately that there exist finite measures, say β and ν , and ǫ > 0 such that Q m 0 ( x, dy ) ≤ β ( dy ) , Q m 0 ( x, dy ) ≥ ǫ ν ( dy ) , ∀ x ∈ X . (1.7) In the ca se that U = 0 (i.e Q = M is a proba bilistic kernel) and M is ψ -irreducible and aperio dic, this type of minoriza tion ov er t he en tire s ta te space X implies uniform ergo dicity of Q , which is in turn equiv a lent to Q satisfying a F oster-Lyapuno v drift condition with a b ounded drift function [Meyn and T weedie, 20 09, Theorem 16.2.2]. In the scenar io o f present interest, wher e in genera l U 6 = 0 , o ne may take V : X → [1 , ∞ ) to b e defined by V ( x ) = 1 , for all x , and then when (1.3) ho lds , it is trivially true that there exis ts δ ∈ (0 , 1) and b < ∞ s uch that Q satisfies the multiplic ative drift condition, Q  e V  ≤ e V (1 − δ )+ b I X , (1.8) where I X is the indicato r function on X . Q may then a lso be viewed as a b ounded linear op era tor on the space of real-v alued and b ounded functions o n X endow ed with the ∞ -nor m, which is nor m-equiv alent [in the sense of Meyn and T weedie , 200 9, p.393 ] to k ϕ k e V := sup x ∈ X | ϕ ( x ) | exp V ( x ) , with V any b ounded weigh ting function. As explained in the next section, the in ter est in writing (1.7 )-(1.8) is that conditions expres sed 4 in this manner hav e na tural generalisatio ns in the co nt ext o f weigh ted ∞ -norm function spaces with po ssibly unbounded V . 1.3 Setting and Main Result Del Moral [2004, (e.g. Chapter 4 and Section 12.4)] and Del Moral and Doucet [200 4] address the setting in which { Q n ; n ≥ 1 } is co ns idered a s a semigroup of b ounded linear op er ators on the Ba na ch space of r eal-v alued and b ounded functions on X , endow ed with the ∞ -norm, and Del Mora l a nd Miclo [2003] address the L 2 setting, connecting stabilit y pro p e rties of the measur es { γ n,x } and their no r mal- ized count erparts to the sp ectral theory o f b ounded linear op er a tors on Banach spaces. Kon toyiannis and Meyn [200 3, 200 5] hav e developed multiplicativ e ergo dic a nd sp ectra l theo ries of op erator s of the for m Q in the setting of weighte d ∞ - no rm spaces; a function space setting which has a l- ready proved to b e very fruitful for the s tudy of gener al state-spa ce Markov chains [Meyn and T weedie , 2009, Chapter 16] without reversibility assumptions. The r eader is referr ed to [Kon toyiannis and Meyn, 2003, 2005] for extensive historical pers p ective on this sp ectral theory and related topics, including (of particular relev ance in the present co nt ext) the theory of non-negative op er ators due to Nummelin [2004, Cha pter 5]. The work of [Kon toyiannis and Mey n, 2003, 2005, Meyn, 2006] is gea red to wards large deviation theory for sample path ergo dic averages n − 1 P n − 1 k =0 U ( X k ) under the tr ansition M and in that context it is na tural to state assumptions on M and U sepa rately . By contrast, when studying the particle systems describ ed ab ov e, we are not directly conce r ned with such sa mple paths, but rather the relationship b etw een the pr op erties of the particle approximations  γ N n,x  and their exact counter- parts { γ n,x } . Some of the results of Kon toyiannis and Meyn [20 03, 200 5] will b e a pplied to this effect, but starting from ass umptions expresse d dire c tly in terms of Q which reflect the scenario o f interest. The co re as sumptions in the present work (see Section 2 .2 for pre c is e statements) are that for some constants m 0 ≥ 1 , δ ∈ (0 , 1 ) and all d ≥ 1 large enough, Q m 0 ( x, dy ) ≥ ǫ d ν d ( dy ) , ∀ x ∈ C d , (1.9) Q  e V  ≤ e V (1 − δ )+ b d I C d , (1.10) with V un bo unded and C d := { x : V ( x ) ≤ d } ⊂ X a sublevel s et. It is noted that one recov ers the minorization and drift of (1.7) -(1.8) in the case that V is b ounded and C d = X . W e will also inv oke a densit y assumption which is w eaker than the upper bo und in ( 1 .7). It will be illustrated through examples in Section 4 that (1 .9)-(1.10) can be sa tisfied in circumstances which allow M to be non- ergo dic. F urthermor e, it will also be demonstrated that, in co ntrast to (1.3), conditions (1.9)-(1.10) can be sa tisfied with U not b ounded ab ov e, sub ject to strong eno ugh a ssumptions o n M and a r estriction on the growth rate of the p o s itive par t of U . The main result o btained in the present work (Theo rem 3.2 in Section 3) is a bo und of the form: N > c 1 ( n + 1) ≥ φ ( x ) = ⇒ E N x   γ N n,x (1) γ n,x (1) − 1 ! 2   ≤ c 2 4 N ( n + 1) v 2+ ǫ ( x ) h 2 0 ( x ) , 5 with φ ( x ) := c 1  1 B 1 log  B 2 0 v ( x ) h 0 ( x )  + 1  , where v ( x ) = e V ( x ) , B 0 , B 1 , c 1 , c 2 are co nstants which ar e indep endent of N , n and x and for a real n um ber a we denote as ⌈ a ⌉ the smallest int e g er j such that j ≥ a . In this display h 0 is the eigenfunction asso ciated with the principal eigenv alue of Q a nd the co nstant B 1 is directly related to the s ize o f the sp e ctral gap of Q . V erification of the existence of h 0 along with v arious other sp ectral q uantit ies plays a central role in the pr o ofs. W e note that Del Mo ral and Doucet [200 4], Céro u et a l. [2011] also consider the case in which exp U ( x ) may touch zero and the for mer ar e also directly concerned with approximation of the eigen- v alue λ co rresp onding to h 0 via the empirical probabilit y measures  η N n  . These issues are beyond the scop e o f the present article but the study o f these and r elated issues in a more general time- inhomogeneous s etting is underw ay . It is also rema rked that Cérou et al. [201 1] consider a mo re general t yp e o f particle system, whic h in volves an accept/reject ev olution mechanism. The approa ch taken here is also a pplica ble in that c o ntext, but for simplicity of presen tation w e only cons ider the selection-mutation transition in (1.2). The r emainder o f the paper is structured a s follows. Section 2 is lar gely exp ositor y: it intro duces v arious sp ectra l definitions and the main assumptions o f the pres e nt work and go es on to show how these assumptions v alidate the applica tio n of multi plicative ergo dicity results of Konto yiannis a nd Meyn [2005]. It is stress ed that muc h of the co nt ent of this sectio n is included in or der to mak e clear the simila r ities and differences b etw een the setting of interest and the main s tated a ssumptions a nd results of Kont oyiannis a nd Meyn [2005]. Section 3 deals with the v ariance b o unds for the particle approximations. Numerical examples are given in Section 4. Many o f the pr o ofs of the results in Section 2 a re in Appendix A. Some pro ofs and lemma s for the res ults in Section 3 can be found in Appendix B. 2 Multiplicati v e Er go dicit y 2.1 Notations and Con v entions