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TheAnnalsofAppliedProbability 2008,Vol.18,No.6,2320–2336 DOI:10.1214/08-AAP524 (cid:13)c InstituteofMathematicalStatistics,2008 CENTRAL LIMIT THEOREM FOR THE SOLUTION OF THE KAC EQUATION 9 0 By Ester Gabetta1 and Eugenio Regazzini2 0 2 Universita` degli Studi di Pavia and Universita` degli Studi di Pavia n a We prove that the solution of the Kac analogue of Boltzmann’s J equation can be viewed as a probability distribution of a sum of a 6 random number of random variables. This fact allows us to study 1 convergence to equilibrium by means of a few classical statements pertaining to the central limit theorem. In particular, a new proof ] of the convergence to the Maxwellian distribution is provided, with R a rate information both under the sole hypothesis that the initial P energy is finite and under the additional condition that the initial . h distribution has finite moment of order 2+δ for some δ in (0,1]. t Moreover, it is provedthat finitenessofinitial energy is necessary in a orderthatthesolutionofKac’sequationcanconvergeweakly.While m this statement may seem to be intuitively clear, to our knowledge [ thereis noproof of it as yet. 1 v 1. Introduction and presentation of new results. 4 6 1.1. Introduction. Marc Kac studied Boltzmann’s derivation of a basic 4 2 equation of kinetic theory by simplifying the problem to an n-particle sys- . tem in one-dimension and, under suitable conditions, he got the following 1 0 analogue of the Boltzmann equation: 9 ∂ 1 0 f(v,t)= f(vcosθ wsinθ,t) v: ∂t 2π ZR×[0,2π){ − (1) f(vsinθ+wcosθ,t) Xi  × f(v,t)f(w,t) dwdθ, r f(v,0)=f (v) (t>0,v R),− } a  0 ∈ Received October 2006; revised October 2007. 1SupportedinpartbyMinisterodell’Istruzione, dell’UniversitàedellaRicerca(MIUR Grant 2006/015821). 2SupportedinpartbyMinisterodell’Istruzione, dell’UniversitàedellaRicerca(MIUR Grant 2006/134526). AMS 2000 subject classifications. 60F05, 82C40. Key words and phrases. Central limit theorem, Kac’s equation, Kologorov distance, Wild’s sum. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Probability, 2008,Vol. 18, No. 6, 2320–2336. This reprint differs from the original in pagination and typographic detail. 1 2 E. GABETTA AND E. REGAZZINI where f and f(,t) denote the probability density functions of the velocity 0 · of each particle at time 0 and at time t, respectively. This problem admits a unique solution within the class of all probability density functions on R. See, for example, Kac (1956), Kac (1959), McKean (1966), Cercignani (1975) and Diaconis and Saloff-Coste (2000). Bobylëv (1984) proved that the Fourier transform φ(,t) of the solution · f(,t) of (1) must satisfy · ∂ 1 2π φ(ξ,t)= φ(ξsinθ,t)φ(ξcosθ,t)dθ φ(ξ,t), (2) ∂t 2π −  Z0 φ(ξ,0)=φ0(ξ) (t>0,ξ R), ∈ φ being the Fourier transform of f . Clearly, problem (2) is well defined 0 0 for arbitrary (not necessarily absolutely continuous) probability measures µ(,t) and µ on the class B(R) of all Borel subsets of R, provided that 0 · φ(,t) and φ are thought of as Fourier–Stieltjes transforms of µ(,t) and µ , 0 0 · · respectively. Thesolutionof(2)—whichexistsandisuniquewithintheFourier–Stieltjes transformsofallprobabilitymeasureson B(R)—canbeexpressedbymeans of the transform of the Wild series [see Wild (1951)], that is, (3) φ(ξ,t)= e−t(1 e−t)n−1qˆ+(ξ;φ ) (t 0,ξ 0), − n 0 ≥ ≥ n≥1 X where qˆ+ can be found by recursion as n n−1 1 qˆ+(ξ;φ )= qˆ+ (ξ;φ ) qˆ+(ξ;φ ) (n=2,3,...), n 0 n 1 n−j 0 ◦ j 0 − jX=1 with qˆ :=φ . The symbol g g , where g and g are characteristic func- 1 0 1 2 1 2 ◦ tions, designates the Wild product 1 2π g g (ξ)= g (ξcosθ)g (ξsinθ)dθ (ξ R). 