math.PR 2026-03-16 0

Normal approximation for the polynomial functionals of correlated random field sampling along random walk path in dimension $1+1$

Let $ξ$ be the stationary occupation field generated by a Poisson system of independent simple symmetric random walks on $\mathbb Z$ in space--time dimension $1+1$. For a finite set $A\subset\mathbb Z$, we consider the classical fixed-region observab…

Authors: Ao Huang, Guanglin Rang, Zhonggen Su

NORMAL APPR O XIMA TION F OR THE POL YNOMIAL FUNCTIONALS OF CORRELA TED RANDOM FIELD SAMPLING ALONG RANDOM W ALK P A TH IN DIMENSION 1 + 1 A O HUANG, GUANGLIN RANG, AND ZHONGGEN SU Abstract. Let ξ b e the stationary o ccupation field generated by a Poisson system of indep enden t simple symmetric random walks on Z in space–time dimension 1 + 1 . F or a finite set A ⊂ Z , we consider the classical fixed-region observ ables W N ( A ) , the cum ulative o ccupation of A up to time N , and D N ( A ) , the num ber of distinct particles visiting A up to time N . W e pro ve quantitativ e cen tral limit theorems for b oth observ ables, with W asserstein rate of order N − 1 / 4 . In addition, we introduce an indep endent nearest-neighbour random walk S = ( S n , n ≥ 0) on Z with non-zero drift and sample the field along this ballistic path. F or a fixed polynomial observ able φ ( x ) = P k j =0 β j x j , β k  = 0 , of degree k ∈ N , we consider the partial sums Y N ,φ = P N n =1 φ ( ξ ( n, S n )) . W e pro ve a W asserstein bound of order N − 1 / 2 for the normal approximation of the standardized Y N ,φ . T o the b est of our knowledge, this is the first quan titativ e normal approximation result for p olynomial functionals of the P oisson o ccupation field sampled along a random walk path. The drift induces an effective decorrelation of the sampled environmen t, leading to a substantial improv ement o ver fixed-region sampling. The proofs rely on a representation of ξ as a Poisson functional on path space and on the Malliavin–Stein metho d for Poisson functionals. Key w ords: dynamic random environmen t; Mallia vin–Stein metho d; occupation field; Poisson p oin t pro cess; Poisson random walks; quantitativ e CL T. MSC2020 sub ject classification: Primary 60F05; Secondary 60G55, 60H07, 60J10, 60K35. 1. Introduction and main resul ts Bac kground. W e study quantitativ e normal appro ximations for functionals of a strongly cor- related random field in space–time dimension 1 + 1 . The field arises as the o ccupation field of a P oisson system of indep endent simple symmetric random w alks (SSR W) on Z , started from i.i.d. P oisson initial particles. Suc h Poisson systems go bac k at least to the work of Derman [ 7 ], and ha v e since app eared in a v ariet y of con texts as one of the simplest examples of infinite particle systems; see, e.g., Liggett [ 21 ] and Kipnis–Landim [ 14 ] for background. Besides b eing a natural Mark o vian mo del with an explicit in v ariant measure, this occupation field has b een used as a dynamic random environmen t in a num b er of works; see, e.g., [ 11 , 8 , 10 , 13 , 5 ]. In particular, it has serv ed as a random p oten tial or catalyst through which other particles mov e; see, e.g., Shen–Song–Sun–Xu [ 29 ] and the references therein. W e no w turn to the discrete-time mo del in v estigated in this pap er. Mo del. Throughout the pap er w e write N 0 = { 0 , 1 , 2 , . . . } , N = { 1 , 2 , 3 , . . . } and let Z denote the set of all integers. Let Z ∼ N (0 , 1) b e a standard Gaussian random v ariable. W e now recall the discrete-time mo del used throughout the pap er. The follo wing definition is adapted from [ 6 , 29 ]. Definition 1.1 (Poisson field of indep enden t walks) . A t time n = 0 , we start with ξ (0 , x ) p articles at e ach site x ∈ Z , wher e the family { ξ (0 , x ) , x ∈ Z } c onsists of indep endent Poisson r andom variables with me an λ > 0 . Each p article then p erforms an indep endent simple symmetric r andom walk on Z . W e denote by ξ ( n, x ) the numb er of p articles at p osition x and time n ∈ N 0 . Mor e 1 2 AO HUANG, G. RANG, AND Z. SU pr e cisely, (1.1) ξ ( n, x ) = X y ∈ Z ξ (0 ,y ) X i =1 1 { X y,i n = x } , n ∈ N 0 , x ∈ Z , wher e X y ,i = ( X y ,i n , n ∈ N 0 ) is the i -th r andom walk starting fr om y at time 0 . It is w ell known (see, for example, [ 14 , 7 ]) that this infinite particle system admits the i.i.d. pro duct Poisson measure with mean λ as an inv arian t and ergo dic measure, so that for eac h fixed n ∈ N 0 the random v ariables { ξ ( n, x ) , x ∈ Z } are again i.i.d. Poisson with mean λ . In particular, the field { ξ ( n, x ) : n ∈ N 0 , x ∈ Z } is stationary in time and homogeneous in space. On the other hand, it exhibits strong correlation in the time direction: the v alue of ξ ( n, x ) dep ends on the common history of many particles and is therefore highly dep enden t across differen t n . This strong space–time dep endence is one of the main difficulties in establishing quantitativ e limit theorems. In tw o earlier pap ers, P ort [ 26 , 27 ] inv estigated the corresp onding Poisson system of indep enden t Mark o v chains and introduced, for a finite set A ⊂ Z , tw o classical fixed-region observ ables. T o mak e this more precise, let A ⊂ Z b e a finite and nonempty set and define ξ ( n, A ) := X x ∈ A ξ ( n, x ) , n ∈ N 0 , the total n um b er of particles in A at time n . In our notation, his cum ulativ e o ccupation functional is defined by W N ( A ) := N X n =1 ξ ( n, A ) = N X n =1 X x ∈ A ξ ( n, x ) , N ∈ N . He also considered the num b er of distinct particles that visit A up to time N , D N ( A ) := X y ∈ Z ξ (0 ,y ) X i =1 1 {∃ 1 ≤ m ≤ N : X y,i m ∈ A } . P ort pro v ed strong la ws of large n um b ers and cen tral limit theorems for W N ( A ) and D N ( A ) . F ollowing Port’s work, Co x and Griffeath [ 5 ] studied contin uou s-time analogues and established large deviation principles for o ccupation-time t yp e functionals of Poisson systems of indep endent random walks on Z d . Ho wev er, these results are essentially qualitative and do not provide explicit rates of conv ergence. Moreo v er, the observ ables considered there inv olv e only finitely man y spatial sites. Main results. In the present work w e revisit the same Poisson field and fo cus on normal ap- pro ximation with explicit rates for functionals that dep end on a muc h larger num ber of space–time p oin ts, including functionals sampled along an indep enden t random w alk path. Our first main result giv es W asserstein b ounds for the classical fixed-region observ ables W N ( A ) and D N ( A ) . W e write d W for the W asserstein distance; see Section 2.2 for the definition. Theorem 1.2 (Fixed-region o ccupation and range-t yp e functionals) . L et F ( A ) N = W N ( A ) − E W N ( A ) p V ar( W N ( A )) , G ( A ) N = D N ( A ) − E D N ( A ) p V ar( D N ( A )) . Then ther e exists a c onstant C 1 , C 2 > 0 , dep ending only on λ and A , such that: (a) d W  F ( A ) N , Z  ≤ C 1 N − 1 / 4 , N ≥ 1 . NORMAL APPRO XIMA TION FOR THE POL YNOMIAL FUNCTIONALS OF RANDOM W ALK 3 (b) d W  G ( A ) N , Z  ≤ C 2 N − 1 / 4 , N ≥ 1 . Remark 1.3. After c ompleting this work, we notic e d that Jar amil l o–Muril lo-Salas [ 12 ] develop a Stein–Me cke appr o ach for Poisson-driven p article systems and obtain quantitative Gaussian ap- pr oximations for o c cup ation-time typ e additive functionals of the form A r [ ψ ] = Z r 0 ⟨ µ η t , ψ ⟩ d t, Z r [ ψ ] = A r [ ψ ] − E [ A r [ ψ ]] σ r . In p articular, in the c ase of one-dimensional uniformly el liptic diffusions and ψ ≥ 0 , their Pr op osi- tion 5.2 yields a W asserstein b ound of or der r − 1 / 4 for Z r [ ψ ] . This is c onsistent with the r ate N − 1 / 4 in The or em 1.2 (a) for the discr ete-time o c cup ation functional W N ( A ) , which c an b e viewe d as a lattic e analo gue of such o c cup ation-time additive functionals (formal ly c orr esp onding to ψ = 1 A and r ≈ N ). On the other hand, the fr amework in Jar amil lo–Muril lo-Salas [ 12 ] is tailor e d to additive func- tionals of the ab ove time-inte gr ate d form and the c orr esp onding moment b ounds r ely on c onditional density estimates for the underlying motion. The distinct-visitor c ount D N ( A ) in The or em 1.2 (b) is a hitting-typ e functional (it r e c or ds whether a tr aje ctory visits A at le ast onc e up to time N ) and is not of the form R r 0 ⟨ µ η t , ψ ⟩ d t . Ther efor e, their r esults do not dir e ctly apply to D N ( A ) without further extensions. In order to go b ey ond a fixed finite region A , w e in tro duce an independent biased (simple asymmetric) nearest–neighbour random walk S = ( S n , n ∈ N 0 ) on Z , indep enden t of the Poisson field ξ . More precisely , S 0 = 0 and its increments ∆ n := S n − S n − 1 are i.i.d. with P S (∆ n = +1) = p , P S (∆ n = − 1) = q := 1 − p , p ∈ (0 , 1) , p  = 1 2 , so that S has non–zero drift v := E S [∆ 1 ] = p − q  = 0 and is ballistic. Sampling along this random path allows us to explore the random