Let X be a state space and B ( X ) b e an as s o ciated coun ta bly gener ated σ -algebr a. W e are typically inter- ested in the ca se X = R d x , d x ≥ 1 , but our results are readily applicable in the context of more general non-compact state-s pa ces. F or a weighting function v : X → [1 , ∞ ) , and ϕ a measura ble real-v alued function on X , define the no r m k ϕ k v := sup x ∈ X | ϕ ( x ) | /v ( x ) a nd let L v := { ϕ : X → R ; k ϕ k v < ∞} be the corr esp onding Banach spa ce. Throughout, when dea ling w ith weigh ting functions w e employ an low er/upp er - case conv ention for exp onentiation and write interc hang eably v ≡ e V . F or K a kernel on X × B ( X ) , a function ϕ and a measur e µ denote µ ( ϕ ) := R ϕ ( x ) µ ( dx ) , K ϕ ( x ) := R K ( x, dy ) ϕ ( y ) and µK ( · ) := R µ ( dx ) K ( x, · ) . Let P be the collection of pro bability measures on ( X , B ( X )) , and for a given weigh ting function v : X → [1 , ∞ ) let P v denote the subset o f such measure s 6 µ such that µ ( v ) < ∞ . F o r n ≥ 0 the n - fold iterate of K is deno ted: K 0 := I d, K n := K . . . K | {z } n times , n ≥ 1 . The induced op er a tor norm o f a linear op er ator K acting L v → L v is 9 K 9 v := sup  k K ϕ k v k ϕ k v ; ϕ ∈ L v , k ϕ k v 6 = 0  = sup {k K ϕ k v ; ϕ ∈ L v , | ϕ | ≤ v } . The sp ectrum of K as a n op er ator on L v , denoted by S v ( K ) , is the s et of complex z such that [ I z − K ] − 1 do es not exis t as a bo unded linear op era tor o n L v . The c o rresp o nding sp ectra l radius of K , denoted by ξ v ( K ) , is given by ξ v ( K ) := sup {| z | ; z ∈ S v ( K ) } = lim n →∞ 9 K n 9 1 /n v , where the limit alwa y s exists by s ubadditive arg ument s , but may b e infinite. The following definitions are from Kon toyiannis and Meyn [200 5]. • A p ole z 0 ∈ S v ( K ) is of fin ite multiplicity n if – for some ǫ 1 > 0 we have { z ∈ S v ( K ); | z − z 0 | ≤ ǫ 1 } = { z 0 } , – and the ass o ciated pro jection op era tor J := 1 2 π i Z ∂ { z : | z − z 0 |≤ ǫ 1 } [ I z − K ] − 1 d z , can b e expressed as a finite linear co mbination of some { s i } ⊂ L v and { ν i } ⊂ P v , J = n − 1 X i,j =0 m i,j [ s i ⊗ ν j ] , where [ s i ⊗ ν j ] ( x, dy ) = s i ( x ) ν j ( dy ) . • K a dmits a sp e ctr al gap in L v if there exists ǫ 0 > 0 such that S v ( K ) ∩ { z : | z | ≥ ξ v ( K ) − ǫ 0 } is finite and contains only p oles of finite mult iplicity . • K is v -uniform if it admits a sp ectral gap and there exists a unique p ole λ ∈ S v ( K ) of mu ltiplicit y 1, satisfying | λ | = ξ v ( K ) . • K ha s a discr ete sp e ctrum if for any compa ct set B ⊂ C \ { 0 } , S v ( K ) ∩ B is finite a nd contains only p o les of finite multiplicit y . • K is v -sep ar able if for a ny ǫ > 0 there exists a finite rank op erator b K ( ǫ ) such that 9 K − b K ( ǫ ) 9 v ≤ ǫ 2.2 Multiplicativ e Ergo dic Theorem In this section w e present the ma in assumptions and state some results from Konto yiannis a nd Meyn [2005] (see also Kon toyiannis and Meyn [20 03]). 7 2.2.1 Assumptions ( H1 ) The semigr oup { Q n ; n ≥ 1 } is ψ -ir r educible and ap erio dic (see Meyn [2 0 06, Section 2.1]). ( H2 ) There exis ts an un b o unded V : X → [1 , ∞ ) , constants m 0 ≥ 1 , δ ∈ (0 , 1) and d ≥ 1 with the following prop er ties: F or each d ≥ d and C d := { x ∈ X ; V ( x ) ≤ d } , • there exists ǫ d ∈ (0 , 1 ] a nd ν d ∈ P v such that C d is ( m 0 , ǫ d , ν d ) -small for Q , i.e., Q m 0 ( x, · ) ≥ I C d ( x ) ǫ d ν d ( · ) , ∀ x ∈ X , (2.1) with ν d ( C d ) > 0 . F urthermore Q m 0 ( C d ) ( x ) > 0 for all x ∈ X . • there exists b d < ∞ such that the following multiplicativ e drift condition holds, Q  e V  ≤ e V (1 − δ )+ b d I C d . (2.2) ( H3 ) U : X → R is such that U + := max ( U, 0) ∈ L V . ( H4 ) There exists t 0 ≥ 1 and for ea ch d ≥ d there exists a meas ur e β d , such that β d  e V  < ∞ and P x  X t 0 ∈ A, τ C c d > t 0  ≤ β d ( A ) , x ∈ C d , A ∈ B ( X ) , where P x denotes the law of the Marko v chain { X n } with transition M a nd τ A := inf { n ≥ 1 : X n ∈ A } . R emark 2.1 . W e take c a re to emphasize the follo wing differe nce s and similarities b etw e e n the a b ove assumptions and the setting of Konto yiannis and Me y n [200 5]. • Assumption (H2) equa tion (2.2) applies directly to the Q kernel, whereas Kon toyiannis and Meyn [2005] imp ose a multiplicativ e dr ift condition on M . The key issue is tha t the multiplicativ e dr ift condition for Q is the essential and implicit ingredient o f Lemma B.4 o f Kon toyiannis and Meyn [2005], and as we shall see in Section 4, under the conditions that U is b ounded ab ov e but not bounded b elow, assumption (H2) can hold without g eometric drift assumptions on M . A related phenomenon is co nsidered by Meyn [2006] in o rder to obtain “one- sided” large deviation principles for ergo dic sa mple-path av era ges for the chain w ith transition M . • Assumption (H2) requires the sublevel sets of V to b e small for Q and this is exploited in Lemma A.1. The explicit m 0 -step minorisation condition ma kes it easy to b ound below the spectr al radius of Q , see Lemma 2.1. I n the s e tting of Konto yiannis and Meyn [2003] the spec tr al r adius of Q is b ounded b elow by 1 as U is a ssumed cen ter ed with respect to the inv a r iant probability distribution for M . In the pres ent co nt e x t, this centering assumption is unnatural, esp ecially as we wan t to co nsider s ome situations where such an in v ar iant probability do es not exist. 8 • Assumption (H3) is weaker than the corr esp onding ass umption in the statement of [Konto yiannis a nd Meyn, 2005, Theorem 3.1]. How ever, (H3) coincides with the first part of [Kon toyiannis and Meyn, 2005, Equatio n 73], which combin ed with (H1), (H2) a nd (H4) in Lemma 2 .2 b elow, is enough to prove that Q has a discrete sp ectrum in L v . • As shown in [Kon toyiannis and Meyn, 200 5, Theorem 3.4] and [Kon toyiannis and Meyn, 20 03], a MET can b e pro ved without (H4), but at the cost o f restrictions on the cla s s of functions to which U b elongs which a re a little unwieldy . 2.2.2 Results W e now give a collection of results which are used to prov e the MET, Theorem 2.2. The proo fs a re given in Appendix A. It is remarked that the steps in the pro of of Theo rem 2.2 are effectively the same as part of the pro o f of Theorem 3.1 of Kon toyiannis and Me yn [2005], how ever, our starting assumptions are stated differently . The following preparator y lemma esta blis hes that the F eynman-Kac formula (1.1) is well defined and presents b ounds on the spec tr al radius of Q . Lemma 2. 