1 2 1 2 ◦ 2π ∈ Z0 Gettingdowntotheapproachtoequilibriumofthesolutionof(1)astgoes to infinity, according to Boltzmann’s classical research, the entropy of f(,t) · should increase to its upper bound, log(σ√2πe) with σ2 = v2f (v)dv, R 0 while f tends to the Maxwellian function (viz., the Gaussian density with R zero mean and variance σ2) 1 g (v)= e−v2/(2σ2) (v R). σ σ√2π ∈ McKean (1966) argues that the Wild representation suggests a simpler ex- planation: the central limit theorem for Maxwellian molecules. With the aim CENTRAL LIMIT THEOREM FOR THE SOLUTIONOFTHE KACEQUATION 3 of demonstrating the solidity of his argument, he starts by proving a new expression for qˆ+, that is, n (4) qˆ+(ξ;φ )= p (γ)c (ξ;φ ), n 0 n γ 0 γ∈XG(n) where c denotes the n-fold Wild product of φ with itself performed ac- γ 0 cording to an algebraic structure schematized by the element γ of a class G(n) of random trees with n leaves. p () is a probability on the sub- n · sets of G(n). See McKean (1967), Carlen, Carvalho and Gabetta (2000), Carlen, Carvalho and Gabetta(2005),Bassetti, Gabetta and Regazzini(2007). Then, considering the form of c , with the aid of the Lindeberg version of γ the central limit theorem, McKean proves the following statement on the weak convergence of the probability distribution function C , corresponding γ to c , toward the Gaussian distribution function G (v)= v g (x)dx: γ σ −∞ σ Set σ2 := v2f (v)dv and let v 3f (v)dv be finite. Then, for any R 0 R| | 0 R δ>0, there are constants c=c(δ,f ), c =c (δ,f ) and a positive integer R R0 1 1 0 n =n (δ,f ) such that 0 0 0 p γ G(n):sup C (v) G (v) >δ c(δ,f )n8/(3π)−1 n γ σ 0 (cid:18)(cid:26) ∈ v∈R| − | (cid:27)(cid:19)≤ (5) (n n ), 0 ≥ which leads to (6) sup F(v,t) G (v) 0 (as t + ), σ v∈R| − |→ → ∞ whereF(,t)denotestheprobabilitydistributionfunctionwhichcorresponds · to the solution φ(,t) of (2). · 1.2. Presentation of new results. The study of necessary and sufficient conditions under which (6) holds true, together with some hints to rate of convergence, is the main scope of the present paper. We will prove the following: Theorem 1. Let µ be a nondegenerate probability measure on B(R) 0 and let F(,t) be the probability distribution function corresponding to the · solution φ(,t) of the Kac equation (2). Then · sup F(v,t) G (v) 0 (as t + ) σ v∈R| − |→ → ∞ holds true if and only if σ2:= x2µ (dx) is finite. R 0 R 4 E. GABETTA AND E. REGAZZINI It is wellknown that (6) is valid when the initial energy is finite. See, for example, Carlen and Lu (2003). As far as the necessity of this condition is concerned, while it cannot be doubted from a physical intuitive standpoint, it seems that no proof of it has been advanced as yet. Moreover, our ap- proach leads to state a rather precise quantitative evaluation of the rate of convergence. Thisresultiscontained inthenexttheorem,whereF andF 0 0,d are probability distribution functions defined by F (x):=µ (( ,x]), 0 0 −∞ F (x):=µ ([ x,+ )) (x R). 0,d 0 − ∞ ∈ Theorem 2. Let µ be a nondegenerate probability measure on B(R), 0 σ2:= x2µ (dx) be finite and let a, p be fixed numbers in (0,1) and (2,+ ), 0 ∞ respectively. Then, there is a strictly positive constant A such that R sup F(x,t) G (x) AM(t)1/5+ 1sup F (x) F (x)e−t, x∈R| − σ |≤ 2x∈R| 0 − 0,d | where M(t)= u2µ (du) e−B1t e−B2t 0 Z|u|>σ(xt)a−1 ∨ ∨ for every t t :=inf t: u2µ (du) 1 with ≥ 0 { |u|>σ(xt)a−1 0 ≤ } x :=exp tcpR, t {− } 1 2α p B :=acp, B :=1 2α cp, c 0, − , 1 2 p − − ∈ p (cid:18) (cid:19) 1 2π α = cosθ pdθ. p 2π | | Z0 Moreover, if m = x2+δµ (dx)<+ for some δ in (0,1], then 2+δ 0 | | ∞ m 1 sup F(x,t) G (x) RC 2+δe−t(1−2α2+δ)+ sup F (x) F (x)e−t, x∈R| − σ |≤ δ σ2+δ 2x∈R| 0 − 0,d | where C is a universal constant (Berry–Esseen constant). δ Constant A can be easily obtained by looking at the proof of Theorem 2 in Section 3. The proofs of Theorems 1 and 2 rest on an idea which goes back to McKean (1966). In thepresent paperwe go deep into that idea by providing acompleteproofofthenextbasictheorem,inwhichq (n):=e−t(1 e−t)n−1, t n = 1,2,...; u∞ is the probability measure on B([0,2π)∞) wh−ich makes the coordinates of [0,2π)∞ independent and uniformly distributed, and µ∞ 0 meets the same conditions with [0,2π) and u replaced