environmen t at spatial lo cations that change with time and t ypically separate linearly in time. Motiv ated in part b y [ 29 ], we study p olynomial functionals sampled along the tra jectory . F or a fixed p olynomial observ able φ : N 0 → R , φ ( x ) = k X j =0 β j x j , β k  = 0 , of degree k ∈ N , we study the partial sums (1.2) Y N ,φ := N X n =1 φ ( ξ ( n, S n )) . F rom the p ersp ectiv e of random media, the case φ ( x ) = x corresp onds to the o ccupation time of a particle moving through a dynamic random environmen t, while more general p olynomial ob- serv ables naturally enco de nonlinear fluctuation statistics and app ear, for instance, in p erturbative expansions of partition functions for p olymers in such en vironments. W e denote by (Ω , F , P ) the underlying probabilit y space supp orting the Poisson field ξ and all random w alks, and b y E the asso ciated exp ectation, regardless of whether we consider only the field ξ or the joint randomness of ξ and S ; when necessary , a subscript will indicate the corresp onding probabilit y or exp ectation. W e write P n ( x ) = P S ( S n = x ) , x ∈ Z , n ∈ N 0 , 4 AO HUANG, G. RANG, AND Z. SU for its n -step transition probabilities. More generally , if S 0 = y , P S ( S n = x | S 0 = y ) = P n ( x − y ) , x, y ∈ Z . Conditionally on the initial configuration { ξ (0 , x ) , x ∈ Z } , the family of particle walks { X y ,i : y ∈ Z , 1 ≤ i ≤ ξ (0 , y ) } is indep endent, and each X y ,i = ( X y ,i n , n ∈ N 0 ) is a simple symmetric random walk on Z starting from y . W e denote by Q n ( x ) = P  X 0 ,i n = x  , x ∈ Z , n ∈ N 0 , the n -step transition probabilities of the particle walks. Hence, for general starting p oin t y , P  X y ,i n = x   X y ,i 0 = y  = Q n ( x − y ) , x, y ∈ Z . In our setting, Q n corresp onds to the simple symmetric random walk kernel gov erning the particle tra jectories in the Poisson field, whereas P n is the transition k ernel of the biased sampling walk S . W e keep differen t symbols to emphasize that S is indep enden t of the Poisson field ξ and of all the particle walks X y ,i . Our second main theorem concerns the functionals sampled along the random walk path ( 1.2 ). Theorem 1.4 (Annealed normal approximation for Y N ,φ under drifted sampling) . Under drifte d sampling v  = 0 , for every fixe d p olynomial φ , define the normalize d functional F N ,φ := Y N ,φ − E [ Y N ,φ ] σ N ,φ , σ 2 N ,φ := V ar( Y N ,φ ) , Then ther e exists a c onstant C φ ∈ (0 , ∞ ) , dep ending on the fixe d p olynomial φ , such that for al l N ≥ 1 , d W  F N ,φ , Z  ≤ C φ N − 1 / 2 . Mor e over, ther e exists a c onstant c φ ∈ (0 , ∞ ) such that σ 2 N ,φ = c φ N + O (1) , N → ∞ . Mor e pr e cisely, (1.3) c φ = k X q =1 c 2 φ,q q ! λ q  1 + 2 X t ≥ 1 a ( q ) t  , a ( q ) t := E S  Q t ( S t ) q  , wher e ( c φ,q ) q ≥ 1 ar e the Poisson–Charlier c o efficients of the c enter e d single-site observable φ ( ξ ( n, x )) − E [ φ ( ξ ( n, x ))] , as in L emma 4.6 . Remark 1.5. (1) Why fixe d-r e gion sampling is slower. The c ovarianc e identity ( 2.28 ) implies that, for a fixe d finite set A ⊂ Z , Co v( ξ ( n, A ) , ξ ( n + t, A )) de c ays at the same p olynomial r ate as Q t (0) in dimension one. Henc e the c orr elations of ( ξ ( n, A )) n ≥ 1 ar e not summable, and this long memory is r efle cte d in the slower W asserstein r ate N − 1 / 4 in The or em 1.2 . (2) Why b al listic p ath-sampling r e c overs the classic al N − 1 / 2 r ate. A long the drifte d walk S , the r elevant c orr elation c o efficients ar e a ( q ) t = E S [ Q t ( S t ) q ] . NORMAL APPRO XIMA TION FOR THE POL YNOMIAL FUNCTIONALS OF RANDOM W ALK 5 Sinc e | S t | ≈ | v | t with exp onential ly smal l deviations and Q t ( x ) satisfies a sub-Gaussian he at-kernel b ound, one has a ( q ) t ≤ C t − q / 2 e − ct , for suitable c onstants c, C ∈ (0 , ∞ ) dep ending on q and the drift p ar ameter. In p articu- lar, P t ≥ 1 a ( q ) t < ∞ for every fixe d q , which yields line ar varianc e gr owth σ 2 N ,φ ≍ N and ultimately the W asserstein r ate N − 1 / 2 in The or em 1.4 . (3) On the c o efficients c φ,q . The c onstants c φ,q ar e deterministic and dep end only on the fixe d p olynomial φ and on λ . They ar e the c o efficients in the Poisson–Charlier exp ansion of the c enter e d single-site observable and c an b e c ompute d fr om the ortho gonal pr oje ction formula in L emma 4.6 . (4) A nne ale d natur e. The or em 1.4 is anne ale d, in the sense that it aver ages jointly over the Poisson field and the sampling walk. Theorems 1.2 (a) and 1.4 th us exhibit a clear dic hotom y betw een fixed-region sampling and ballistic sampling. While W N ( A ) retains the long temp oral memory of the o ccupation field in one dimension, sampling along a drifted path effectively decorrelates the en vironmen t and yields the t ypical N − 1 / 2 W asserstein rate for the entire family of p olynomial observ ables Y N ,φ . T o the b est of our knowledge, these are the first quantitativ e normal approximation results for such f unctionals of the Poisson field sampled along a random walk path. Pro of strategy . Our pro ofs are based on Mallia vin calculus on P oisson space com bined with Stein’s metho d which go es back to [ 23 ], often referred to as the Malliavin–Stein approach and is also employ ed in [ 9 , 16 , 28 , 18 ]. The key observ ation is that the o ccupation field ξ can be regarded as a functional of a P oisson random measure on path space, so that W N ( A ) , D N ( A ) and Y N ,φ b elong to the L 2 space of a Poisson random measure. This allows us to apply the general normal appro ximation b ounds av ailable for P oisson functionals. Roughly sp eaking, our approach has three steps. 1. W e realize the o ccupation field as a functional of a P oisson random measure on path space; see ( 2.26 ). F or W N ( A ) and D N ( A ) this yields first-chaos represen tations suc h as ( 3.1 ) and ( 3.15 ), while for the path-sampled p olynomial functionals Y N ,φ w e obtain a finite conditional Poisson–Charlier c haos expansion; see ( 4.12 ). 2. W e determine the correct v ariance scale from the asso ciated cov ariance sums. In particular, fixed-region sampling in dimension one leads to V ar( W N ( A )) ≍ N 3 / 2 and V ar( D N ( A )) ≍ N 1 / 2 , whereas drifted path sampling yields linear v ariance growth through the summability of the av er- aged correlations a ( q ) t ; see ( 4.14 ). 3. W e apply Poisson Malliavin–Stein b ounds. F or the path-sampled obs erv ables, this requires an additional annealed-b y-conditional reduction; see ( 4.23 ). The main tec hnical input is then the con trol of the asso ciated contraction k ernels and fluctuation terms; see Lemma 4.11 . The main tec hnical difficulty lies in con trolling the con tributions of higher–order chaos terms and their correlations o v er large time interv als. Nevertheless, the Poisson structure of the mo del and the explicit represen tation of the o ccupation field make it p ossible to carry out these computations and to derive the quantitativ e b ounds stated in Theorems 1.2 and 1.4 . Discussions. 1. Our techniques can potentially b e extended to higher dimensions. The restriction to the one-dimensional setting simplifies the notation and allo ws us to av oid certain tec hnicalities. F or the fixed-region observ ables, the v ariance asymptotics and hence the resulting quantitativ e rates are exp ected to dep end on the dimension through the decay of the underlying heat k ernel. F or 6 AO HUANG, G. RANG, AND Z. SU the path-sampled polynomial functionals under drifted sampling, the same mec hanism based on ballistic separation and summable av eraged correlations in Subsection 4.1 should remain applicable in higher dimensions under analogous assumptions. 2. The discrete-time setting is adopted here mainly for clarity of notation an d to k eep the tec hnical core transparent. The same strategy should also apply to contin uous-time analogues, after replacing the discrete path space by a suitable càdlàg path space and the discrete kernels by the corresp onding transition semigroup. 3. The particle motion in the environmen t need not b e simple symmetric random walk. What is used in the pro ofs is not the exact form of the SSR W kernel, but rather the Poisson path-space realization together with suitable heat-k ernel b ounds, and, where needed, lo cal limit estimates. In this sense, the present metho d should b e adaptable to more general symmetric or cen tered walks, and to other translation-inv ariant motions for whic h comparable kernel estimates are av ailable. 