1 . A s s u me ( H 2). Then for al l x ∈ X , n ≥ 1 , ϕ ∈ L v , | γ n,x ( ϕ ) | < ∞ , (2.3) and for al l d ≥ d , ǫ d ν d ( C d ) ≤ ξ v ( Q ) < ∞ , (2.4) wher e d is as in (H 2). T o cla rify how ass umptions (H1)-(H4) connect with the r e sults of K ontoyiannis and Meyn [200 5] we next present a lemma regarding the v -separ ability of Q which is a stepping s tone to the MET. Observe that the multipl icative drift condition (H2) implies that Q can b e appr oximated in norm to arbitrary precision by truncation to the s ublevel sets of V , in the sense that for any r ≥ d , I C c r Q  e V  ≤ e V − δr , (2.5) and then with b Q ( r ) := I C r Q , it follows immediately that 9 Q − b Q ( r ) 9 v ≤ e − δr . In the following lemma , which combines [Kon toyiannis and Meyn, 2005, Lemmata B.3-B.5 ] and is included here for complete- ness, the density assumption (H4) plays a key ro le in esta blishing that iterates of this truncation o f Q can b e approximated by a finite r a nk kernel. Lemma 2. 2 . A s s u me ( H 1)-(H4). Then Q 2 t 0 +2 is v -sep ar able, wher e t 0 is as in (H 4). The following theorem makes a key connection b etw een v -separa bility and a discrete spec trum. Theorem 2. 1. [Kontoyiannis and Meyn, 2005, The or em 3.5] If the li n e ar op er ator Q : L v → L v is b ounde d and Q t 0 : L v → L v is v -sep ar able for some t 0 ≥ 1 , then Q has a di scr ete sp e ctrum in L v . 9 Under (H2) Q is indeed b ounded, s o has a discrete spectrum in L v and then b y definition it also admits a sp ectral gap in L v . F or any θ > ξ v ( Q ) we may co nsider the resolven t op erato r defined by R θ := [ I θ − Q ] − 1 = ∞ X k =0 θ − k − 1 Q k , ( 2 .6) W e can now sta te and prov e the MET: Theorem 2.2 . A s s u me (H1)-(H 4). Then λ = ξ v ( Q ) is a maximal and isolate d eigenvalue for Q . F or any d ≥ d and θ > ξ v ( Q ) , the op er ator H θ ,d define d by H θ ,d := h I λ θ −  R θ − θ ( − m 0 − 1) ǫ d I C d ⊗ ν d i − 1 = ∞ X k =0 λ − k − 1 θ  R θ − θ ( − m 0 − 1) ǫ d I C d ⊗ ν d  k , (2.7) is b ounde d as an op er ator on L v , with λ θ := ( θ − ξ v ( Q )) − 1 . The function h 0 ∈ L v and me asur e µ 0 ∈ P v define d by h 0 := H θ ,d ( I C d ) µ 0 H θ ,d ( I C d ) , µ 0 := ν d H θ ,d ν d H θ ,d (1) . (2.8) ar e indep en dent of θ , d and satisfy Qh 0 = λh 0 , µ 0 Q = λµ 0 , µ 0 ( h 0 ) = 1 . F urthermor e, ther e exist c onstant s B 0 < ∞ and B 1 > 0 such that for any ϕ ∈ L v , any n ≥ 1 and any x ∈ X ,   λ − n γ n,x ( ϕ ) − h 0 ( x ) µ 0 ( ϕ )   ≤ k ϕ k v B 0 e − nB 1 v ( x ) . (2.9) Pr o of. W e give only a sketc h pro o f, as it is esse ntially that of Theorem 3 .1 o f Kont oyiannis and Meyn [2005]. As established in Lemma 2 .1, under our ass umptions 0 < ξ v ( Q ) < ∞ . F urthermore the semigroup asso c iated with Q is ψ -irr educible, and as obser ved a b ov e Q is b ounded on L v , has a discrete sp e ctrum a nd therefore admits a spec tr al gap in L v . Propo s ition 2 .8 of Konto yiannis a nd Meyn [2005] therefore applies. Thus Q is v -uniform and λ = ξ v ( Q ) is a ma ximal and isolated eigenv a lue. By the minorization co ndition of (H2) one can obtain a minoriza tio n condition for R θ of (2.6): R θ ( x, dy ) ≥ θ ( − m 0 − 1) ǫ d I C d ( x ) ν d ( dy ) , which holds for any d ≥ d , θ > ξ v ( Q ) . Therefore by the ar gument in Kon toyiannis and Meyn [2005][Pro of o f Prop osition 2.8], for any θ > ξ v ( Q ) and d ≥ d , the s pe c tr al r adiue o f [ R θ − θ ( − m 0 − 1) ǫ d I C d ⊗ ν d ] is strictly less than λ θ = ( θ − ξ v ( Q )) − 1 . Thu s H θ ,d is b ounded a s an op er ator on L v and the sum in (2.7) conv er ges in the op er ator norm. Then also by [Kon toyiannis and Mey n, 2005][Prop osition 2.8], H θ ,d ( I C d ) ∈ L v is an eigenfunc- tion for Q with eigenv alue λ = ξ v ( Q ) . By similar ar guments to Konto yiannis and Meyn [200 3][proof of Prop o s ition 4.5 ] it is easily verfied that ν d H θ ,d is an eigenmeasure . The normaliza tion to h 0 and µ 0 is 10 justified by the finiteness, under our assumptions, of the ass o ciated qua nt ities. By [Kon toyiannis a nd Meyn, 2003][Theorem 3.3 part (iii), see also co mmen ts on p.332] h 0 and µ 0 constructed using any θ , d ar e resp ectiv aly the ψ -e ssentially unique eigenfunction and unique eig enmeasure satisfying µ 0 ( X ) = 1 , µ 0 ( h 0 ) = 1 , hence the lack of dep endence o n θ , d . T o obtain (2.9) o ne may define the t wiste d k er nel: ˇ P ( x, dy ) := λ − 1 h − 1 0 ( x ) Q ( x, dy ) h 0 ( y ) , (2.10) which can b e seen to b e well defined as a Marko v kernel, a s λ is strictly po sitive and finite a nd (H2) implies h 0 is everywhere finit e and strictly p ositive. F ur thermore one observes immediately that ˇ P admits ˇ π , defined by ˇ π ( ϕ ) = µ 0 ( h 0 ϕ ) /µ 0 ( h 0 ) = µ 0 ( h 0 ϕ ) , as an inv aria nt proba bilit y distribution. By Lemma A.1 in Appe ndix A one ca n apply Theo rem 3.4 of Kon toyiannis and Meyn [2005] to the Markov chain asso ciated to the twisted kernel, (in the notatio n o f of Theo rem 3.4 of Konto yiannis a nd Meyn [2005], take g ≡ ϕ/h 0 , F ≡ 0 ). This r esults in the b ound (2.9), which completes the pro o f. R emark 2.2 . Up on dividing thr o ugh by h 0 , the equation (2.9) of the MET may b e view ed as a pro ba- bilistic, geometr ic ergo dic theorem for the t wisted chain asso ciated to the kernel (2.10) and the mo dified test function ϕ/h 0 , with a na tur a lly mo dified drift function ˇ v = e ˇ V prop ortiona l to v /h 0 . See Lemma A.1 in Appe ndix A. R emark 2.3 . The constant B 1 in equation (2.9) is directly related to the size of the sp ectra l gap of Q , see [Kon toyiannis and Meyn, 20 03, Pro of of Theorem 4.1]. 3 Non-Asymptotic V a riance 3.1 T ens or Pro duct F unctionals The v arious tensor product functionals considered in the rema inder of this pap er require some addi- tional notation. F or a meas urable function F on X 2 and a weight ing function v : X → [1 , ∞ ) , we define the nor m k F k v, 2 := sup x,y ∈ X 2 | F ( x, y ) | / ( v ( x ) v ( y )) and denote L v, 2 := n F : X 2 → R ; k F k v, 2 < ∞ o the corresp o nding function s pa ce. F or tw o functions ϕ 1 , ϕ 2 ∈ L v , we denote by ϕ 1 ⊗ ϕ 2 ∈ L v, 2 the tensor pro duct function defined b y ϕ 1 ⊗ ϕ 2 ( x, x ′ ) := ϕ 1 ( x ) ϕ 2 ( x ′ ) . Let K : X × B ( X ) → R + be a kernel on X . The t wo-fold tensor pro duct op er a tor corresp o nding to K is defined, for any F ∈ L v, 2 , by K ⊗ 2 ( F ) ( x, x ′ ) := Z X 2 K ( x, dy ) K ( x ′ , dy ′ ) F ( y , y ′ ) . The iterated op erator notation of the previous section is car r ied over so that K ⊗ 2 0 := I d, K ⊗ 2 n := K ⊗ 2 . . . K ⊗ 2 | {z } n times , n ≥ 1 . Corresp o nding to the pa r ticle empirical meas ures of section 1.1, for n ≥ 1 , we introduce the tensor 11 pro duct empirical measures (or 2-fo ld V − statistic):  η N n  ⊗ 2 := 1 N 2 X 1 ≤ i,j ≤ N δ ( ζ i n ,ζ j n ) ,  γ N n,x  ⊗ 2 := γ N n,x (1) 2  η N n  ⊗ 2 . F ollowing the definition of Cérou et a l. [2 011], the coa les cent integral op er a tor D , acting on func- tions on X 2 , is defined by D ( F ) ( x, x ′ ) = F ( x, x ) , ( x, x ′ ) ∈ X 2 . F or any 0 ≤ s ≤ ( n + 1) , w e denote by I n,s := { ( i 1 , ..., i s ) ∈ N s 0 ; 0 ≤ i 1 < . . . < i s ≤ n } the set of coalescent time configurations over a horizo n of leng th n + 1 and for ( i 1 , ..., i s ) ∈ I n,s and x ∈ X , the nonegative mea sure Γ ( i 1 ,...i s ) n,x on  X 2 , B  X 2  , and its no rmalised counterpart ¯ Γ ( i 1 ,...,i s ) n,x , are defined b y Γ ( i 1 ,...,i s ) n,x := γ ⊗ 2 i 1 ,x D Q ⊗ 2 i 2 − i 1 D . . . Q ⊗ 2 i s − i s − 1 D Q ⊗ 2 n − i s , ¯ Γ ( i 1 ,...i s ) n,x := Γ ( i 1 ,...,i s ) n,x γ n,x (1) 2 , (3.1) for s ≥ 1 , and for s = 0 , Γ ( ∅ ) n,x ( F ) := γ ⊗ 2 n,x ( F ) and ¯ Γ ( ∅ ) n,x ( F ) := η ⊗ 2 n ( F ) . W e r efer the re ader to Cérou et al. [201 1, Section 3] for a helpful visual representation of the integrals in the transp or t equation (3.1). W e ha ve alr e ady chec ked in Lemma 2.1 that the F eynman-Ka c form ula (1.1) is w ell defined under our ass umptions in the L v setting, whic h v alidates the denominator of (3 .1). When Theorem 2.2 holds, w e will deno te b y ˇ E x exp ectation with resp ect to the law of the twisted Marko v chain  ˇ X n ; n ≥ 0  , i.e that with trans ition kernel ˇ P as in equa tion (2.10) and initialised from ˇ X 0 = x . 3.2 Non-Asymptotic V aria nce In this section we give our main res ult. The pro of is detailed in section 3 .3. The following additional assumption imp oses so me further restrictions on the function class co nsidered, but this is not ov er ly demanding, considering that we will b e dealing with coalesc ed tenso r pro duct quant ities. ( H5 ) Let V and ¯ d b e a s in ass umption (H2). There ex ists 0 < ǫ 0 < ǫ and for all d ≥ ¯ d , there e xists b ∗ d < ∞ such that Q  e (1+ ǫ ) V  ≤ e (1+ ǫ ) V − (1+ ǫ 0 ) V + b ∗ d I C d . The following theorem is due to Cér ou et al. [201 1]. Theorem 3.1. [Cér ou et al., 20 11 , Pr op osition 3. 4] F or any n ≥ 1 , x ∈ X and N ≥ 1 the fol lowing exp ansion holds: E N x   γ N n,x (1) γ n,x (1) − 1 ! 2   = n +1 X s =1  1 − 1 N  ( n +1) − s 1 N s X ( i 1 ,...,i s ) ∈I n,s h ¯ Γ ( i 1 ,...i s ) n,x (1 ⊗ 1) − 1 i , (3.2) wher e E N x denotes exp e ctation w.r.t. the law of the N -p article syst em. 12 A full pro of is not pr ovided here. How ever, we no te that we may wr ite E N x   γ N n,x (1) γ n,x (1) − 1 ! 2   = E N x h  γ N n,x  ⊗ 2 (1 ⊗ 1) i γ n,x (1) 2 − 1 , (3.3) where the equality is due to the lack of bias pro p erty E N x  γ N n,x (1)  = γ n,x (1) and the definition of  γ N n,x  ⊗ 2 . In s umma r y , the pro of of Theo rem 3.1 in volves recursive calculation of the exp ectation on the r ight o f (3.3), follow ed by o rganisa tion o f the resulting terms in to the for m (3.2). The reader is directed to [Cérou et al., 20 11] for the details. It is remar ked that there is a different error decomp os ition in [Chan and Lai, 201 1], which can ho ld to an y order under appro priate regularity conditions; one w ould conjecture that this decompo sition can als o b e treated, but this is not considered here. The main result of this section is the follo wing theorem, whose pro o f is p ostp oned. Theorem 3.2. Assume (H1)-(H5). Then ther e ex ists c 1 < ∞ and c 2 < ∞ dep ending only on t he quantities in (H1)-(H 5) such that for al l x ∈ X , N > c 1 ( n + 1) ≥ φ ( x ) = ⇒ E N x   γ N n,x (1) γ n,x (1) − 1 ! 2   ≤ c 2 4 N ( n + 1) v 2+ ǫ ( x ) h 2 0 ( x ) , with φ ( x ) := c 1  1 B 1 log  B 2 0 v ( x ) h 0 ( x )  + 1  , and wher e B 0 and B 1 ar e as in The or em 2. 2 . 3.3 Construction of the Pro of In the following Section, w e detail the argument to pr ov e Theorem 3.2. T o that end, w e present the essence o f the a rgument with Propo sition 3.1 and Lemma 3.1 below; the pro ofs of whic h are in Appendix B along with some supp orting r esults. The pro o f of Theor em 3.2 is constructed in the following manner. By Theorem 3 .1 w e hav e the decomp osition (3.2) in ter ms of the op e r ators n ¯ Γ ( i 1 ,...i s ) n,x o . The pro of in Cér o u et al. [2011] fo cuses upo n controlling these expressions via the regularity co nditions mentioned in section 1.2; our pro o f will do the same, except under (H1)-(H5). Throughout the remainder of this pa pe r , let V ∗ : X → [1 , ∞ ) is defined by V ∗ ( x ) := V ( x ) (1 + ǫ ) − log h 0 ( x ) + log k h 0 k v (1+ ǫ ) , (3.4) where ǫ is as in (H5). W e pro ceed with the following key prop o sition. 13 Prop ositio n 3.1. Assume (H 1)-(H 5). Then ther e ex ists c < ∞ dep ending only on the quant ities in (H1)-(H5) s u ch that for al l n ≥ 1 , 0 ≤ s ≤ n + 1 , ( i 1 , ...i s ) ∈ I n,s , F ∈ L v 1 / 2 , 2 and x ∈ X , ¯ Γ ( i 1 ,...i s ) n,x ( F ) ≤ k F k v 1 / 2 , 2 c s +1 v ( x ) h 0 ( x ) ˇ E x h Q k ∈{ i 1 ,...,i s − 1 } v  ˇ X k  v ∗  ˇ X i s  i ˇ E x  1 /h 0  ˇ X n  2 , (3.5) with the c onventions that the pr o duct in t he numer ator is unity when s ≤ 1 , and in the c ase of s = 0 , i s = 0 . In the ab ove display, v is as in (H 2), h 0 ∈ L v is t he eigenfunction as in The or em 2.2 and v ∗ = e V ∗ is as in (3.4) . This res ult of Propo sition 3.1 connects the op era to rs n ¯ Γ ( i 1 ,...i s ) n,x o with exp ectations of the Lyapuno v functions v and v ∗ and the eigenfunction, w.r .t. the twisted chain. Giv en this result, one needs to control the n umera tor and denominato r. The latter ca n be achieved b y the MET of Theorem 2.2 and the former via the following: Lemma 3.1. Assume (H1)-(H5). Then ther e exists c < ∞ dep ending only on t he quantities in (H 1)- (H5) su ch that for any n ≥ 1 , 1 ≤ s ≤ n + 1 , ( i 1 , ..., i s ) ∈ I n,s , ˇ E x   Y k ∈{ i 1 ,...,i s } v  ˇ X k  v ∗  ˇ X n +1    ≤ c s +1 v ∗ ( x ) , ∀ x ∈ X , (3.6) wher e v ∗ is as in (3.4) . W e now pro ceed with the pro o f o f Theorem 3.2. Pr o of. [Proof of Theorem 3.2] By Proposition 3.1 and Lemma 3.1 w e hav e that there exists a finite constant c dep ending only o n the quantities in (H1)-(H5) such tha t ¯ Γ ( i 1 ,...i s ) n,x (1 ⊗ 1) ≤ c s +1 v ( x ) h 0 ( x ) v ∗ ( x ) 1 ˇ E x  1 /h 0  ˇ X n  2 . (3.7) Using the fact that ˇ E x  1 /h 0  ˇ X n  = γ n,x (1) / [ λ n h 0 ( x )] w e app eal to (2.9) of the MET of Theorem 2.2 as follows. Without loss o f genera lit y , it c an be assumed that B 0 > 1 . Then for a ll x ∈ X n ≥  1 B 1 log  B 2 0 v ( x ) h 0 ( x )  ⇒ 1 − B 0 e − B 1 n v ( x ) h 0 ( x ) ≥ B 0 − 1 B 0 ⇒ ˇ E x  1 /h 0  ˇ X n  ≥ B 0 − 1 B 0 . (3.8) Throughout the remainder o f the pro o f the left-most inequal it y in (3.8) is a s sumed to hold. Then combining (3.8) with (3.7) and recalling the definition o f v ∗ we hav e that there exists c 0 < ∞ such that ¯ Γ ( i 1 ,...i s ) n,x (1 ⊗ 1) ≤ c 0 c s +1 v 2+ ǫ ( x ) h 2 0 ( x ) . Pro ceeding by the essentially the same a rgument as in [Cérou et a l., 201 1, Pro of of Theorem 5.1], w e 14 use the identi t y: n +1 X s =1 X ( i 1 ,...,i s ) ∈I n,s Y j ∈{ i 1 ,...,i s } a j = " n Y s =0 (1 + a s ) # − 1 , which holds for any n ≥ 1 and { a s ; s ≥ 0 } , to establish via Theorem 3.1 that E N x   γ N n,x (1) γ n,x (1) − 1 ! 