by R and µ , respec- 0 tively. CENTRAL LIMIT THEOREM FOR THE SOLUTIONOFTHE KACEQUATION 5 Theorem 3. For each t>0, there are a probability space (Ω,F,P ) and t random variables ν˜ :Ω N, t → γ˜:Ω G:= G(n), → n [ θ˜:=(θ˜ ,θ˜ ,...):Ω [0,2π)∞, 1 2 → x˜:=(x˜ ,x˜ ,...):Ω R∞ 1 2 → with joint distribution P ν˜ =n,γ˜=γ,θ˜ A,x˜ B =q (n)p (γ) (γ)u∞(A)µ∞(B) t{ t ∈ ∈ } t n 1G(n) 0 (n N,γ G,A B([0,2π)∞),B B(R∞)) ∈ ∈ ∈ ∈ such that ν˜t V := π (γ˜,θ˜)x˜ t j j j=1 X has probability distribution µ(,t), that is, the distribution corresponding to · the solution φ(,t) of (2). · Apart from the definition of functions π , that we postpone to Section 2, j where a physical interpretation is given, Theorem 3 allows us to understand theconnectionbetweenconvergencetoequilibriumofµ(,t)andcentrallimit · theorem: Indeed, µ(,t) is the distribution of V , that is, a sum of random t · variables.Withrespecttoordinaryapplicationsofthecentrallimittheorem, here we have a random number (ν˜) of summands, and these summands t have (not stochastically independent) random coefficients (π , j=1,...). j But these difficulties can be avoided through a careful utilization of the features of the joint distribution of (ν˜,γ˜,θ˜,x˜). This way, one can provide t completeproofsofTheorems1and2throughsimpleadaptationsofpowerful classicalresultsfromprobabilitytheory.Actually,wearepursuingtheobject of tracing to the above very same kind of ideas the study of the trend to equilibrium (with rate information) under the most important (weak or strong) modes of convergence, both for the Kac model and for other models suchasan“inelastic”versionof(1)–(2)introducedinPulvirenti and Toscani (2004), and the Boltzmann equation for Maxwellian molecules in case of spatially homogeneous initial data with uniform collision kernel [see, e.g., Carlen and Lu (2003)]. Itiswelltopausehereandconsiderwhatwillbeinvolvedinthearguments usedtoproveTheorems1and2.First,wewillprovideaproofforTheorem2 under the sole hypothesis that the initial energy is finite. It is apparent that 6 E. GABETTA AND E. REGAZZINI this covers also the sufficiency part in Theorem 1. The line of reasoning, to obtain the rate of convergence in Theorem 2, consists in adapting the argumentgenerallyusedintheproofoftheclassicalLindeberg–Fellerversion of the central limit theorem. As far as the necessity part in Theorem 1 is concerned—that is, convergence in distribution of V implies that σ2 is t finite—we will resort to a method used in Fortini, Ladelli and Regazzini (1996) to prove central limit theorems for arrays of partially exchangeable random variables. The method rests on the fact that Theorem 3 entails conditional independence of the summands in the definition of V , given t (ν˜,γ˜,θ˜). After denoting conditional distribution of V , given (ν˜,γ˜,θ˜), by t t t Λ , the next step consists in proving that convergence in distribution of V , ν˜t t as t + , implies that any increasing and diverging to infinity sequence → ∞ of positive terms t ,t ,... contains a subsequence (t ) for which 1 2 n′ (7) the distribution of Λ weakly converges to the distribution of Λ, ν˜tn′ Λ being some (random) probability measure. Then, one combines (7) with the Skorokhod–Dudley representation to transform (7) into a statement about (almost sure) weak convergence of a suitably defined random distribution Λ∗ toward a random probability measure Λ∗, where Λ∗ has ν˜∗ ν˜∗ tn′ tn′ the distribution of Λ , and Λ∗ has the distribution of Λ. At this stage, ν˜tn′ the central limit theorem is employed to deducenecessary conditions for the convergenceofΛ∗ .Finally,oneconcludesbyshowingthattheseconditions ν˜∗ tn′ boil down to the existence of a bounded variance for the initial distribution µ . 0 As to organization of the rest of the paper,Section 2 includes, in addition to some necessary preliminary concepts and notation, a proof of Theorem 3. In Section 3 the reader can find the proofs of Theorems 1 and 2. The Appendix contains the proofs of a few preparatory propositions. 