4. F or the path-sampled observ ables, the role of the drift is structural rather than cosmetic. In the drifted regime, the a veraged correlations are summable and the v ariance is asymptotically linear, whic h is the mechanism b ehind the N − 1 / 2 rate in Theorem 1.4 . By contrast, under symmetric sampling the correlation decay is only p olynomial, and App endix A shows that the v ariance gro wth then dep ends on the first non-v anishing Poisson–Charlier co efficien t of the observ able: it is of order N 3 / 2 in rank one, N log N in rank tw o, and N from rank three onw ard. The app endix also explains wh y the conditional-v ariance step in the drifted Malliavin–Stein pro of no longer closes in the rank- one symmetric regime. General notation. F or the reader’s conv enience, w e collect some notation used throughout the pap er. • C , c denote generic p ositiv e constan ts that may change from line to line. When needed, w e indicate parameter dep endence by writing, for example, C := C ( λ, φ ) or c := c ( k ) . • F or quantities a = a ( x ) and b = b ( x ) ≥ 0 in the relev an t limiting regime (t ypically x = N → ∞ or x = t → ∞ ), w e write a ≲ b or a = O ( b ) if | a ( x ) | ≤ C b ( x ) for some constant C > 0 and all sufficiently large x , and a ≍ b if b oth a ≲ b and b ≲ a hold. W e write a = o ( b ) if a ( x ) /b ( x ) → 0 as x → ∞ . • W e write a n ∼ b n if a n /b n → 1 in the regime under consideration. Organization of the pap er. The rest of the pap er is organized as follows. In Section 2 w e collect the necessary background on Mallia vin calculus on Poisson space and Stein’s metho d, and we state a general normal approximation b ound for square–integrable Poisson functionals that will be used throughout the pap er. In Section 3 w e apply this b ound to the linear functional W N ( A ) , D N ( A ) and prov e Theorem 1.2 . Section 4 establishes the annealed quantitativ e CL T for the path-sampled observ ables Y N ,φ and pro v es Theorem 1.4 . App endix A con tains additional argumen ts of Theorem 1.4 . 2. Preliminaries 2.1. Mallia vin calculus on Poisson space. In this section, w e briefly in troduce some basic notions and prop erties on Mallia vin calculus on P oisson space. F or more details on Mallia vin calculus for P oisson functionals, we refer to [ 22 , 18 , 23 , 17 , 19 ] and the references therein. P oisson random measure. Let ( X , X ) b e a standard Borel space, which is equipp ed with a σ -finite measure µ . By η we denote a P oisson random measure on X with con trol measure µ , which NORMAL APPRO XIMA TION FOR THE POL YNOMIAL FUNCTIONALS OF RANDOM W ALK 7 is defined on an underlying probability space (Ω , F , P ) . Actually , η can b e realized by η = n X j =0 δ γ j , γ j ∈ X and n ∈ N 0 ∪ {∞} , where δ γ is Dirac measure at γ ∈ X . Here we say Poisson random measure η with control measure µ means: η = { η ( B ) : B ∈ X 0 } is a collection of random v ariables indexed by the elemen ts of X 0 = { B ∈ X : µ ( B ) < ∞} suc h that η ( B ) is P oisson distributed with mean µ ( B ) for eac h B ∈ X 0 , and for all n ∈ N , the random v ariables η ( B 1 ) , . . . , η ( B n ) are independent whenev er B 1 , . . . , B n are disjoint sets from X 0 . T o this end, let N σ = N ( X ) b e the space of integer-v alued σ -finite measures on X equipp ed with the smallest sigma-field N making the mappings ν → ν ( B ) measurable for all B ∈ X . W e remark that the P oisson random measure η is a random element in the space N σ . The distribution of η (on the space N σ ) will b e denoted b y P η . F or more details see [ 4 , Chapter VI] and [ 19 ]. L 1 - and L 2 -spaces. F or n ∈ N \ { 0 } , denote b y L 1 ( µ n ) and L 2 ( µ n ) the space of integrable and square-in tegrable functions with resp ect to µ n , resp ectiv ely . The scalar pro duct and the norm in L 2 ( µ n ) are denoted by ⟨· , ·⟩ n and ∥ · ∥ n , resp ectiv ely . F rom no w on, w e will omit the index n as it will alw a ys be clear from the context. Moreo v er, let us denote by L 2 ( P η ) the space of square-in tegrable functionals of a Poisson random measure η . Finally , w e denote b y L 2 ( P , L 2 ( µ )) the space of join tly measurable mappings h : Ω × X → R suc h that Z Ω Z X h ( ω , z ) 2 µ ( dz ) P ( dω ) < ∞ (recall that (Ω , F , P ) is our underlying probability space). No w we come to the chaos decomposition of square integrable functionals. Chaos decomp osition of L 2 functionals. F or any g ∈ L 2 ( µ ) , the sto c hastic integral of g with resp ect to the comp ensated Poisson measure ˆ η := η − µ is denoted by I 1 ( g ) = R X g ( γ ) ˆ η (d γ ) . F or n ≥ 2 and g n ∈ L 2 ( µ n ) , the multiple Wiener-Itô integral of g n is given b y I n ( g n ) = X J ⊂ [ n ] ( − 1) n −| J | Z X n g n ( γ 1 , . . . , γ n ) η ( | J | ) (d γ J ) µ n −| J | (d γ J C ) . (2.1) Esp ecially , when g n = g ⊗ n with some g ∈ L 2 ( µ ) , ( 2.1 ) is reduced to I n ( g n ) = n X k =0 ( − 1) n − k n k ! η ( k ) ( g ⊗ k )[ µ ( g )] n − k , (2.2) where the η ( k ) , for k ≥ 2 , is called k-th factorial measure of η . F or B ∈ X , we hav e that η ( k ) ( B k ) = η ( B )( η ( B ) − 1) · · · ( η ( B ) − k + 1) . F or every f ∈ L 2 ( P η ) , there exists a a uniquely determined symmetric function g n ∈ L 2 ( µ n ) , n ≥ 0 , (with the conv en tion L 2 ( µ 0 ) = R ) such that (2.3) f ( η ) = ∞ X n =0 I n ( g n ) , where the series conv erges in L 2 . Moreov er, (2.4) E  I n ( g n ) I m ( g m )  = n ! 1 { n = m } ⟨ g n , g m ⟩ L 2 ( µ n ) . 8 AO HUANG, G. RANG, AND Z. SU The representation ( 2.3 ) is called the chaotic expansion of F and we say that F has a finite chaotic expansion if only finitely many of the functions g n are non-v anishing. In particular, ( 2.3 ) together with the orthogonality of multiple sto c hastic in tegrals leads to the v ariance formula (2.5) V ar( f ) = ∞ X n =1 n ! ∥ g n ∥ 2 . Mallia vin op erators. In order to obtain the co efficien ts g n for a giv en functional f , w e need to in tro duce Mallia vin deriv ativ e. The deriv ative of measurable function f : N σ → R , in the direction of γ ∈ X , denoted by D γ f , on N σ is defined by D γ f ( η ) = f ( η + δ γ ) − f ( η ) , η ∈ N σ , (2.6) where the Dirac measure δ γ is defined by δ γ ( B ) = 1 B ( γ ) , B ∈ X . By iterating pro cedure, one can define D n γ n ,...,γ 1 f for n ≥ 2 and ( γ 1 , . . . , γ n ) ∈ X n recursiv ely by D n γ n ,...,γ 1 f = D 1 γ n D n − 1 γ n − 1 ,...,γ 1 f , with D 1 f = D f , D 0 f = f . The following formula is obvious, D n γ n ,...,γ 1 f ( η ) = X J ⊂ [ n ] ( − 1) n −| J | f ( η + X j ∈ J δ γ j ) . (2.7) F rom this w e kno w the op erator D n γ n ,...,γ 1 is symmetric in ( γ 1 , . . . , γ n ) . F or f ∈ L 2 ( P η ) , w e define a mapping T n f : X n → R via T n f ( γ 1 , . . . , γ n ) = E [ D n γ n ,...,γ 1 f ] . (2.8) F or any f ∈ L 2 ( P η ) , the ab o v e T n f ∈ L 2 s ( µ n ) , where the subscript "s" in L 2 s ( µ n ) stands for the function is symmetric in its arguments, and the chaos decomp osition ( 2.3 ) can b e written as f ( η ) = ∞ X k =0 I k ( T k f ) k ! . (2.9) Supp ose that f ∈ L 2 ( P η ) admits the decomp osition as ( 2.3 ). W e call f ∈ dom( D ) , the domain of op erator D , if ∞ X n =1 nn ! ∥ g n ∥ 2 n < ∞ . In this case, one has P η a.e. that D γ f = ∞ X n =1 nI n − 1 ( g n ( γ , · )) , γ ∈ X . (2.10) F or f ∈ L 2 ( P η ) satisfying ( 2.3 ), w e call f b elonging to the domain of Ornstein-Uhlenbeck op erator L , denoted by f ∈ dom( L ) , if { g n } ∞ n =0 satisfies ∞ X n =1 n 2 n ! ∥ g n ∥ 2 n < ∞ . In this case, one has Lf = − ∞ X n =1 nI n ( g n ) (2.11) NORMAL APPRO XIMA TION FOR THE POL YNOMIAL FUNCTIONALS OF RANDOM W ALK 9 and its inv erse is denoted by L − 1 . In terms of the chaos expansion of a cen tred random v ariable f ∈ L 2 ( P η ) , i.e. E ( f ) = 0 , it is giv en by L − 1 F = − ∞ X n =1 1 n I n ( g n ) . (2.12) Finally , let γ 7→ h ( γ ) b e a random function on X with chaos expansion h ( γ ) = h 0 ( γ ) + ∞ X n =1 I n ( h n ( γ , · )) with symmetric functions h n ( γ , · ) ∈ L 2 ( µ n ) such that ∞ X n =0 ( n + 1)! ∥ h n ∥ 2 < ∞ , (let us write h ∈ dom( δ ) if this is satisfied), the Skorohod integral δ ( h ) of h is defined as δ ( h ) = ∞ X n =0 I n +1 ( e h n ) , where e h n is the canonical symmetrization of h n as a function of n + 1 v ariables. The next lemma summarizes a relationships b et ween the op erators D , δ and L , the classical and a mo dified integration-b y-parts-form ula as well as an isometric formula for Skorohod integrals. Lemma 2.1. (i) It holds that f ∈ dom(L) if and only if f ∈ dom(D) and Df ∈ dom( δ ) , in which c ase δ ( D f ) = − Lf . (2.13) (ii) W e have the inte gr ation-by-p arts formula E [ f δ ( h )] = E ⟨ D f , h ⟩ (2.14) for every f ∈ dom(D) and h ∈ dom( δ ) . (iii) Supp ose that f ∈ L 2 ( P η ) (not ne c essarily assuming that f b elongs to the domain of D ), that h ∈ dom( δ ) has a finite chaos exp ansion and that D γ 1 ( f > x ) h ( γ ) ≥ 0 for any x ∈ R and µ -almost al l γ ∈ X . Then E [ 1 ( f > x ) δ ( h )] = E ⟨ D 1 ( f > x ) , h ⟩ . (2.15) (iv) If h ∈ dom( δ ) it holds that E [ δ ( h ) 2 ] = E Z X h ( γ 1 ) 2 µ ( dγ 1 ) + E Z X Z X ( D γ 2 h ( γ 1 ))( D γ 1 h ( γ 2 )) µ ( dγ 1 ) µ ( dγ 2 ) . (2.16) W e refer the reader to Lemma 2.1 in [ 9 ] for more details. Moreov er, we refer to [ 20 ] for a path wise in terpretation of the Skorohod in tegral. Con tractions. F or in tegers q 1 , q 2 ≥ 1 , let r ∈ { 0 , . . . , min ( q 1 , q 2 ) } , ℓ ∈ { 1 , . . . , r } , and let f 1 ∈ L 2 ( µ q 1 ) and f 2 ∈ L 2 ( µ q 2 ) b e symmetric functions. The contraction kernel f 1 ⋆ ℓ r f 2 on X q 1 + q 2 − r − ℓ acts on the tensor pro duct f 1 ⊗ f 2 first by iden tifying r v ariables and then in tegrating out ℓ among them. More formally , f 1 ⋆ ℓ r f 2 ( γ 1 , . . . , γ r − ℓ , t 1 , . . . , t q 1 − r , s 1 , . . . , s q 2 − r ) = Z X ℓ f 1 ( γ 1 , . . . , γ ℓ , γ 1 , . . . , γ r − ℓ , t 1 , . . . , t q 1 − r ) × f 2 ( γ 1 , . . . , γ ℓ , γ 1 , . . . , γ r − ℓ , s 1 , . . . , s q 2 − r ) µ ℓ (d( γ 1 , . . . , γ ℓ )) . 