2   ≤ c 0 c v 2+ ǫ ( x ) h 2 0 ( x ) n +1 X s =1  1 − 1 N  ( n +1) − s 1 N s X ( i 1 ,...,i s ) ∈I n,s c s = c 0 c v 2+ ǫ ( x ) h 2 0 ( x )  1 − 1 N  n +1 "  1 + c N − 1  n +1 − 1 # ≤ c 0 c v 2+ ǫ ( x ) h 2 0 ( x ) "  1 + c N − 1  n +1 − 1 # . Then exactly as in [Céro u et al., 2011, Pro of of Cor ollary 5.2], N > 1 + c ( n + 1) ⇒  1 + c N − 1  n +1 − 1 ≤ 2 N − 1 c ( n + 1) ≤ 4 N c ( n + 1) . This completes the pro of. 4 Examples This section giv es some discussion and examples of circumstances in whic h the assumptions can b e satisfied. In particular w e focus o n the drift as s umption of (H2). It seems natural to consider tw o general cases: those in whic h it is not assumed, or it is assumed, that the Markov kernel M itself satisfies a multiplicativ e drift condition. 4.1 Cases without a m ultiplicativ e drift assumption on M In this situation, the decay of the p otential function plays a key role in establishing the m ultiplicativ e drift condition, illustrated as follows. Lemma 4.1. A ssume that ther e exists V : X → [1 , ∞ ) u nb ounde d such t hat 9 M 9 v < ∞ and for al l d ≥ 1 , C d is (1 , ǫ d , ν d ) -smal l for M , with ν d ( C d ) > 0 and M ( C d )( x ) > 0 for al l x . If for al l d ≥ 1 , inf x ∈ C d U ( x ) > −∞ , and ther e exists d 1 such that sup x ∈ C d 1 U ( x ) < ∞ and for some δ 1 ∈ (0 , 1) , sup x ∈ C c d 1 U ( x ) /V ( x ) ≤ − δ 1 , assumption (H2) is satisfie d. Pr o of. W e have Q  e V  ( x ) ≤ exp ( V ( x ) + U ( x ) + log 9 M 9 v ) , ∀ x ∈ X . As V is unbounded, for a ny δ ∈ (0 , δ 1 ) there exists d large enough such that for a ll x ∈ X and d ≥ d , I C c d ( x ) Q  e V  ( x ) ≤ exp ( V ( x )(1 − δ )) , I C d ( x ) Q  e V  ( x ) ≤ ex p  d + sup y ∈ C d U ( y ) + log 9 M 9 v  , 15 which is enoug h to v erify the drift part of (A2). The minorization condition with m 0 = 1 and the Q ( C d )( x ) > 0 pa rt are direct as U ( x ) is bo unded b elow on C d . In the extensive literatur e on Lyapunov drift for Mar ko v kernels there are several co nditions which immediately guarantee the existence of v such that 9 M 9 v < ∞ . F or exa mple, a ny M satisfying the po lynomial drift condition of Jar ner and Rob erts [20 02] automatically s atisfies 9 M 9 v < ∞ for the same v up to a fa c to r of e . How ever, erg o dicity of M is not necessary , as illustrated in the following simple example. 4.1.1 Gaussian Random W alk Let X := R a nd U and M b e defined by U ( x ) := − x 2 , M ( x, dy ) := 1 √ 2 π exp − ( y − x ) 2 2 ! dy , where dy denotes Lebesgue mea sure. T aking ψ as Leb esg ue measur e, the ψ -ir r educibilit y and ap eri- o dicity of { Q n ; n ≥ 1 } is immediate. F o r the drift a nd mino rization conditions of (H2), elementary manipulations s how that equa tion (2.2 ) holds with V ( x ) = x 2 / (2 (1 + δ 0 )) + 1 for suitable δ 0 > 0 and s olutions of the minoriza tion condition (2.1) are also e asily obtained. Condition (H3) is trivially satisfied beca use U is non-p ositive. The density assumption (H4) is satisfied with β d prop ortiona l to the restriction of Leb esg ue mea s ure to C d . Assumption (H5) holds for ǫ s mall enough and ǫ 0 = ǫ/ 2 . It is generally not ea sy to obtain or estimate v a lues for the constants in Theor em 3.2. In a ll the n umerical examples which follow, we consider a fixed v alue of N and consider the r elative v ariance as a function of the n a nd the initial condition x . The numerical results of F igure 4.1 sho w estimates o f E N x   γ N n,x (1) γ n,x (1) − 1 ! 2   with fixed N = 2000 , for v ario us x and n , with in ea ch ca se the exp e c ta tion a pproximated by averaging over 2 × 10 4 independent simulations o f the particle system. F or this mo del γ n,x (1) ca n b e co mputed analytically , and this exact v a lue was used in the estimates. The linear g rowth of the relative v ariance and its dependence on the initial p oint x is apparent from the figure. 4.2 Cases with a m ultiplicativ e drift assumption on M The following Le mma shows that condition (H2) holds for suitable U when M itself sa tisfies a multi- plicative drift condition. Lemma 4 .2. Assu me that t her e exists V : X → [1 , ∞ ) unb ounde d, δ 1 > 0 , d 1 ≥ 1 and for e ach d ≥ d 1 ther e exists b d < ∞ such that M  e V  ≤ e V (1 − δ 1 )+ b d I C d , (4.1) and the set C d = { x ; V ( x ) ≤ d } is (1 , ǫ d , ν d ) -smal l for M , with ν d ( C d ) > 0 and M ( C d )( x ) > 0 fo r al l x . Then if U + ∈ L V , lim r →∞   I C c r U +   V = 0 and for al l finite d , inf x ∈ C d U ( x ) > − ∞ , assumption (H2) holds. 16 −4 −2 0 2 4 0 0.1 0.2 x 0 −10 −5 0 5 10 0 0.05 0.1 0.15 0.2 x 0 0 20 40 60 80 100 0 0.05 0.1 0.15 0.2 n 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 n Figure 4.1 : T op: Gaussian ra ndo m walk mo del. Bottom: erg o dic autoreg ression mo del. Left: Relative v ariance vs . initial condition x 0 , at times  , n = 20 ; × , n = 40 ; ∗ , n = 60 ; ⋄ , n = 8 0 ; ◦ , n = 100 . Right : Relative v ariance vs. n , from initial co nditions (dashed) x 0 = 0 , (solid - top) x 0 = 4 , (solid - b ottom) x 0 = 10 . 17 Pr o of. Due to the dr ift condition (4.1), for a ny δ ∈ (0 , δ 1 ) , Q  e V  ≤ exp  V (1 − δ ) − ( δ 1 − δ ) V + U + + b d I C d  , and due to lim r →∞   I C c r U +   V = 0 , there exists d such that for all d ≥ d , Q  e V  ≤ exp  V ( 1 − δ ) + ¯ b d I C d  , where ¯ b d := b d + d k U + k V , which verifies the drift part of (H2). The minoriza tion co ndition with m 0 = 1 and Q ( C d )( x ) > 0 pa rt are direct as U ( x ) is bo unded b elow on C d . 4.2.1 Ergo dic Autoregression Let X := R a nd U and M b e defined by U ( x ) := | x | , M ( x, dy ) := 1 √ 2 π exp − ( y − αx ) 2 2 ! dy , for fixed | α | < 1 . Element ary manipulations then show that, for δ 0 > 0 and d large enough, M satisfies (4.1) with V ( x ) = x 2 / (2 (1 + δ 0 )) + 1 . As p er the random w alk example, M readily admits minorization on the sublevel sets C d . The po tenti al function U clea r ly satisfies (H3). Lemma 4.2 shows that (H2) is satisfied. The densit y assumption (H4) is s a tisfied for β d prop ortiona l to Lebesg ue measur e restric ted to C d . Again it is straightforward to chec k that (H5) is satisfied for ǫ > 0 small enough and ǫ 0 = ǫ/ 2 . Figure 4 .1 also shows estimates of the relative v aria nce obtained b y simulation fo r this mo del with α = 0 . 4 and using N = 10 4 particles, av eraged over 10 4 independent realizations. Again the linea r growth o f the v ar iance is apparent, but there a pp e a rs to be less v ariation with resp ect to the initial condition than in the random walk example. 4.2.2 Co x-Ingerso l l-Ros s Pro ces s The Cox-Ingersoll- Ro ss (CIR) pro ces s, [Cox et a l., 19 8 5], is a diffu sion pro cess that is t ypically us ed in financial a pplications to ca pture mean-re verting b e haviour and state-dep e nden t volatilit y , which is thought to o ccur in many real scenario s. The pro cess is defined via the sto chastic different ial equation: dX t = θ ( µ − X t ) dt + σ p X t dW t where { W t } is standard Br ownian motion, θ > 0 is the mea n-reversion rate, µ > 0 is the level of mean-reversion and σ > 0 is the volatilit y . W e assume that 2 θ µ σ 2 > 1 so that the pro ces s is stationa ry and never touches zer o. Throughout the r emainder of section 4.2.2, for ∆ > 0 we deno te b y M ∆ the tr ansition probability from a ny time t to t + ∆ of the CIR pro ces s with parameters θ, µ, σ . The following lemma iden tifies a drift function for M ∆ , exhibiting a tra de- off betw een g r owth r ate of the drift function sp ecified by a parameter s , the parameter s o f the CIR pro cess a nd the time step size ∆ . 18 Lemma 4 .3. F or s > 0 and ∆ > 0 , c onsider the c andidate drift function V : R + → [1 , ∞ ) , define d by V ( x ) := 1 + 4 θ sx σ 2 (1 − e − θ ∆ ) . (4.2) Then s u bje ct to t he c onditions: s ∈  0 , 1 − e − θ ∆ 2  , δ ∈  0 , 1 − e − θ ∆ 1 − 2 s  , d ≥ 1 − 2 θ µ log (1 − 2 s ) /σ 2 1 − e − θ ∆ / (1 − 2 s ) − δ =: d , (4.3) the fol lowing mu lt iplic ative drift c ondition is satisfie d: M ∆  e V  ≤ e V (1 − δ )+ b d I C d , with V as in (4.2) and b d := de − θ ∆ 1 − 2 s − 2 θ µ σ 2 log (1 − 2 s ) + 1 . Pr o of. F or t ≥ 0 define c t := 2 θ σ 2 (1 − e − θ t ) , κ := 4 θ µ σ 2 , and the scaled pro cess Z t := 2 c t X t . Conditional on X 0 = x , Z t has a non-central chi-square distribution with degree o f freedom κ and non-centralit y parameter taking the v alue 2 c t xe − θ t [Cox et al., 1985]. W e then hav e for a ny x ∈ X , M ∆  e V  ( x ) = E x [exp ( sZ ∆ )] exp(1) = exp  2 c ∆ xs  e − θ ∆ 1 − 2 s  − κ 2 log (1 − 2 s ) + 1  ≤ exp  V ( x )  e − θ ∆ 1 − 2 s  − κ 2 log (1 − 2 s ) + 1  . where the equalities hold due to the existence of the moment g enerating function E x [exp ( sZ t )] , for s < 1 / 2 , which is satisfied under the co nditions o f (4.3). Under these conditions we also then have fo r d ≥ d and x / ∈ C d , M ∆  e V  ( x ) ≤ exp  V ( x ) (1 − δ ) − d  1 − e − θ ∆ 1 − 2 s − δ  − κ 2 log (1 − 2 s ) + 1  ≤ exp [ V ( x ) (1 − δ )] , and for x ∈ C d , M  e V  ( x ) ≤ exp  d  e − θ ∆ 1 − 2 s  − κ 2 log (1 − 2 s ) + 1  = exp ( b d ) . W e will consider as an example the case where the Marko v chain { X n } is the skeleton o f the CIR pro cess ov er a discrete time grid of spacing ∆ and U ( x ) := α lo g x for some fixed α . Lemmata 4 .2 and 4.3 establish that (H2)-(H3) are satisfied and one ca n c heck (H4 )-(H5) are satisfied s imilarly to the 19 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10 −7 n Figure 4.2: Co x- Ingersoll- Ross pro cess. Relative v ariance vs. n , f r om initial conditions x 0 = 0 . 1 (dashed), x 0 = 1 (solid - bottom), x 0 = 3 (dot-dashed), x 0 = 10 (so lid top). previous example. Figure 4.2 displays estimates of the relative v ariance for this mo del, computed via sim ulation, when ∆ = 0 . 0 1 , (i.e. M ≡ M 0 . 01 ), α = 0 . 01 , θ = 10 , µ = 1 , and σ = 0 . 1 . This was obtained using N = 10 3 particles, averaged ov er 3 × 10 3 independent re alizations. Again the linear growth of the r elative v ariance is prese nt for different initial conditions. Note one may interpret γ 100 ,x (1) as the geometric mean E x [ Q 99 k =0 X 1 / 100 k ] , which can b e used for prediction in a v ariety o f financial applications. 5 Summary In this pap er we hav e es tablished a linea r-in- n b o und o n the non-a symptotic v ariance asso cia ted with particle approximations of time-homo geneous F e y nman-Kac fo rmulae, under assumptions that can b e verified on non-co mpact state- s paces. There ar e several po ssible extensions to this work. Firstly , to cons ider non-homogeneo us F eynman- Kac formulae, which o ccur ro utinely in a pplications such as filter ing and B ay esian statistics. Secondly , an impor tant developing area in the analysis of sequential Monte Carlo methods is the case when the dimension of the state-s pa ce ca n be very lar ge [Bes kos et al., 2011]. Suc h analysis has relied on classical geometric drift conditions and it would be in ter esting to consider the role of m ultiplicative drift conditions in this context. 20 A c kno wledg ement s W e w o uld like to thank the as so ciate editor and the referee for some very useful comments that have lead to considerable improv ements in the paper . The first and third authors ackno wledge the assistance of the Londo n Mathematical So ciety for their funding, via a r esearch in pair s grant. The second author was suppor ted by the EPSR C pr ogra mme gra nt on Co ntrol F or Energy a nd Sustainability EP/G066477 /1. A Pro ofs and Auxiliary Results for Section 2 Pr o of. [Proof of Le mma 2.1] Fix any d ≥ d . The upp er bo und of (2.3) is an immediate consequence of the inequality Q  e V  /e V ≤ e b d , implied by (2.2). F or the upper b ound of (2.4), use the standa rd inequality ξ v ( Q ) ≤ 9 Q 9 v and then also due to the drift condition in (2.2), 9 Q 9 v < ∞ . Now consider the lower bo und. It is claimed that for any k ≥ 3 and 1 ≤ j ≤ k − 1 , Q km 0  e V  ( x ) ≥ Q ( k − j ) m 0 ( I C d ) ( x ) ǫ j d ν d ( C d ) j − 1 ν d  e V  , ∀ x ∈ X , (A.1) where m 0 is as in (H2). F or each k , the claim is verified by induction in j ; fix k ≥ 3 arbitrarily . F or j = 1 , Q km 0  e V  ( x ) ≥ Q ( k − 1) m 0  I C d Q m 0  e V  ( x ) ≥ Q ( k − 1) m 0 ( I C d ) ( x ) ǫ d ν d  e V  which initializes the induction. Now assume that (A.1) holds a t r ank 1 ≤ j < k − 1 . Then at rank j + 1 , a pplying the induction hypothesis Q km 0  e V  ( x ) ≥ Q ( k − j − 1) m 0 ( I C d Q m 0 ( I C d )) ( x ) ǫ j d ν d ( C d ) j − 1 ν d  e V  ≥ Q ( k − j − 1) m 0 ( I C d ) ( x ) ǫ j +1 d ν d ( C d ) j ν d  e V  , ∀ x ∈ X , where (2.1) has b een applied, th us the cla im is verified. Now applying (A.1) with j = k − 1 gives, Q km 0  e V  ( x ) e V ( x ) ≥ Q m 0 ( I C d ) ( x ) e V ( x ) ǫ k − 1 d ν d ( C d ) k − 2 ν d  e V  > 0 , ∀ x ∈ X , which implies that 9 Q km 0 9 1 / ( km 0 ) v ≥ ǫ 1 − 1 / ( km 0 ) d ν d ( C d ) 1 − 2 / ( km 0 ) ν d  e V  1 / ( km 0 )  sup x ∈ X Q ( I C d ) ( x ) e V ( x )  1 / ( km 0 ) . T aking k → ∞ is enough to verify (2 .4), as lim n →∞ 9 Q n 9 1 /n v alwa ys e xists by subadditivity . Pr o of. [Proof of Le mma 2.2] Set r ≥ d arbitra r ily and let b Q ( r ) := I C r Q . F or n ≥ 1 , denote by b Q ( r ) n the n -fold iterate of b Q ( r ) . 