10 AO HUANG, G. RANG, AND Z. SU In addition, we define f 1 ⋆ 0 r f 2 ( γ 1 , . . . , γ r , t 1 , . . . , t q 1 − r , s 1 , . . . , s q 2 − r ) = f 1 ( γ 1 , . . . , γ r , t 1 , . . . , t q 1 − r ) f 2 ( γ 1 , . . . , γ r , s 1 , . . . , s q 2 − r ) . Besides of the contraction f 1 ⋆ ℓ r f 2 , we will also deal with their canonical symmetrizations f 1 e ⋆ ℓ r f 2 . They are defined as ( f 1 e ⋆ ℓ r f 2 )( x 1 , . . . , x q 1 + q 2 − r − ℓ ) = 1 ( q 1 + q 2 − r − ℓ )! X π ( f 1 ⋆ ℓ r f 2 )( x π (1) , . . . , x π ( q 1 + q 2 − r − ℓ ) ) , where the sum runs ov er all ( q 1 + q 2 − r − ℓ )! p ermutations of { 1 , . . . , q 1 + q 2 − r − ℓ } . Pro duct formula. Let q 1 , q 2 ≥ 1 b e integers and f 1 ∈ L 2 ( µ q 1 ) and f 2 ∈ L 2 ( µ q 2 ) b e symmetric functions. In terms of the con tractions of f 1 and f 2 in tro duced in the previous paragraph, one can express the pro duct of I q 1 ( f 1 ) and I q 2 ( f 2 ) as follows: I q 1 ( f 1 ) I q 2 ( f 2 ) = min( q 1 ,q 2 ) X r =0 r ! q 1 r ! q 2 r ! r X ℓ =0 r ℓ ! I q 1 + q 2 − r − ℓ ( f 1 e ⋆ ℓ r f 2 ) . (2.17) see [ 24 , Prop osition 6.5.1]. 2.2. Mallia vin-Stein b ound. Besides Mallia vin calculus, our pro of of Theorem 1.2 and 1.4 rests up on Stein’s metho d that go es bac k to Stein [ 30 , 31 ] and is a p o w erful to ol for proving limit theorems. F or a detailed and more general introduction into this topic, we refer to [ 2 , 3 , 31 ]. Probabilit y distances. T o measure the distance b etw een the distributions of t wo random v ariables X and Y defined on a common probability space (Ω , F , P ) , one often uses distances of the form d H ( X , Y ) = sup h ∈H   E h ( X ) − E h ( Y )   , where H is a suitable class of real-v alued test functions (note that we sligh tly abuse notation b y writing d ( X , Y ) instead of d ( L ( X ) , L ( Y )) ). Prominent examples are the class H W of Lipsc hitz functions with Lipsc hitz constan t bounded by one and the class H K of indicator functions of in terv als ( −∞ , x ] with x ∈ R . The resulting distances d W := d H W and d K := d H K are u sually called W asserstein and Kolmogoro v distance. W e notice that d W ( X n , Y ) → 0 or d K ( X n , Y ) → 0 as n → ∞ for a sequence of random v ariables X n implies conv ergence of X n to Y in distribution. These metrics pro vide quan titative v ersions of the central limit theorem: bounds of the form d ( X , Z ) = O ( N − α ) , where Z ∼ N (0 , 1) is standard normal, describ e the rate of conv ergence of the distribution of X to Z . Stein’s metho d. A standard Gaussian random v ariable Z is c haracterized by the fact that for ev ery absolutely con tinuous function f : R → R for which E [ | Z f ( Z ) | ] < ∞ it holds that E  f ′ ( Z ) − Z f ( Z )  = 0 . The W asserstein distan ce b et ween tw o R d -v alued v ariables F and Z , denoted by d W ( F , Z ) , is defined by d W ( F , Z ) = sup {| E h ( F ) − E h ( Z ) | : ∥ h ∥ Lip ≤ 1 } , (2.18) where, for all functions h : R d → R , ∥ h ∥ Lip = sup x,y ∈ R d ,x  = y | h ( x ) − h ( y ) | ∥ x − y ∥ R d . NORMAL APPRO XIMA TION FOR THE POL YNOMIAL FUNCTIONALS OF RANDOM W ALK 11 By Stein metho d, if F is square in tegrable and Z ∼ N (0 , 1) standard normal distribution, then the W asserstein distance ( 2.18 ) has the following Stein b ound: d W ( F , Z ) ≤ sup f ∈ F W | E f ′ ( F ) − E F f ( F ) | , (2.19) where F W = { f ∈ C 1 : ∥ f ′ ∥ ∞ ≤ p 2 /π , ∥ f ′′ ∥ ∞ ≤ 2 } and C 1 is the collection of all contin uously differen tiable functions on R . The follo wing result giv es the upp er bound of the W asserstein distance d W ( F , Z ) in terms of Mallia vin calculus , see [ 1 , Theorem 7] for details. Lemma 2.2. A l l notations and assumptions as b efor e. L et F ∈ dom(D) with E [ F ] = 0 , then we have d W ( F , Z ) ≤ r 2 π E h | 1 − ⟨ D F, − D L − 1 F ⟩ L 2 ( µ ) | i + Z X E h | D γ F | 2 | D γ L − 1 F | i µ (d γ ) ≤ r 2 π E h (1 − ⟨ D F, − D L − 1 F ⟩ L 2 ( µ ) ) 2 i + Z X E h | D γ F | 2 | D γ L − 1 F | i µ (d γ ) , (2.20) wher e we use the standar d notation ⟨ D F, − D L − 1 F ⟩ L 2 ( µ ) = − Z X ( D z F ) × ( D z L − 1 F ) µ (d z ) . 2.3. F unctional of P oisson random fields. T o prov e Theorem 1.2 and 1.4 , we need to use the b ound in ( 2.20 ). W e will sho w that ξ ( · , · ) is a functional of Poisson random measure b elo w. It will b e useful to view ξ ( · , · ) as a subpro cess of a P oisson p oint pro cess on a space of simple random walk trajectories as follows. Denote (2.21) W =  w = ( w ( n )) n ∈ N 0 : w ( n ) ∈ Z , | w ( n + 1) − w ( n ) | = 1 ∀ n ∈ N 0  , as the set of simple random w alk tra jectories on Z . Endow W with the sigma-algebra W generated b y the canonical pro jections Z n : W → Z , Z n ( w ) = w ( n ) , n ∈ N 0 . A partition of W into disjoint measurable sets is given b y { W x } x ∈ Z , where W x = { w ∈ W : w (0) = x } . W e introduce the space ¯ Ω of p oint measures on W as (2.22) ¯ Ω = n η = P j ∈ Z + δ w j : w j ∈ W ∀ j ∈ Z + , | η ( W x ) | < ∞ ∀ x ∈ Z o . Recall Definition 1.1 , and define a random p oin t measure η ∈ ¯ Ω by (2.23) η = X z ∈ N X 1 ≤ i ≤ ξ (0 ,z ) δ X z,i , where δ w stands for the Dirac measure concen trated at the p oin t w ∈ W . It is straigh tforward to c hec k that, under P , η is a Poisson p oint process on W with intensit y measure µ , where (2.24) µ ( · ) = λ X x ∈ Z P x ( · ) , and P x is the la w on W , with support on W x . Note that under whic h Z ( · ) = ( Z n ( · )) n ∈ N 0 is distributed as a simple symmetric random walk on Z . F or a nonnegative measurable functional F : W → [ 0 , ∞ ] . F or F ∈ L 1 ( µ ) , w e use the P oisson in tegral ⟨ η , F ⟩ := Z W F ( w ) η (d w ) = X j ≥ 1 F ( w j ) ∈ [0 , ∞ ] . 12 AO HUANG, G. RANG, AND Z. SU The Laplace functional (Campb ell formula) of η is (2.25) E h e −⟨ η ,F ⟩ i = exp  − Z W (1 − e − F ( w ) ) µ (d w )  , F ≥ 0 . F or ( n, x ) ∈ N 0 × Z define the cylinder set C n,x := { w ∈ W : w ( n ) = x } ∈ W , and the indicator functional F n,x ( w ) := 1 C n,x ( w ) = 1 { w ( n ) = x } . Therefore (2.26) ξ ( n, x ) = Z W 1 { w ( n ) = x } η (d w ) = X y ∈ Z ξ (0 ,y ) X k =1 1 { Y y ,k n = x } , whic h is exactly the equation ( 1.1 ) in “Poisson field of indep enden t w alks” construction. Th us ξ ( n, x ) counts how many Poissonian tra jectories pass through x at time n . More generally , for each n ∈ N we obtain a random counting measure on Z by (2.27) ξ n ( B ) := Z W 1 { w ( n ) ∈ B } η (d w ) , B ⊂ Z . The representation ( 2.26 ) and ( 2.25 ) immediately yield the standard facts used later. • Marginals. Since µ ( C n,x ) = λ X y ∈ Z P y  Z n = x  = λ X y ∈ Z P 0  Z n = x − y  = λ X z ∈ Z P 0  Z n = z  = λ, w e hav e ξ ( n, x ) ∼ P oi( λ ) , E [ ξ ( n, x )] = λ, V ar( ξ ( n, x )) = λ. • Indep endence at fixed time. F or a fixed n , the sets { C n,x } x ∈ Z are pairwise disjoint, hence { ξ ( n, x ) } x ∈ Z are indep enden t; by the previous item they are i.i.d. Poi( λ ) . Equiv alently , ξ n in ( 2.27 ) is a P oisson random measure on Z with intensit y λ #( · ) (where #( · ) is coun ting measure). • T wo-time correlations. F or general measurable A 1 , A 2 ⊂ W , Co v  η ( A 1 ) , η ( A 2 )  = µ ( A 1 ∩ A 2 ) for a Poisson random measure. With A 1 = C n 1 ,x 1 and A 2 = C n 2 ,x 2 w e obtain (2.28) Co v  ξ ( n 1 , x 1 ) , ξ ( n 2 , x 2 )  = µ  C n 1 ,x 1 ∩ C n 2 ,x 2  = λ Q | n 2 − n 1 | ( x 2 − x 1 ) . In particular, v ariables at different times are t ypically not indep enden t. • Laplace functionals of linear statistics. F or any test function χ : Z → [ 0 , ∞ ) , ⟨ ξ n , χ ⟩ := X x ∈ Z χ ( x ) ξ ( n, x ) = Z W χ  Z n ( w )  η (d w ) , and by ( 2.25 ), E h e −⟨ ξ n ,χ ⟩ i = exp  − λ X x ∈ Z  1 − e − χ ( x )   . This is the Laplace functional of an indep enden t pro duct of Poi( λ ) v ariables indexed by Z , confirming the previous item. Let N ( W ) b e the space of lo cally finite coun ting measures on W and equip it with the ev aluation σ -algebra N := σ  # 7→ #( A ) : A ∈ W  . F or every nonnegative measurable f : W → [0 , ∞ ] , the map T f : N ( W ) → [0 , ∞ ] , T f ( η ) := Z W f ( w ) η (d w ) , NORMAL APPRO XIMA TION FOR THE POL YNOMIAL FUNCTIONALS OF RANDOM W ALK 13 is ( N , B ([0 , ∞ ])) -measurable. By the measurability of T F n,x , the map J n,x : N ( W ) → [0 , ∞ ] , J n,x ( η ) := η ( C n,x ) , is ( N , B ([0 , ∞ ])) -measurable, hence ξ ( n, x ) = J n,x ( η ) is a (measurable) functional of the P oisson measure η . Let ψ : [0 , ∞ ] → R b e Borel measurable . By closure under comp osition, ψ  ξ ( n, x )  = ψ  J n,x ( η )  is a measurable functional of the P oisson measure η . More generally , for any finite family ( n i , x i ) N i =1 and any Borel map Ψ : [0 , ∞ ] N → R , Ψ  ξ ( n 1 , x 1 ) , . . . , ξ ( n N , x N )  = Ψ  J n 1 ,x 1 ( η ) , . . . , J n N ,x N ( η )  is again a measurable Poisson measure functional on N ( W ) . Finally , using ( 2.26 ) we can write W N ( A ) = N X n =1 X x ∈ A ξ ( n, x ) = Z W g N ,A ( w ) η (d w ) , g N ,A ( w ) := N X n =1 1 { w ( n ) ∈ A } . Similarly , other observ ables suc h as the distinct-particle coun t D N ( A ) can also b e written as P oisson functionals. Define the "hitting" set H N ,A := { w ∈ W : ∃ m ∈ { 1 , . . . , N } s.t. w ( m ) ∈ A } . Then D N ( A ) = η ( H N ,A ) , i.e., D N ( A ) counts the num b er of Poissonian trajectories that hit A up to time N . Moreo v er, for the path-sampled p olynomial functional ( 1.2 ), for every fixed tra jectory S = ( S n , n ∈ N 0 ) and k ∈ N , Y N ,φ = N X n =1 φ ( ξ ( n, S n )) = N X n =1 φ ( η ( C n,S n )) is a measurable functional of η . 3. Proof of Theorem 1.2 3.1. Pro of of Theorem 1.2 (a): the functional W N ( A ) . In this section, we pro ve the quanti- tativ e normal appro ximation for the functional W N ( A ) = N X n =1 ξ ( n, A ) , where A ⊂ Z is a finite and nonempty set. W e first determine the Wiener–Itô c haos expansion of W N ( A ) . Recall that ξ ( n, A ) can b e repre- sen ted as a functional of the P oisson random measure η on the path space W via ξ ( n, A ) = Z W 1 { w ( n ) ∈ A } η (d w ) . F or n ∈ { 1 , . . . , N } , define C n,A := { w ∈ W : w ( n ) ∈ A } , 14 AO HUANG, G. RANG, AND Z. SU and set f N ( w ) := N X n =1 1 C n,A ( w ) , w ∈ W . Then, by the Poisson path-space representation, (3.1) W N ( A ) − E [ W N ( A )] = N X n =1  ξ ( n, A ) − µ ( C n,A )  = Z W f N ( w ) ˆ η (d w ) =: I 1 ( f N ) , where ˆ η = η − µ is the comp ensated Poisson measure on W . Hence, by the isometry of first-order Poisson sto c hastic in tegrals, σ 2 N ,W := V ar( W N ( A )) = ∥ f N ∥ 2 L 2 ( µ ) = N X n,m =1 µ  C n,A ∩ C m,A  . (3.2) F or n ≤ m , using the Marko v prop ert y of the simple symmetric random walk and µ = λ P u ∈ Z P u , w e hav e (3.3) µ  C n,A ∩ C m,A  = λ X u ∈ Z P u  w ( n ) ∈ A, w ( m ) ∈ A  = λ X u ∈ Z X x ∈ A P u  w ( n ) = x  P x  w ( m − n ) ∈ A  = λ X x ∈ A P x  w ( m − n ) ∈ A  = λ X x,y ∈ A Q m − n ( y − x ) . Substituting ( 3.3 ) into ( 3.2 ) and separating the diagonal terms yields (3.4) V ar  W N ( A )  = λ " N | A | + 2 N − 1 X n =1 ( N − n ) X x,y ∈ A Q n ( y − x ) # . W e will use the follo wing parit y-corrected local cen tral limit theorem to obtain sharp asymptotics for ( 3.4 ). Lemma 3.1. R e c al l that Q n ( x ) is the tr ansition kernel of the one-dimensional simple symmetric r andom walk. Then sup x ∈ Z      √ n 2 Q n ( x ) − 1 { x ≡ n (mod 2) } 1 √ 2 π e − x 2 / (2 n )      → 0 as n → ∞ . Conse quently, ther e exists a c onstant C < ∞ such that Q n ( x ) ≤ C n − 1 / 2 e − x 2 / (2 n ) ≤ C n − 1 / 2 , x ∈ Z , n ≥ 1 . Pr o of The local limit statemen t is the classical local central limit theorem for lattice distributions; see, e.g., [ 25 , Chapter VI I, Theorem 1]. The upper b ound follows immediately from this asymptotic for all sufficien tly large n , and the finitely many small v alues of n can b e absorb ed into the constant C . W e can now refine ( 3.4 ) as follo ws. Lemma 3.2. A s N → ∞ , (3.5) V ar  W N ( A )  = 8 λ | A | 2 3 √ 2 π N 3 / 2 + o ( N 3 / 2 ) . NORMAL APPRO XIMA TION FOR THE POL YNOMIAL FUNCTIONALS OF RANDOM W ALK 15 In p articular, (3.6) V ar  W N ( A )  ≍ N 3 / 2 . Pr o of Let A − A := { y − x : x, y ∈ A } . Since A − A is finite, Lemma ( 3.1 ) implies that, uniformly in d ∈ A − A , (3.7) Q n ( d ) = 1 { n ≡ d (mod 2) } 2 √ 2 π n + ρ n ( d ) , sup d ∈ A − A √ n | ρ n ( d ) | − − − → n →∞ 0 . Indeed, since A − A is finite, e − d 2 / (2 n ) = 1 + O ( n − 1 ) uniformly in d ∈ A − A , and this error can b e absorb ed into ρ n ( d ) . F or d ∈ A − A , define Σ N ( d ) := N − 1 X n =1 ( N − n ) Q n ( d ) . By ( 3.7 ), Σ N ( d ) = 2 √ 2 π X 1 ≤ n ≤ N − 1 n ≡ d (mod 2) ( N − n ) n − 1 / 2 + N − 1 X n =1 ( N − n ) ρ n ( d ) . Since sup d ∈ A − A √ n | ρ n ( d ) | → 0 and N − 1 X n =1 ( N − n ) n − 1 / 2 ≍ N 3 / 2 , a standard ε -argument yields (3.8) sup d ∈ A − A      N − 1 X n =1 ( N − n ) ρ n ( d )      = o ( N 3 / 2 ) . Next, for each fixed h ∈ { 0 , 1 } , write T N ,h := X 1 ≤ n ≤ N − 1 n ≡ h (mod 2) ( N − n ) n − 1 / 2 . W riting n = 2 m + h , one chec ks b y comparison with the corresp onding Riemann integral that (3.9) T N ,h = 2 3 N 3 / 2 + o ( N 3 / 2 ) , N → ∞ . Therefore, combining ( 3.8 ) and ( 3.9 ), we get (3.10) Σ N ( d ) = 4 3 √ 2 π N 3 / 2 + o ( N 3 / 2 ) , uniformly in d ∈ A − A . Substituting ( 3.10 ) into ( 3.4 ), we obtain V ar  W N ( A )  = 8 λ | A | 2 3 √ 2 π N 3 / 2 + o ( N 3 / 2 ) , whic h prov es ( 3.5 ). □ 16 AO HUANG, G. RANG, AND Z. SU W e now apply the Mallia vin–Stein b ound. Define F ( A ) N := W N ( A ) − E [ W N ( A )] σ N ,W = I 1  f N σ N ,W  . Since F ( A ) N b elongs to the first-order chaos, one has L − 1 F ( A ) N = − F ( A ) N , D w F ( A ) N = 1 σ N ,W f N ( w ) , and D w F ( A ) N is deterministic. This implies: (1) The first term in the Mallia vin–Stein b ound ( 2.20 ) v anishes identically: E h    1 − ⟨ D F ( A ) N , − D L − 1 F ( A ) N ⟩ L 2 ( µ )    i =    1 − ∥ σ − 1 N ,W f N ∥ 2 L 2 ( µ )    = | 1 − 1 | = 0 . (2) The W asserstein distance is con trolled solely by the remainder term inv olving the third momen t: (3.11) d W ( F ( A ) N , Z ) ≤ E Z W | D w F ( A ) N | 3 µ (d w ) = 1 σ 3 N ,W Z W | f N ( w ) | 3 µ (d w ) . W e no w estimate R f 3 N d µ . Using the definition of f N and the idemp otence prop ert y of indicator functions ( 1 k E = 1 E ), we expand the cub e of the sum: f N ( w ) 3 = N X n =1 1 C n,A ( w ) ! 3 = N X n =1 1 C n,A ( w ) + 6 X 1 ≤ n 0 such that for al l n ≥ 1 , P S    S n − v n   ≥ | v | 2 n  ≤ 2 e − c 0 n . In p articular, P S  | S n | ≤ | v | 2 n  ≤ 2 e − c 0 n . Pr o of Apply Ho effding’s inequality to P n i =1 (∆ i − v ) to obtain the first b ound. F or the second, note that on {| S n | ≤ | v | 2 n } , | S n − v n | ≥ | v | n − | S n | ≥ | v | 2 n, so {| S n | ≤ | v | 2 n } ⊆ {| S n − v n | ≥ | v | 2 n } . □ Lemma 4.2 (Exp onen tial momen t deca y under drift) . Fix m ∈ N . Under ( 4.2 ) , ther e exist c onstants c := c ( v ) , C := C ( m ) ∈ (0 , ∞ ) such that for al l r ≥ 1 , E S  Q r ( S r ) m  ≤ C r − m/ 2 e − cr . In p articular, for every fixe d m the se quenc e r 7→ E S [ Q r ( S r ) m ] is summable and r 7→ r E S [ Q r ( S r ) m ] is also summable. 20 AO HUANG, G. RANG, AND Z. SU Pr o of Let θ := | v | / 2 > 0 and split according to E r := {| S r | ≤ θ r } . On E r , sup x Q r ( x ) ≤ C r − 1 / 2 and by Lemma 4.1 , we ha v e E  Q r ( S r ) m 1 E r  ≤ C r − m/ 2 P ( E r ) ≤ C r − m/ 2 e − cr . On E c r w e use Lemma 3.1 : Q r ( S r ) m ≤ C r − m/ 2 exp  − mS 2 r C r  ≤ C r − m/ 2 exp  − mθ 2 C r  ≤ C r − m/ 2 e − cr , hence E [ Q r ( S r ) m 1 E c r ] ≤ C r − m/ 2 e − cr . □ W e also record a multi-con v olution coun ting b ound used rep eatedly . Lemma 4.3. L et M ∈ N and let ( u ( j ) n ) n ≥ 1 , j = 1 , . . . , M , b e nonne gative se quenc es with P n ≥ 1 u ( j ) n < ∞ for every j . Then for al l N ≥ 1 , X n 1 ,...,n M ≥ 1 ( N − n 1 − · · · − n M ) + M Y j =1 u ( j ) n j ≤ N M Y j =1 X n ≥ 1 u ( j ) n . Pr o of Use the fact ( N − n 1 − · · · − n M ) + ≤ N and T onelli’s theorem. □ 4.2. Conditional chaos expansion. In this subsection w e deriv e a conditional P oisson-c haos expansion for Y N ,φ . W e no w work on the pro duct probabilit y space with la w P = P ξ ⊗ P S and exp ectation E . F or a fixed realization of S , write A n := C n,S n , n = 1 , . . . , N , where the cylinder set C n,x := { w ∈ W : w ( n ) = x } . F or later use, note that for all 1 ≤ n, m ≤ N , (4.3) µ ( A n ) = λ, µ ( A n ∩ A m ) = λQ | m − n | ( S m − S n ) . W e b egin with its exp ectation. Lemma 4.4. F or every N ≥ 1 , E ξ [ Y N ,φ | S ] = N E  φ (P oi( λ ))  a.s. In p articular, E [ Y N ,φ ] = N E  φ (P oi( λ ))  . Pr o of Conditionally on S , for each n = 1 , . . . , N , E ξ [ φ ( ξ ( n, S n )) | S ] = X x ∈ Z 1 { S n = x } E ξ [ φ ( ξ ( n, x ))] = E [ φ (Poi( λ ))] , since ξ ( n, x ) ∼ P oi( λ ) for ev ery fixed ( n, x ) . Summing ov er n yields E ξ [ Y N ,φ | S ] = N E [ φ (P oi( λ ))] . T aking exp ectation with resp ect to S gives the unconditional identit y . □ T o expand single-site observ ables in P oisson chaoses, we use the orthogonal Charlier basis on P oi( λ ) together with an Itô-t yp e iden tity for indicator kernels. Let C std n ( x ; λ ) denote the standard Charlier p olynomial as in [ 15 ], and define the corresp onding monic Charlier p olynomial by C n ( x ; λ ) := ( − λ ) n C std n ( x ; λ ) , n ≥ 0 . Then C 0 ( x ; λ ) ≡ 1 , C 1 ( x ; λ ) = x − λ. NORMAL APPRO XIMA TION FOR THE POL YNOMIAL FUNCTIONALS OF RANDOM W ALK 21 Moreo v er, b y rescaling the standard form ulas [ 15 , Equation (9.14.1)–(9.14.3)], the monic Charlier p olynomials satisfy the three-term