21 Then under (H3), b Q ( r ) t 0 +1 ( x, A ) = E x " t 0 Y n =0 I C r ( X n ) exp ( U ( X n )) I A ( X t 0 +1 ) # ≤ exp  rt 0   U +   V  E x " t 0 Y n =0 I C r ( X n ) exp ( U ( X t 0 )) I A ( X t 0 +1 ) # , ∀ x ∈ X , A ∈ B ( X ) , and therefore under (H4), b Q ( r ) t 0 +1 ( x, A ) ≤ β ∗ r ( A ) := e xp  rt 0   U +   V  Z C r β r ( dy ) Q ( y , A ) , ∀ x ∈ X , A ∈ B ( X ) . (A.2) Lemma B3 of [Kon toyiannis a nd Meyn, 2005] then implies that b Q ( r ) 2 t 0 +2 is v -separ able. In or der to establish that Q 2 t 0 +2 is v -sepa rable, we will prov e that 9 Q 2 t 0 +2 − b Q ( r ) 2 t 0 +2 9 v can b e made a rbitrarily small through suitable choice o f r . B y dec o mp o sing the difference Q 2 t 0 +2 − b Q ( r ) 2 t 0 +2 in a telescoping fashion and a pplying the sub-additive and sub-mult iplica tive prop erties o f the op erato r norm we obtain: 9 Q 2 t 0 +2 − b Q ( r ) 2 t 0 +2 9 v ≤ 2 t 0 +1 X n =0 9 b Q ( r ) 2 t 0 +2 − ( n +1) Q n +1 − b Q ( r ) 2 t 0 +2 − n Q n 9 v , ≤ 9 Q − b Q ( r ) 9 v 2 t 0 +1 X n =0 9 b Q ( r ) 2 t 0 +2 − ( n +1) 9 v 9 Q n 9 v . (A.3) Now for any n ≥ 0 , sup r 9 b Q ( r ) n 9 v ≤ 9 Q n 9 v < ∞ , where the final inequa lit y follo ws from equation (2.3) o f Lemma 2.1, and by (2.5) we hav e 9 Q − b Q ( r ) 9 v → 0 as r → ∞ . Therefor e it follows from (A.3) that 9 Q 2 t 0 +2 − b Q ( r ) 2 t 0 +2 9 v → 0 as r → ∞ , so we conclude that Q 2 t 0 +2 is v -separ able. This c ompletes the pro of. The following lemma co nsiders the twisted kernel ˇ P defined in (2.10). Lemma A.1 . Ass u me (H1)-(H4). Then t her e exists δ 0 ∈ (0 , δ ) , d 0 ≥ 1 and for any d ≥ d 0 , ther e exists ˇ b d < ∞ such that ˇ P  e ˇ V  ≤ e ˇ V − δ 0 V + ˇ b d I C d , (A.4) sup x ∈ C d e ˇ V ( x ) < ∞ , (A.5) wher e ˇ V : X → [1 , ∞ ) is define d by ˇ V ( x ) := V ( x ) − log h 0 ( x ) + log k h 0 k v . F urthermor e, ther e exists ρ < 1 , dep ending only on d 0 and δ 0 , and for any d ≥ d 0 ther e exists ˇ b ′ d < ∞ such that ˇ P  e ˇ V  ≤ ρe ˇ V + ˇ b ′ d I C d . (A.6) Pr o of. Under the assumptions of the lemma, we hav e already seen via [Kon toyiannis and Meyn, 2 005, Prop osition 2.8] that the twisted kernel is well defined. Firs t consider, (A.4); under (H2), setting 22 δ 0 ∈ (0 , δ ) , for any d ≥ d , ˇ P  e V h 0  = λ − 1 h − 1 0 Q  e V  ≤ exp ( V − log h 0 − δ 0 V − ( δ − δ 0 ) V − log λ + b d I C d ) . As V is unbounded, there exists d 0 such that for all d ≥ d 0 , equation (A.4) holds with ˇ b d := b d − log λ . F or (A.5) by itera tion of the eigenfunction equation, we have that for a ny d ≥ d 0 , h 0 ( x ) = λ − m 0 Q m 0 ( h 0 ) ( x ) ≥ ǫ d ν d ( h 0 ) , ∀ x ∈ C d where we apply the minoriza tion pa rt of (H2) to obtain the inequality . It remains to establish (A.6). First c o nsidering the ca se x / ∈ C d , (A.4) implies that ˇ P  e ˇ V  ( x ) ≤ e ˇ V ( x ) − δ 0 V ( x ) ≤ e ˇ V ( x ) − δ 0 d so that (A.6) holds with ρ := e − δ 0 d 0 . F or x ∈ C d , equation (A.4) shows that (A.6) with ˇ b ′ d := exp( d − lo g ǫ d − lo g ν d ( h 0 ) + ˇ b d + lo g k h 0 k v ) . B Pro ofs and Auxiliary Results for Section 3 In this a ppendix we deta il the pr o ofs and auxiliary results tha t a re used in Section 3 . The pr o ofs and results are pro vided in a logical order ; that is , eac h result at most depends on the pr eceding one(s). In particular, the pro o f of Lemma 3.1 follows the pr o of of Lemma B.1. Lemma B. 1. A ssume (H 1)-(H5).Then ther e exists ¯ ρ < 1 , d 0 ≥ 1 and for any d ≥ d 0 ther e exists ¯ b d < ∞ and ¯ b ′ d < ∞ such that ˇ P  e V ∗  ≤ e V ∗ − V + ¯ b d I C d (B.1) ˇ P  e V ∗  ≤ ¯ ρe V ∗ + ¯ b ′ d I C d , (B.2) wher e V ∗ is as in e qu ation (3.4). Pr o of. Under the ass umptions o f the lemma, Theorem 2.2 holds, the eigenfunction h 0 ∈ L v , and the t w is ted k ernel is well defined. Then under (H5), w e have for any d ≥ d , ˇ P  e V (1+ ǫ ) h 0  = λ − 1 h − 1 0 Q  e V (1+ ǫ )  ≤ exp ( V (1 + ǫ ) − log h 0 − V − ǫ 0 V − log λ + b ∗ d I C d ) . As V is unbounded, there e x ists d 0 such that for all d ≥ d 0 , equation (B.1) holds with ¯ b d := b ∗ d − log λ . The pro of of (B.2) then follows exactly as in the pr o of o f Lemma A.1. Pr o of. [Proof o f Lemma 3.1] W e first consider so me b ounds on itera tes of the twisted kernel. Standard iteration of the g eometric drift condition in equation (B.2) s hows that there exists a finite cons ta nt c 1 such that sup n ≥ 0 ˇ P n ( v ∗ ) ( x ) ≤ c 1 v ∗ ( x ) , x ∈ X , (B.3) 23 and then due to the multiplicativ e dr ift condition in equation (B.1), sup n> 0 v ( x ) ˇ P n ( v ∗ ) ( x ) = sup n ≥ 0 v ( x ) ˇ P ˇ P n − 1 ( v ∗ ) ( x ) ≤ c 1 v ( x ) ˇ P ( v ∗ ) ( x ) ≤ cv ∗ ( x ) , x ∈ X , (B.4) where c := c 1 e ¯ b d . In order to prov e (3.6) first fix arbitra rily n ≥ 1 , 1 ≤ s ≤ n + 1 and ( i 1 , . . . , i s ) ∈ I n,s . The pro of is via a backward inductive a rgument through the coalesc ent time indices. Assume that a t rank 1 < j < s , v ( x ) ˇ E x   Y k ∈{ i j +1 − i j ,...,i s − i j } v  ˇ X k  v ∗  ˇ X n +1 − i j    ≤ c s +1 − j v ∗ ( x ) . (B.5) Assuming (B.5) is true, then a t rank j − 1 , v ( x ) ˇ E x   Y k ∈{ i j − i j − 1 ,...,i s − i j − 1 } v  ˇ X k  v ∗  ˇ X n +1 − i j − 1    = v ( x ) Z ˇ P i j − i j − 1 ( x, dx ′ ) v ( x ′ ) ˇ E x ′   Y k ∈{ i j +1 − i j ,...,i s − i j } v  ˇ X k  v ∗  ˇ X n +1 − i j    ≤ c s +1 − j v ( x ) Z ˇ P i j − i j − 1 ( x, dx ′ ) v ∗ ( x ′ ) ≤ c s +1 − ( j − 1) v ∗ ( x ) , where the final inequality is due to equation (B.4). F urthermor e v ( x ) ˇ E x  v ∗  ˇ X n +1 − i s  = v ( x ) ˇ P n +1 − i s ( v ∗ ) ( x ) ≤ cv ∗ ( x ) , where the inequality is ag ain due to (B.4) and therefor e a t rank j = s − 1 , v ( x ) ˇ E x   Y k =( i s − i s − 1 ) v  ˇ X k  v ∗  ˇ X n +1 − i s − 1    = v ( x ) Z ˇ P i s − i s − 1 ( x, dx ′ ) v ( x ′ ) ˇ E x ′  v ∗  ˇ X n − i s  ≤ cv ( x ) Z ˇ P i s − i s − 1 ( x, dx ′ ) v ∗ ( x ′ ) ≤ c 2 v ∗ ( x ′ ) . The ab ov e arguments prove that (B.5) holds at ra nk j = 1 a nd the proo f o f the Lemma is then a ls o complete as n + 1 , 1 ≤ s ≤ n + 1 and ( i 1 , . . . , i s ) ∈ I n,s were arbitrary . Lemma B.2. Assume (H 1)-(H5). Then ther e exists c < ∞ dep ending only on the quantities in (H1)-(H5) s u ch that for any n ≥ 1 and ϕ : X → R + 0 , λ − 2 n D Q ⊗ 2 n ( ϕ ⊗ v ) ( x, x ′ ) ≤ cv ( x ) h 0 ( x ) ˇ P n  ϕ h 0  ( x ) , ( x, x ′ ) ∈ X , (B.6) 24 wher e v is as in (H2), and λ and h 0 ∈ L v ar e r esp e ctively the eigenvalue and eigenfunction as in The or em 2.2 . Pr o of. By standard iteration o f the geo metric drift c o ndition in equation (A.6) of Lemma A.1, there is a finite constant c suc h tha t sup n ≥ 0 ˇ P n ( ˇ v ) ( x ) ≤ c ˇ v ( x ) , x ∈ X . (B.7) Then due to the definition o f the t wisted kernel and ˇ v (see Lemma A.1), there exis ts a consta nt c such that for any n ≥ 1 , and ϕ : X → R + 0 , λ − 2 n Q ⊗ 2 n ( ϕ ⊗ v ) ( x, x ′ ) = h 0 ( x ) h 0 ( x ′ ) ˇ P ⊗ 2 n  ϕ h 0 ⊗ v h 0  ( x, x ′ ) ≤ ch 0 ( x ) h 0 ( x ′ ) ˇ P ⊗ 2 n  ϕ h 0 ⊗ ˇ v  ( x, x ′ ) ≤ ch 0 ( x ) ˇ P n  ϕ h 0  ( x ) v ( x ′ ) , ( x, x ′ ) ∈ X 2 , (B.8) where the final inequality is due to (B.7). Lemma