recurrence (4.4) ( x − λ ) C r ( x ; λ ) = C r +1 ( x ; λ ) + r C r ( x ; λ ) + r λ C r − 1 ( x ; λ ) , r ≥ 1 . Let A ∈ W satisfy µ ( A ) = λ and set N A := η ( A ) ∼ Poi( λ ) , then the orthogonality relation for the standard Charlier p olynomials, rewritten in terms of Poisson exp ectation, yields (4.5) E [ C r ( N A ; λ )] = 0 , r ≥ 1 , and (4.6) E [ C r ( N A ; λ ) C s ( N A ; λ )] = 1 { r = s } r ! λ r , r , s ≥ 0 . Lemma 4.5 (Ch arlier–Itô iden tit y for indicator kernels) . F or every r ≥ 1 , (4.7) I r  1 ⊗ r A  = C r ( N A ; λ ) . In p articular, I 1 ( 1 A ) = N A − λ = C 1 ( N A ; λ ) , I 2 ( 1 ⊗ 2 A ) = N 2 A − (2 λ + 1) N A + λ 2 = C 2 ( N A ; λ ) . Pr o of Set P r := I r ( 1 ⊗ r A ) for r ≥ 0 with the conv en tion P 0 := I 0 (1) = 1 . Applying the Poisson pro duct form ula ( 2.17 ) with q 1 = 1 , q 2 = r , f 1 = 1 A and f 2 = 1 ⊗ r A yields, for all r ≥ 1 , (4.8) I 1 ( 1 A ) I r ( 1 ⊗ r A ) = I r +1 ( 1 ⊗ ( r +1) A ) + r I r ( 1 ⊗ r A ) + r λ I r − 1 ( 1 ⊗ ( r − 1) A ) . Since I 1 ( 1 A ) = N A − λ , this can b e rewritten as ( N A − λ ) P r = P r +1 + r P r + r λ P r − 1 , with P 0 = 1 and P 1 = N A − λ . This coincides with the defining recursion and initial conditions of the monic Charlier p olynomials. Hence P r = C r ( N A ; λ ) for all r ≥ 1 , whic h prov es ( 4.7 ). □ W e now expand the cen tered single-site p olynomial observ able φ ( ξ ( n, x )) − E [ φ ( ξ ( n, x ))] . Lemma 4.6 (Single-site p olynomial c haos expansion and co efficien ts) . Ther e exist unique c o effi- cients c φ, 1 , . . . , c φ,k such that (4.9) φ ( N A ) − E [ φ ( N A )] = k X q =1 c φ,q I q  1 ⊗ q A  . Mor e over, these c o efficients ar e given by (4.10) c φ,q = E  φ ( N A ) C q ( N A ; λ )  q ! λ q , 1 ≤ q ≤ k . In p articular, c φ,k = β k  = 0 . Pr o of Since { C q ( · ; λ ) } q ≥ 0 form a complete orthogonal system in L 2 (P oi( λ )) and φ is a p olynomial of degree k , there exist unique co efficien ts d φ,q suc h that φ ( N A ) − E [ φ ( N A )] = k X q =1 d φ,q C q ( N A ; λ ) . 22 AO HUANG, G. RANG, AND Z. SU T aking the inner pro duct with C q ( N A ; λ ) and using ( 4.6 ) gives d φ,q = E [ φ ( N A ) C q ( N A ; λ )] E [ C q ( N A ; λ ) 2 ] = E [ φ ( N A ) C q ( N A ; λ )] q ! λ q . Setting c φ,q := d φ,q yields ( 4.10 ). By Lemma 4.5 , C q ( N A ; λ ) = I q ( 1 ⊗ q A ) , so the ab o v e expansion translates in to ( 4.9 ). Finally , since φ has degree k with leading co efficien t β k and C k ( · ; λ ) is monic of degree k , we ha v e c φ,k = β k  = 0 . □ W e can now sum the single-site expansion along the sampled path to obtain a conditional Poisson- c haos expansion for Y N ,φ . F or fixed S and q = 1 , . . . , k , define (4.11) f q ,S ( w 1 , . . . , w q ) := N X n =1 q Y j =1 1 A n ( w j ) , ( w 1 , . . . , w q ) ∈ W q . Eac h f q ,S is symmetric. Moreov er, ∥ f q ,S ∥ 2 L 2 ( µ ⊗ q ) = N X n,m =1 µ ( A n ∩ A m ) q ≤ N 2 λ q < ∞ , so f q ,S ∈ L 2 s ( µ ⊗ q ) . Conditionally on S , applying Lemma 4.6 with A = A n and summing ov er n = 1 , . . . , N yields Y N ,φ − E ξ [ Y N ,φ | S ] = k X q =1 c φ,q I q ( f q ,S ) . Since Lemma 4.4 shows that E ξ [ Y N ,φ | S ] = E [ Y N ,φ ] , we ha v e (4.12) Y N ,φ − E [ Y N ,φ ] = k X q =1 c φ,q I q ( f q ,S ) , for P S -a.e. realization of S, as an identit y in L 2 ( P ξ ) . 4.3. V ariance gro wth: σ 2 N ,φ ≍ N . T o normalize the c haos expansion, we will show that σ 2 N ,φ gro ws linearly in N in this subsection. F or q ≥ 1 and t ≥ 1 define the drift-av eraged sequence (4.13) a ( q ) t := E S  Q t ( S t ) q  . Lemma 4.7 (An nealed v ariance) . Under ( 4.2 ) , ther e exists c φ ∈ (0 , ∞ ) such that σ 2 N ,φ = V ar( Y N ,φ ) = c φ N + O (1) , henc e σ 2 N ,φ ≍ N . Mor e pr e cisely, (4.14) c φ = k X q =1 c 2 φ,q q ! λ q  1 + 2 X t ≥ 1 a ( q ) t  . Pr o of Since E ξ [ Y N ,φ | S ] = N E [ φ (P oi( λ ))] is deterministic, the law of total v ariance yields σ 2 N ,φ = V ar( Y N ,φ ) = E S  V ar ξ ( Y N ,φ | S )  . NORMAL APPRO XIMA TION FOR THE POL YNOMIAL FUNCTIONALS OF RANDOM W ALK 23 By ( 4.12 ) and orthogonality of Poisson c haoses, (4.15) V ar ξ ( Y N ,φ | S ) = k X q =1 c 2 φ,q q ! ∥ f q ,S ∥ 2 L 2 ( µ ⊗ q ) . W e compute ∥ f q ,S ∥ 2 explicitly . Expanding ( 4.11 ) and using pro duct structure of µ ⊗ q , ∥ f q ,S ∥ 2 L 2 ( µ ⊗ q ) = Z W q  N X n =1 q Y j =1 1 A n ( w j )  2 µ ⊗ q ( d  w ) = N X n,m =1 Z W q q Y j =1 1 A n ∩A m ( w j ) µ ⊗ q ( d  w ) = N X n,m =1 µ ( A n ∩ A m ) q . Note µ ( A n ∩ A m ) = λQ | m − n | ( S m − S n ) , hence (4.16) ∥ f q ,S ∥ 2 L 2 ( µ ⊗ q ) = λ q N X n,m =1 Q | m − n | ( S m − S n ) q = λ q  N + 2Σ ( q ) S  , where Σ ( q ) S := X 1 ≤ n 0 . □ 4.4. Annealed Mallia vin–Stein b ound. Our goal is to b ound the W asserstein distance b et w een the la w of F N ,φ and the standard Gaussian law under the join t measure P = P ξ ⊗ P S . Unlik e the fixed-region functionals W N ( A ) and D N ( A ) , the v ariable F N ,φ dep ends on t wo indep enden t sources of randomness: the P oisson field η and the sampling w alk S . Therefore, under the annealed la w, F N ,φ is not a P oisson functional of η alone, so the standard Poisson Malliavin–Stein b ound ( 2.20 ) cannot b e applied directly . The k ey observ ation is that, for P S -a.e. fixed realization of S , the expansion ( 4.12 ) is a finite Poisson-c haos expansion in L 2 ( P ξ ) . Hence the usual P oisson Malliavin– Stein inequalit y applies conditionally on S . The only additional step is then to a v erage these conditional b ounds ov er the s ampling walk in order to recov er an annealed estimate. Throughout this subsection, the Mallia vin op erators act only in the P oisson direction, with S regarded as fixed. 24 AO HUANG, G. RANG, AND Z. SU Lemma 4.8 (An nealed-b y-conditional distances b ound) . F or any N ≥ 1 , d H  L ( F N ,φ ) , L ( Z )  ≤ E S h d H  L ξ ( F N ,φ | S ) , L ( Z )  i , wher e Z ∼ N (0 , 1) is indep endent of ( η , S ) . Pr o of F or any h ∈ H , we hav e    E [ h ( F N ,φ )] − E [ h ( Z )]    =    E S  E ξ [ h ( F N ,φ ) | S ] − E [ h ( Z )]     (4.18) ≤ E S h   E ξ [ h ( F N ,φ ) | S ] − E [ h ( Z )]   i (4.19) ≤ E S h d H  L ξ ( F N ,φ | S ) , L ( Z )  i . (4.20) T aking the supremum o v er all h ∈ H yields the claim. □ F or P S -a.e. fixed realization of S , the expansion ( 4.12 ) is a finite P oisson-c haos expansion in L 2 ( P ξ ) . Hence, applying ( 2.10 ) and ( 2.12 ) path wise, we obtain for µ -a.e. w ∈ W , (4.21) D w F N ,φ = 1 σ N ,φ k X q =1 c φ,q q I q − 1  f q ,S ( w , · )  , and (4.22) − D w L − 1 F N ,φ = 1 σ N ,φ k X q =1 c φ,q I q − 1  f q ,S ( w , · )  . Applying the conditional Poisson Mallia vin–Stein b ound and then av eraging o v er S via Lemma 4.8 yields (4.23) d W ( F N ,φ , Z ) ≤ r 2 π E h    1 − ⟨ D F N ,φ , − D L − 1 F N ,φ ⟩ L 2 ( µ )    i + E Z W | D w F N ,φ | 2 | D w L − 1 F N ,φ | µ ( dw ) . Set (4.24) Θ N ,φ := ⟨ D F N ,φ , − D L − 1 F N ,φ ⟩ L 2 ( µ ) , and define T ( φ ) 1 := E  | 1 − Θ N ,φ |  , T ( φ ) 2 := E Z W | D w F N ,φ | 2 | D w L − 1 F N ,φ | µ ( dw ) . Then ( 4.23 ) b ecomes (4.25) d W ( F N ,φ , Z ) ≤ r 2 π T ( φ ) 1 + T ( φ ) 2 . In the remainder of this subsection, we first consider the basic observ able φ ( x ) = x , whic h already captures the main route of the pro of in a particularly transparent form. In this case only the first P oisson c haos app ears, whereas the general p olynomial case follows the same strategy with a more in v olved finite-c haos expansion. NORMAL APPRO XIMA TION FOR THE POL YNOMIAL FUNCTIONALS OF RANDOM W ALK 25 4.4.1. The c ase φ ( x ) = x . In this subsection, we sp ecialize the general annealed normal approxi- mation argument to the linear functional Y N ,x = N X n =1 ξ ( n, S n ) . F or φ ( x ) = x , w e ha v e c x, 1 = 1 and f 1 ,S ( w ) = P N n =1 1 A n ( w ) . T o apply the annealed Malliavin–Stein b ound, we first identify th e correct v ariance scale, then write do wn the corresp onding Malliavin op erators, and finally estimate the tw o terms app earing in the b ound. W e start with the v ariance normalization. Lemma 4.7 yields (4.26) σ 2 N ,x = c 1 N + O (1) , c 1 = λ  1 + 2 X t ≥ 1 a (1) t  ∈ (0 , ∞ ) , σ 2 N ,x ≍ N . This shows that σ 2 N ,x gro ws linearly in N , which will b e rep eatedly used when conv erting momen t b ounds in to the final N − 1 / 2 rate. Next w e record the first-order chaos represen tation and the asso ciated Mallia vin op erators. Since for φ ( x ) = x we hav e Y N ,x − E [ Y N ,x ] = I 1 ( f 1 ,S ) by ( 4.12 ), it follo ws that F N ,x = I 1 ( f 1 ,S ) σ N ,x . Consequen tly , for µ -a.e. w ∈ W , using D w I 1 ( g ) = g ( w ) and L − 1 I 1 ( g ) = − I 1 ( g ) , (4.27) D w F N ,x = 1 σ N ,x f 1 ,S ( w ) , and (4.28) − D w L − 1 F N ,x = D w F N ,x . F rom ( 4.24 ), we ha ve Θ N ,x = ⟨ D F N ,x , − D L − 1 F N ,x ⟩ L 2 ( µ ) = Z W | D w F N ,x | 2 µ ( dw ) . By ( 4.27 ) and ( 4.16 ), yields (4.29) Θ N ,x = 1 σ 2 N ,x N X n,m =1 µ ( A n ∩ A m ) = 1 σ 2 N ,x  λN + 2 λ Σ (1) S  . Note that E [ Θ N ,x ] = 1 since E [ λN + 2 λ Σ (1) S ] = σ 2 N ,x b y ( 4.17 ). W e b egin with b ounding T x 1 . T o control T x 1 , we need a v ariance b ound for the correlation sum Σ (1) S . The following estimate will also b e used later in the general p olynomial case. Lemma 4.9. Fix q ∈ { 1 , . . . , k } . Under the drift assumption ( 4.2 ) , ther e exists C := C ( q ) < ∞ such that for al l N ≥ 1 , V ar S (Σ ( q ) S ) ≤ C N , wher e Σ ( q ) S = P 1 ≤ n 0 , dep ending only on k , suc h that (4.49) ρ u ≤ c 1 u − 1 / 2 e − c 0 u , u ≥ 1 . In particular, ρ a + b ≤ C a − 1 / 2 e − c 0 a e − c 0 b , ρ b + c ≤ C b − 1 / 2 e − c 0 b e − c 0 c , and similarly ρ a + b + c ≤ C a − 1 / 2 e − c 0 a e − c 0 b e − c 0 c . It follows from ( 4.48 ) that there exist summable sequences u (1) , u (2) , u (3) on N (dep ending only on k ) suc h that (4.50) E [ T ( n, m, n ′ , m ′ )] ≤ C u (1) a u (2) b u (3) c . NORMAL APPRO XIMA TION FOR THE POL YNOMIAL FUNCTIONALS OF RANDOM W ALK 31 Summing ( 4.50 ) o ver all quadruples with four distinct indices yields, up to a combinatorial factor at most 4! that we absorb into C , N X n,m,n ′ ,m ′ =1 |{ n,m,n ′ ,m ′ }| =4 E [ T ( n, m, n ′ , m ′ )] ≤ C X a,b,c ≥ 1 ( N − a − b − c ) + u (1) a u (2) b u (3) c . By Lemma 4.3 with M = 3 , the right-hand side is b ounded by C N . Step 2: degenerate configurations. Assume no w that the set { n, m, n ′ , m ′ } has cardinality at most three. Then one or more of the time gaps v anish, and the corresp onding transition k ernels reduce to Q 0 (0) = 1 . Consequen tly , the same argument as ab o v e yields b ounds in v olving at most tw o p ositiv e gaps (there are three distinct times), or at most one p ositive gap (there are only t w o distinct times), or none gap (all equal). Th us these contributions are b ounded by expressions of the form C X a,b ≥ 1 ( N − a − b ) + e u (1) a e u (2) b or C X a ≥ 1 ( N − a ) + b u a or C N , for suitable summable sequences e u (1) , e u (2) , b u depending only on k . Applying Lemma 4.3 with M = 2 or M = 1 shows that all degenerate configurations also contribute at most C N . Com bining the distinct-time and degenerate cases, w e obtain N X n,m,n ′ ,m ′ =1 E [ T ( n, m, n ′ , m ′ )] ≤ C N . T ogether with ( 4.46 ), this prov es E  ∥ ¯ H q 1 ,q 2 ,r,ℓ,S ∥ 2 L 2 ( µ ⊗ d )  ≤ C N . Since ∥ H q 1 ,q 2 ,r,ℓ,S ∥ 2 ≤ ∥ ¯ H q 1 ,q 2 ,r,ℓ,S ∥ 2 , the claimed estimate follows. □ W e can now close the estimate for T ( φ ) 1 . Using ( 4.39 ) and Lemma 4.11 , E  V ar ξ (Θ N ,φ | S )  ≤ C σ 4 N ,φ N . By ( 4.40 ) and Lemma 4.10 , V ar S  E ξ [Θ N ,φ | S ]  = V ar S  V S σ 2 N ,φ  = 1 σ 4 N ,φ V ar S ( V S ) ≤ C σ 4 N ,φ N . Therefore, by total v ariance, V ar(Θ N ,φ ) ≤ C σ 4 N ,φ N . Since σ 2 N ,φ ≍ N (Lemma 4.7 ), w e hav e σ 4 N ,φ ≍ N 2 and thus V ar(Θ N ,φ ) ≤ C / N . Plugging in to ( 4.35 ) gives (4.51) T ( φ ) 1 ≤ C N − 1 / 2 . Estimate of T ( φ ) 2 . W e no w b ound T ( φ ) 2 in ( 4.23 ). Using the elementary inequality | P m j =1 x j | 3 ≤ m 2 P m j =1 | x j | 3 together with the finiteness of the sums in ( 4.21 )–( 4.22 ), there exists C := C ( φ ) < ∞ suc h that for 32 AO HUANG, G. RANG, AND Z. SU all w , | D w F N ,φ | 2 | D w L − 1 F N ,φ | ≤ C σ 3 N ,φ k X q =1    I q − 1  f q ,S ( w , · )     3 . (4.52) In tegrating ov er w and taking exp ectation yields (4.53) E Z W | D w F N ,φ | 2 | D w L − 1 F N ,φ | µ ( dw ) ≤ C σ 3 N ,φ k X q =1 T q ,N , where T q ,N := E Z W    I q − 1  f q ,S ( w , · )     3 µ ( dw ) . Th us it suffices to sho w that T q ,N = O ( N ) for eac h fixed q ≤ k . W e will do so b y combining a momen t b ound for m ultiple integrals. Lemma 4.12 (F ourth moment b ound via con tractions) . Fix an inte ger q ≥ 1 . L et h ∈ L 2 s ( µ ⊗ q ) b e symmetric and assume that, for every 0 ≤ r ≤ q and 0 ≤ ℓ ≤ r , the c ontr action kernel h e ⋆ ℓ r h b elongs to L 2 ( µ ⊗ (2 q − r − ℓ ) ) . Then ther e exists C := C ( q ) < ∞ (dep ending only on q ) such that (4.54) E ξ  I q ( h ) 4  ≤ C q X r =0 r X ℓ =0 (2 q − r − ℓ )!   h e ⋆ ℓ r h   2 L 2 ( µ ⊗ (2 q − r − ℓ ) ) . Pr o of By the pro duct formula ( 2.17 ) with q 1 = q 2 = q and f 1 = f 2 = h , I q ( h ) 2 = q X r =0 r ! q r ! 2 r X ℓ =0 r ℓ ! I m ( r,ℓ )  h e ⋆ ℓ r h  , m ( r , ℓ ) := 2 q − r − ℓ. Since differen t pairs ( r, ℓ ) may yield the same c haos order m , hence the corresp onding m ultiple in tegrals need not b e orthogonal. W e therefore group by chaos order. F or eac h m ∈ { 0 , 1 , . . . , 2 q } set G m := X 0 ≤ r ≤ q , 0 ≤ ℓ ≤ r : m ( r,ℓ )= m α r,ℓ  h e ⋆ ℓ r h  , α r,ℓ := r ! q r ! 2 r ℓ ! . Then I q ( h ) 2 = P 2 q m =0 I m ( G m ) . By orthogonality of different c haos orders, E ξ  I q ( h ) 4  = 2 q X m =0 E ξ  I m ( G m ) 2  = 2 q X m =0 m ! ∥ G m ∥ 2 L 2 ( µ ⊗ m ) . F or each fixed m , the sum defining G m con tains only finitely many terms dep ending on q . Hence, using ∥ P M i =1 u i ∥ 2 2 ≤ M P M i =1 ∥ u i ∥ 2 2 and absorbing the (finite) com binatorial factors in to C , we obtain m ! ∥ G m ∥ 2 2 ≤ C X r,ℓ : m ( r,ℓ )= m m ! ∥ h e ⋆ ℓ r h ∥ 2 2 . Summing ov er m yields ( 4.54 ). □ Corollary 4.13 (Third moment b ound via contractions) . Under the assumptions of L emma 4.12 , ther e exists C := C ( q ) < ∞ (dep ending only on q ) such that (4.55) E ξ  | I q ( h ) | 3  ≤ C ∥ h ∥ L 2 ( µ ⊗ q ) q X r =0 r X ℓ =0   h e ⋆ ℓ r h   2 L 2 ( µ ⊗ (2 q − r − ℓ ) ) ! 1 / 2 . NORMAL APPRO XIMA TION FOR THE POL YNOMIAL FUNCTIONALS OF RANDOM W ALK 33 Pr o of By Cauch y–Sc h w arz, E ξ  | I q ( h ) | 3  ≤ E ξ  I q ( h ) 2  1 / 2 E ξ  I q ( h ) 4  1 / 2 = p q ! ∥ h ∥ 2 E ξ  I q ( h ) 4  1 / 2 . Inserting ( 4.54 ) and absorbing constants in to C . □ Lemma 4.14. Fix q ∈ { 1 , . . . , k } and write q ′ := q − 1 . Under ( 4.2 ) , ther e exists C := C ( φ ) < ∞ such that for al l N ≥ 1 , T q ,N := E Z W    I q ′  f q ,S ( w , · )     3 µ ( dw ) ≤ C N . Pr o of W e treat q = 1 and q ≥ 2 separately . Case q = 1 . Then q ′ = 0 and I q ′ = I 0 is the iden tit y . Since f 1 ,S ( w ) = P N n =1 1 A n ( w ) , the b ound T 1 ,N = O ( N ) follows as in the low-order case b y expanding f 3 1 ,S and using the summability P t ≥ 1 a (1) t < ∞ from Lemma 4.2 . Case q ≥ 2 . Fix S and w ∈ W and set (4.56) h w := f q ,S ( w , · ) ∈ L 2 s ( µ ⊗ q ′ ) , h w ( u 1 , . . . , u q ′ ) = N X n =1 1 A n ( w ) q ′ Y j =1 1 A n ( u j ) . Since h w is a finite sum of indicator tensors of sets with finite µ -mass, all its contractions h w e ⋆ ℓ r h w b elong to the required L 2 spaces. Here and below, for the scalar contraction corresp onding to ( r , ℓ ) = ( q ′ , q ′ ) , we use the conv en tion L 2 ( µ ⊗ 0 ) = R . Applying Corollary 4.13 with q = q ′ to h = h w (conditionally on S ), w e obtain (4.57) E ξ  | I ′ q ( h w ) | 3 | S  ≤ C q ′ ∥ h w ∥ L 2 ( µ ⊗ q ′ ) q ′ X r =0 r X ℓ =0   h w e ⋆ ℓ r h w   2 2 ! 1 / 2 . In tegrating ( 4.57 ) with resp ect to µ ( dw ) and using Cauch y–Sc h w arz in µ ( dw ) yields Z W E ξ  | I ′ q ( h w ) | 3 | S  µ ( dw ) ≤ C q ′  Z W ∥ h w ∥ 2 2 µ ( dw )  1 / 2  Z W X r,ℓ ∥ h w e ⋆ ℓ r h w ∥ 2 2 µ ( dw )  1 / 2 . (4.58) T aking E S and using Cauch y–Sc h warz in P S giv es (4.59) T q ,N ≤ C q ′  E Z W ∥ h w ∥ 2 2 µ ( dw )  1 / 2  E Z W X r,ℓ ∥ h w e ⋆ ℓ r h w ∥ 2 2 µ ( dw )  1 / 2 . W e first ev aluate the factor E R ∥ h w ∥ 2 2 µ ( dw ) . A direct computation using ( 4.56 ) yields Z W ∥ h w ∥ 2 2 µ ( dw ) = N X n,m =1 µ ( A n ∩ A m ) q ′ +1 = N X n,m =1 µ ( A n ∩ A m ) q . Since µ ( A n ∩ A m ) = λQ | m − n | ( S m − S n ) , taking expectation and using stationary incremen ts we obtain E Z W ∥ h w ∥ 2 2 µ ( dw ) = λ q  N + 2 N − 1 X t =1 ( N − t ) a ( q ) t  = O ( N ) , b ecause P t ≥ 1 a ( q ) t < ∞ b y Lemma 4.2 . It remains to b ound E R ∥ h w e ⋆ ℓ r h w ∥ 2 2 µ ( dw ) uniformly ov er 0 ≤ ℓ ≤ r ≤ q ′ . Since symmetrization is an av erage ov er p erm utations, we hav e ∥ h w e ⋆ ℓ r h w ∥ 2 ≤ ∥ h w ⋆ ℓ r h w ∥ 2 , so it suffices to consider the 34 AO HUANG, G. RANG, AND Z. SU unsymmetrized contraction. Using ( 4.56 ), bilinearity of ⋆ ℓ r , and the explicit con traction of indicator tensors (as in Lemma 4.11 ), one obtains h w ⋆ ℓ r h w = N X n,m =1 1 A n ∩A m ( w ) µ ( A n ∩ A m ) ℓ 1 ⊗ ( r − ℓ ) A n ∩A m ⊗ 1 ⊗ ( q ′ − r ) A n ⊗ 1 ⊗ ( q ′ − r ) A m . Expanding the L 2 -norm squared, integrating in w , and using the pro duct structure of µ ⊗ giv es Z W ∥ h w ⋆ ℓ r h w ∥ 2 2 µ ( dw ) = N X n,m,n ′ ,m ′ =1 µ ( A n ∩ A m ) ℓ µ ( A n ′ ∩ A m ′ ) ℓ (4.60) × µ ( A n ∩ A m ∩ A n ′ ∩ A m ′ ) r − ℓ +1 µ ( A n ∩ A n ′ ) q ′ − r µ ( A m ∩ A m ′ ) q ′ − r . The resulting quadruple sum is of exactly the same form as in the pro of of Lemma 4.11 ; only the exp onents of the intersection masses change from ( ℓ + 1 , ℓ + 1 , r − ℓ, q 1 − 1 − r, q 2 − 1 − r ) to ( ℓ, ℓ, r − ℓ + 1 , q ′ − r , q ′ − r ) , and all these exp onents remain b ounded by k . Therefore the same Hölder-and-gap argument applies verbatim. Consequen tly , E Z W ∥ h w ⋆ ℓ r h w ∥ 2 2 µ ( dw ) ≤ C N , uniformly ov er ( r , ℓ ) , and hence also E R W ∥ h w e ⋆ ℓ r h w ∥ 2 2 µ ( dw ) ≤ C N . Com bining these t wo bounds in ( 4.59 ) yields T q ,N ≤ C N . □ Com bining ( 4.53 ) with Lemma 4.14 , w e get (4.61) T ( φ ) 2 ≤ C σ 3 N ,φ k X q =1 T q ,N ≤ C N σ 3 N ,φ . Since σ 