B.3. Assume (H 1)-(H5). Then ther e exists c < ∞ dep ending only on the quantities in (H1)-(H5) s u ch that for any m ≥ 1 , n ≥ 0 and ( x, x ′ ) ∈ X 2 , λ − 2( m + n ) D Q ⊗ 2 m D Q ⊗ 2 n  v 1 / 2 ⊗ v 1 / 2  ( x, x ′ ) ≤ cv ( x ) h 0 ( x ) ˇ E x  v ∗  ˇ X m  . Pr o of. Throughout the pro of c is a finite co ns tant whose v alue may change on each app ear ance. When n = 0 , λ − 2( m + n ) D Q ⊗ 2 m D Q ⊗ 2 n  v 1 / 2 ⊗ v 1 / 2  ( x, x ′ ) = λ − 2( m + n ) Q ⊗ 2 m  D  v 1 / 2 ⊗ v 1 / 2  ( x, x ) = λ − 2( n + m ) Q ⊗ 2 m ( v ⊗ 1) ( x, x ) ≤ cv ( x ) h 0 ( x ) ˇ P m  v h 0  ( x ) ≤ cv ( x ) h 0 ( x ) ˇ P m ( v ∗ ) ( x ) = cv ( x ) h 0 ( x ) ˇ E x  v ∗  ˇ X m  , where the first inequa lity is due to Lemma B.2 and the second inequality is due to the definition of v ∗ . Now consider the ca se n ≥ 1 . W e ha ve λ − 2 n D Q ⊗ 2 n  v 1 / 2 ⊗ v 1 / 2  ( x, x ′ ) ≤ cv ( x ) h 0 ( x ) ˇ P n  v (1+ ǫ 0 ) h 0  ( x ) ≤ cv ( x ) h 0 ( x ) ˇ P n ( v ∗ ) ( x ) ≤ cv ( x ) h 0 ( x ) ˇ E x  v ∗  ˇ X n  v ( x ′ ) , where we hav e used v ≥ 1 , Lemma B.2 with ϕ = v , the definition of v ∗ and ag ain v ≥ 1 . A further 25 application of Lemma B.2 with ϕ ( x ) = v ( x ) h 0 ( x ) ˇ E x  v ∗  ˇ X n  and an application of Lemma 3.1 yields: λ − 2( m + n ) D Q ⊗ 2 m D Q ⊗ 2 n  v 1 / 2 ⊗ v 1 / 2  ( x, x ) ≤ c 2 v ( x ) h 0 ( x ) ˇ E x  ˇ E ˇ X m  v  ˇ X 0  v ∗  ˇ X n  ≤ c 2 v ( x ) h 0 ( x ) ˇ E x  v ∗  ˇ X m  . This completes the pro of. Pr o of. [Proof of Pro p osition 3.1] The starting p oint o f the pro o f is to write, using the definition of the t w is ted k ernel, ¯ Γ ( i 1 ,...i s ) n,x ( F ) = λ − 2 n Γ ( i 1 ,...,i s ) n,x ( F ) λ − 2 n γ n,x (1) 2 = λ − 2 n Γ ( i 1 ,...,i s ) n,x ( F ) h 2 0 ( x ) ˇ E x  1 /h 0  ˇ X n  2 . Thu s in order prov e (3.5), we need to prov e λ − 2 n h − 2 0 ( x )Γ ( i 1 ,...,i s ) n,x ( F ) ≤ k F k v 1 / 2 , 2 c s +1 v ( x ) h 0 ( x ) ˇ E x   Y k ∈{ i 1 ,...,i s − 1 } v  ˇ X k  v ∗  ˇ X i s    , (B.9) for each n ≥ 1 , 0 ≤ s ≤ n + 1 and ea ch p ossible co nfiguration of the coalescent time indices ( i 1 , ..., i s ) ∈ I n,s . W e will consider fir s t the case s > 1 and then s ≤ 1 . Throughout the r e mainder of the pr o of, c denotes a finite and p o sitive constant, whos e v alue ma y change o n each app ear ance but dep ends o nly on the constants in (H1)-(H5). Consider the ca se s > 1 . It is claimed that there ex is ts a finite constant c such that for any n ≥ 1 , ( x, x ′ ) ∈ X 2 , F ∈ L v 1 / 2 , 2 , 1 < s ≤ n + 1 , a nd any ( i 1 , . . . , i s ) ∈ I n,s , λ − 2( n − i 1 ) D Q ⊗ 2 i 2 − i 1 . . . D Q ⊗ 2 i s − i s − 1 D Q ⊗ 2 n − i s  v 1 / 2 ⊗ v 1 / 2  ( x, x ′ ) ≤ c s +1 v ( x ) h 0 ( x ) ˇ E x   Y k ∈{ i 2 − i 1 ,...,i s − 1 − i 1 } v  ˇ X k  v ∗  ˇ X i s − i 1    , (B.10) with the conven tion that the pro duct is equal to unit y when s = 2 . F or a given n , the claim is proved b y backw a rd induction thr o ugh the coalescent time indices. The inductive hypothesis is that at rank 1 ≤ j ≤ s − 1 , λ − 2( n − i j ) D Q ⊗ 2 i j +1 − i j . . . D Q ⊗ 2 i s − i s − 1 D Q ⊗ 2 n − i s  v 1 / 2 ⊗ v 1 / 2  ( x, x ′ ) ≤ c s − j +1 v ( x ) h 0 ( x ) ˇ E x   Y k ∈{ i j +1 − i j ,...,i s − 1 − i j } v  ˇ X k  v ∗  ˇ X i s − i j    , (B.11) with the conv ention that the pro duct equals unit y when j + 1 = s . T o initialise the induction, w e hav e a t rank j = s − 1 that the left hand side of (B.11) is λ − 2( n − i s − 1 ) D Q ⊗ 2 i s − i s − 1 D Q ⊗ 2 n − i s  v 1 / 2 ⊗ v 1 / 2  ( x, x ′ ) , and Lemma B.3 then shows immediately that (B.11) do es indeed hold a t rank s − 1 . W e p oint o ut that 26 the constraint F ∈ L v 1 / 2 , 2 in the statement of the pr op osition is imp osed b eca use in the ca s e i s = n we immediately encoun ter D Q ⊗ 2 n − i s ( v 1 / 2 ⊗ v 1 / 2 ) = D ( v 1 / 2 ⊗ v 1 / 2 ) = v , and we can control in tegrals in volving v using the drift conditions, as in Lemma B.3. If w e were to g ive a separate treatment of Γ ( i 1 ,...,i s ) n,x ( F ) for co alescent time co nfigurations in which i s 6 = n , the constraint on F could b e rela x ed to a larger function class . Pro ceeding with the induction, when the hypo thesis (B.11) h o lds at rank j , we hav e at rank j − 1 : λ − 2( n − i j − 1 ) D Q ⊗ 2 i j − i j − 1 . . . D Q ⊗ 2 i s − i s − 1 D Q ⊗ 2 n − i s  v 1 / 2 ⊗ v 1 / 2  ( x, x ′ ) ≤ c s − j +2 v ( x ) h 0 ( x )   Z ˇ P i j − i j − 1 ( x, dy ) v ( y ) ˇ E y   Y k ∈{ i j +1 − i j ,...,i s − 1 − i j } v  ˇ X k  v ∗  ˇ X i s − i j      = c s − j +2 v ( x ) h 0 ( x ) ˇ E x   Y k ∈{ i j − i j − 1 ,...,i s − 1 − i j − 1 } v  ˇ X k  v ∗  ˇ X i s − i j − 1    , where the inequalit y follows from applying the induction hypo thesis, then m ultiplying by v ( x ′ ) ≥ 1 and then applying Lemma B.2 with ϕ ( x ) the x -dependent part of the r ight ha nd side of (B.11). This concludes the inductive pro o f of (B.10). Consider the case s > 1 , i 1 = 0 . Multiplying the r ight hand side of (B.10) b y v ( ˇ X 0 ) = v ( x ) ≥ 1 and recalling the definition of Γ ( i 1 ,...,i s ) n,x and γ N 0 ,x = δ x , we immediately obtain (B.9), as desired. In the case i 1 > 0 , we multiply (B.10) by v ( x ′ ) and apply Lemma B.2 in a similar fashion as b efor e to yield λ − 2 n D Q ⊗ 2 i 1 D Q ⊗ 2 i 2 − i 1 . . . D Q ⊗ 2 i s − i s − 1 D Q ⊗ 2 n − i s  v 1 / 2 ⊗ v 1 / 2  ( x, x ′ ) ≤ c s +2 v ( x ) h 0 ( x ) ˇ E x   Y k ∈{ i 1 ,...,i s − 1 } v  ˇ X k  v ∗  ˇ X i s    so again we obtain (B.9) a s desired. This completes the trea tment of the case s > 1 . F or the cas e s = 1 , i 1 > 0 , λ − 2 n Q ⊗ 2 i 1 D Q ⊗ 2 n − i 1  v 1 / 2 ⊗ v 1 / 2  ( x, x ) = λ − 2 n D Q ⊗ 2 i 1 D Q ⊗ 2 n − i 1  v 1 / 2 ⊗ v 1 / 2  ( x, x ′ ) ≤ cv ( x ) h 0 ( x ) ˇ E x  v ∗  ˇ X i 1  , where the inequality is due to an a pplication of Lemma B .3. Thus we ha ve (B.9) in the ca se s = 1 , i 1 > 0 . It only remains to a ddress the ca se s = 0 , b ecause for the case s = 1 , i 1 = 0 we observe that Γ ( ∅ ) n,x ( F ) = Γ (0) n,x ( F ) . F or s = 0 we hav e Γ ( ∅ ) n,x ( F ) = γ ⊗ 2 n,x ( F ) = Q ⊗ 2 n ( F ) ( x, x ) ≤ k F k v 1 / 2 , 2 Q ⊗ 2 n ( v ⊗ v )( x, x ) and therefore (recall ˇ v from lemma A.1) λ − 2 n h − 2 0 ( x )Γ ( ∅ ) n,x ( F ) ≤ k F k v 1 / 2 , 2 λ − 2 n h − 2 0 ( x ) Q ⊗ 2 n ( v ⊗ v )( x, x ) ≤ c k F k v 1 / 2 , 2 ˇ P ⊗ 2 n ( ˇ v ⊗ ˇ v ) ( x, x ) ≤ c k F k v 1 / 2 , 2 v ( x ) h 0 ( x ) v ∗ ( x ) . (B.12) 27 where the fina l inequality follows by iteration o f the geometric drift condition (A.6) and the definition of v ∗ . Thus (B.9) holds in the ca se s = 0 . This completes the pro o f of the prop osition. References A. B e s kos, D. Crisan, and A. Jasra . O n the stabilit y of sequential Monte Ca rlo methods in high- dimensions. 2011. F. Cérou, P . Del Moral, and A Guyader. A no nasymptotic v ariance theorem for unnormalized F eynman Kac particle mo dels. 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