2 N ,φ ≍ N (Lemma 4.7 ), we ha v e σ 3 N ,φ ≍ N 3 / 2 and thus T ( φ ) 2 ≤ C N − 1 / 2 . Finally , combining the b ounds for T ( φ ) 1 and T ( φ ) 2 , we obtain T ( φ ) 1 ≤ C N − 1 / 2 , T ( φ ) 2 ≤ C N − 1 / 2 . Hence, by ( 4.23 ), d W ( F N ,φ , Z ) ≤ r 2 π T ( φ ) 1 + T ( φ ) 2 ≤ C N − 1 / 2 . This completes the pro of for the case k ≥ 2 . Appendix A. The symmetric sampling case p = 1 2 A natural question is whether the approach developed in Section 4 con tinues to apply when the sampling walk is symmetric, that is, when p = 1 2 . The purp ose of this app endix is to clarify this p oin t. The structural ingredients of our pro of in the drifted case remain a v ailable under symmetric sampling: the conditional Poisson-c haos expansion, and the annealed Malliavin–Stein framework all con tin ue to hold without essen tial change. The difference arises at the lev el of the quan titativ e estimates. In the drifted regime, the ballistic b eha vior of the sampling w alk yields summable correlation coefficients, whic h is precisely what allo ws the first Malliavin–Stein term to b e shown to decay to zero. In the symmetric regime, b y NORMAL APPRO XIMA TION FOR THE POL YNOMIAL FUNCTIONALS OF RANDOM W ALK 35 con trast, these correlations deca y only polynomially , and the same argumen t no longer yields a deca ying b ound for that term. This should not b e interpreted as indicating that a central limit theorem fails under symmetric sampling. Rather, it shows only that the particular quantitativ e route used in Section 4 do es not, in its presen t form, close in this regime. Nev ertheless, the symmetric case exhibits a non trivial and in teresting v ariance structure, gov erned by the first non-v anishing Poisson–Charlier co efficien t of the observ able, and it is therefore w orthwhile to record the corresp onding calculations explicitly . W e do so b elo w. The question of whether one can nevertheless establish an annealed central limit theorem, or obtain quenc hed W asserstein b ounds, under symmetric sampling is left for future w ork. In this app endix w e discuss the symmetric sampling regime p = 1 2 . T o distinguish it from the drifted sampling walk S used in Section 4 , we denote by b S = ( b S n ) n ≥ 0 an indep enden t simple symmetric random walk (SSR W) on Z . Since b S has the same law as the particle walk, its transition kernel coincides with the particle kernel, and we con tin ue to write Q n ( x ) = P ( b S n = x ) , x ∈ Z , n ∈ N 0 , for this common SSR W kernel. F or a fixed p olynomial observ able φ : N 0 → R , w e define the symmetric-sampling path functional b Y N ,φ := N X n =1 φ  ξ ( n, b S n )  , b σ 2 N ,φ := V ar  b Y N ,φ  . The purp ose of this app endix is tw ofold: (1) to record the v ariance regimes under symmetric sampling for a general fixed p olynomial observ able φ ; (2) to explain why the crude conditional-v ariance step used in the drifted Malliavin–Stein pro of no longer closes in the rank-one symmetric case. A.1. A parit y-corrected lo cal limit theorem and ℓ s -norms of the SSR W k ernel. Recall Lemma 3.1 , we hav e: (A.1) sup x ∈ Z      √ n 2 Q n ( x ) − 1 { x ≡ n (mod 2) } 1 √ 2 π exp  − x 2 2 n       − − − → n →∞ 0 . Equiv alently , (A.2) Q n ( x ) = 1 { x ≡ n (mod 2) } 2 √ 2 π n exp  − x 2 2 n  + o ( n − 1 / 2 ) , uniformly in x ∈ Z . W e now deduce the asymptotics of the ℓ s -norms of Q n . Lemma A.1 ( ℓ s -asymptotics for the SSR W kernel) . Fix an inte ger s ≥ 2 . Then, as n → ∞ , (A.3) X x ∈ Z Q n ( x ) s = κ s n − ( s − 1) / 2 + o  n − ( s − 1) / 2  , κ s := 2 ( s − 1) / 2 π ( s − 1) / 2 √ s . 36 AO HUANG, G. RANG, AND Z. SU In p articular, X x ∈ Z Q n ( x ) 2 ∼ 1 √ π n , X x ∈ Z Q n ( x ) 3 ∼ 2 π √ 3 1 n . Pr o of Fix s ≥ 2 . Let g n ( x ) := 1 { x ≡ n (mod 2) } 2 √ 2 π n exp  − x 2 2 n  . By ( A.2 ), there exists a deterministic sequence ε n ↓ 0 suc h that | Q n ( x ) − g n ( x ) | ≤ ε n n − 1 / 2 for all x ∈ Z . Fix M > 0 and split the sum in to the cen tral region | x | ≤ M √ n and the tail | x | > M √ n . On the central region, uniformly in | x | ≤ M √ n , b oth Q n ( x ) and g n ( x ) are of order n − 1 / 2 . Hence, b y the mean-v alue theorem, | Q n ( x ) s − g n ( x ) s | ≤ C s  Q n ( x ) s − 1 + g n ( x ) s − 1  | Q n ( x ) − g n ( x ) | = o ( n − s/ 2 ) uniformly for | x | ≤ M √ n . Since the n umber of lattice p oints in this region is O ( √ n ) , we obtain (A.4) X | x |≤ M √ n x ≡ n (mod 2) Q n ( x ) s = X | x |≤ M √ n x ≡ n (mod 2) g n ( x ) s + o  n − ( s − 1) / 2  . F or the tail, the standard heat-kernel b ound giv es sup x ∈ Z Q n ( x ) ≲ n − 1 / 2 . Moreo v er, by Ho effding’s inequalit y for the SSR W, there exists c > 0 such that P ( | b S n | ≥ u ) ≤ 2 e − cu 2 /n , u ≥ 0 . Therefore X | x | >M √ n Q n ( x ) s ≤  sup x Q n ( x )  s − 1 X | x | >M √ n Q n ( x ) ≤ C n − ( s − 1) / 2 P ( | b S n | > M √ n ) ≤ C n − ( s − 1) / 2 e − cM 2 . (A.5) An analogous estimate holds for g n : (A.6) X | x | >M √ n g n ( x ) s ≤ C n − ( s − 1) / 2 e − cM 2 . It remains to ev aluate the main term. Since summation ov er one parit y class has mesh size 2 , w e hav e X x ∈ Z x ≡ n (mod 2) g n ( x ) s =  2 √ 2 π n  s X x ∈ Z x ≡ n (mod 2) exp  − sx 2 2 n  =  2 √ 2 π n  s 1 2 Z R exp  − su 2 2 n  du + o ( √ n ) ! =  2 √ 2 π n  s 1 2 r 2 π n s + o ( √ n ) ! = κ s n − ( s − 1) / 2 + o  n − ( s − 1) / 2  . NORMAL APPRO XIMA TION FOR THE POL YNOMIAL FUNCTIONALS OF RANDOM W ALK 37 Com bining this with ( A.4 ), ( A.5 ), and ( A.6 ), and then letting M → ∞ , prov es ( A.3 ). □ F or ℓ ≥ 1 we will rep eatedly use the shorthand (A.7) b a ( ℓ ) t := E  Q t ( b S t ) ℓ  = X x ∈ Z Q t ( x ) ℓ +1 . By Lemma A.1 , (A.8) b a ( ℓ ) t ∼ κ ℓ +1 t − ℓ/ 2 , t → ∞ . A.2. V ariance regimes for general p olynomial observ ables. Let N λ ∼ Poi( λ ) , and write the Poisson–Charlier expansion of the cen tered single-site observ able as (A.9) φ ( N λ ) − E [ φ ( N λ )] = k X ℓ =1 c φ,ℓ C ℓ ( N λ ; λ ) . Define the Poisson–Charlier rank of φ by (A.10) r φ := min { ℓ ∈ { 1 , . . . , k } : c φ,ℓ  = 0 } . The v ariance form ula from Lemma 4.7 sp ecializes under symmetric sampling as follows. Lemma A.2 (V ariance representation under symmetric sampling) . F or b Y N ,φ := N X n =1 φ ( ξ ( n, b S n )) , we have (A.11) V ar  b Y N ,φ  = k X ℓ =1 c 2 φ,ℓ ℓ ! λ ℓ N + 2 N − 1 X t =1 ( N − t ) b a ( ℓ ) t ! , wher e b a ( ℓ ) t is given by ( A.7 ) . Pr o of This is exactly the v ariance formula pro v ed in Lemma 4.7 , with b a ( ℓ ) t = E [ Q t ( b S t ) ℓ ] . Under p = 1 2 , the law of b S t is Q t ( · ) , hence b a ( ℓ ) t = X x ∈ Z Q t ( x ) Q t ( x ) ℓ = X x ∈ Z Q t ( x ) ℓ +1 . This prov es ( A.11 ). □ W e now derive the three v ariance regimes. Recall the elementary asymptotics N − 1 X t =1 ( N − t ) t − 1 / 2 = 4 3 N 3 / 2 + O ( N ) , (A.12) N − 1 X t =1 ( N − t ) t − 1 = N log N + O ( N ) , (A.13) and, for every α > 1 , (A.14) N − 1 X t =1 ( N − t ) t − α ≍ N . 38 AO HUANG, G. RANG, AND Z. SU Lemma A.3 (V ariance regimes by Charlier rank) . L et r φ b e as in ( A.10 ) . (1) If r φ = 1 , then (A.15) V ar  b Y N ,φ  ∼ 8 c 2 φ, 1 λ 3 √ π N 3 / 2 . (2) If r φ = 2 , then (A.16) V ar  b Y N ,φ  ∼ 8 c 2 φ, 2 λ 2 π √ 3 N log N . (3) If r φ ≥ 3 , then (A.17) V ar  b Y N ,φ  ≍ N . Pr o of Insert ( A.8 ) into ( A.11 ). If r φ = 1 , then b a (1) t ∼ κ 2 t − 1 / 2 , κ 2 = 1 √ π . Hence, by ( A.12 ), 2 c 2 φ, 1 λ N − 1 X t =1 ( N − t ) b a (1) t ∼ 2 c 2 φ, 1 λ · 1 √ π · 4 3 N 3 / 2 = 8 c 2 φ, 1 λ 3 √ π N 3 / 2 . All terms with ℓ ≥ 2 are O ( N log N ) = o ( N 3 / 2 ) , proving ( A.15 ). If r φ = 2 , then c φ, 1 = 0 and b a (2) t ∼ κ 3 t − 1 , κ 3 = 2 π √ 3 . Therefore, by ( A.13 ), 2 c 2 φ, 2 2! λ 2 N − 1 X t =1 ( N − t ) b a (2) t ∼ 4 c 2 φ, 2 λ 2 · 2 π √ 3 N log N = 8 c 2 φ, 2 λ 2 π √ 3 N log N . All terms with ℓ ≥ 3 are O ( N ) = o ( N log N ) , proving ( A.16 ). Finally , if r φ ≥ 3 , then for ev ery ℓ ≥ r φ one has b a ( ℓ ) t ≍ t − ℓ/ 2 with ℓ/ 2 > 1 . Hence every summand in ( A.11 ) is of order N by ( A.14 ). F or the lo wer bound, the diagonal contribution already gives V ar  b Y N ,φ  ≥ c 2 φ,r φ r φ ! λ r φ N . Th us ( A.17 ) follo ws. □ Remark A.4. The symmetric r e gime is ther efor e qualitatively differ ent fr om the drifte d r e gime: • r ank-one observables have sup erline ar varianc e of or der N 3 / 2 ; • r ank-two observables have varianc e of or der N log N ; • only fr om r ank thr e e onwar d do es one r e c over line ar varianc e gr owth. In p articular, the varianc e b ehavior under symmetric sampling dep ends on the first non-zer o Poisson– Charlier c o efficient, not mer ely on the de gr e e of the p olynomial. NORMAL APPRO XIMA TION FOR THE POL YNOMIAL FUNCTIONALS OF RANDOM W ALK 39 A.3. Wh y the conditional-v ariance step from the drifted pro of no longer closes. W e no w explain why the conditional-v ariance step used in the pro of of Theorem 1.4 no longer yields a deca ying b ound in the symmetric rank-one regime. F or ℓ ≥ 1 , define the path functional (A.18) b Σ ( ℓ ) N := X 1 ≤ n

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