2. Preliminaries and proof of Theorem 3. The first part of the section contains elements necessary to the definition of the functions π mentioned j in Theorem 3. Recall that, if γ is any McKean tree with n 2 leaves, each ≥ node has either zero or two “children,” a “left child” and a “right child” such as in Figure 1, where a few elements of G(8) are visualized. In each tree of G(n) fix an order on the set of the (n 1) nodes and, accordingly, associate the random variable θ˜ with the kt−h node. See (a) k in Figure 1. Moreover, call 1,2,...,n the n leaves following a left to right order. See (b) in Figure 1. The number of generations which separate leaf j from the “root” node is said to be the depth of j (in symbols, δ˜ ). With j δ˜ (γ) one denotes the depth of the tree γ, that is, min δ˜ (γ),...,δ˜ (γ) if (1) 1 n { } γ G(n). Thecardinality of G(n) is the Catalan number C = 2n−2 /n;see ∈ n n−1 Section 15 of Comtet (1970). (cid:0) (cid:1) CENTRAL LIMIT THEOREM FOR THE SOLUTIONOFTHE KACEQUATION 7 Now, for any leaf j of γ in G(n), look at the path which connects j and the “root” node at the top in ascending order. It consists of δ˜ steps: the j first one from j to its “parent” node, the second from the “parent” to the “grandparent” of j, and so on. Define the product π =π (γ˜,θ˜)=τ(j) τ(j), j j 1 ··· δj where τ(j) equals cosθ˜ if j is a “left child” or sinθ˜ if j is a “right child” δj k k and θ˜ is the element of θ˜ associated to parent node of j; τ(j) equals k δj−1 cosθ˜ or sinθ˜ depending on if the “parent” of j is, in its turn, a “left m m child” or a “right child,” θ˜ being the element of θ˜ associated with the m grandparent of j; and so on. For instance, as to leaf 1 in (a) of Figure 1, π =cosθ˜ cosθ˜ cosθ˜ and, for leaf 6, π =sinθ˜ cosθ˜ sinθ˜ . 1 4 2 1 6 5 3 1 · · · · From the definition of π one obtains j (8) π2=1 j j∈γ X for every γ in G(n), with n=2,3,.... It is worth extending (8) to G(1) by setting π 1 for the sole leaf of γ in G(1). 1 ≡ At this stage one is in a position to specify the form of the n-fold Wild product of φ with itself, corresponding to γ˜ G(n), indicated with c in 0 γ˜ ∈ (4): (9) c (ξ;φ )= φ (π ξ) u∞(dθ) [γ˜ G(ν˜),ξ R]. γ˜ 0 0 j t Z[0,2π)∞ j∈γ˜ ! ∈ ∈ Y See McKean (1966) and McKean (1967). Then, conditionally on γ˜ in G(ν˜ ), t c is a mixture, directed by u∞, of characteristic functions of linear combi- γ˜ nations, with coefficients (π ,...,π )(γ˜,θ), 1 ν˜t Fig. 1. Shaded (unshaded) circles stand for leaves (nodes). 8 E. GABETTA AND E. REGAZZINI of independent random variables x˜ ,...,x˜ . Hence, in view of (3) and (4), 1 ν˜t one recovers the interpretation of µ(,t) stated in Theorem 3 and this com- · pletes the proof of the same theorem. Now we are ready to yield a physical interpretation of this result. If one thinks of each leaf of a tree γ˜ with ν˜ leaves as a particle which collides t with the particle under observation, then the velocity V of this last particle t turns out to be the outcome of ν˜ contributions. The jth contribution to V t t is given by the initial velocity x˜ multiplied by a reducing factor π , which j j depends on the number δ˜ of collisions that particle j experiences before it j collides withthe molecule underobservation, andon thescattering angles θ. The collisions experienced by particle j take place according to the “order” schematized by γ˜. Thereis apreliminarystatement thatplays an importantrolethroughout the rest of the paper. It is drawn from Gabetta and Regazzini (2006) and gives the exact expression of the conditional expectation of ν˜t xδ˜j, given j=1 ν˜ , that is, t P ν˜t Γ(2x+ν˜ 1) (10) E xδ˜j ν˜ = t− , x>0, t t j=1 | ! Γ(2x)Γ(ν˜t) X which yields (11) E ν˜t xδ˜j = q (n)Γ(2x+n−1) =exp t(1 2x) . t t ! Γ(2x)Γ(n) {− − } j=1 n≥1 X X Equalities (10)–(11) can be utilized to discuss the asymptotic behavior (as t + ) of the distribution of the random variable → ∞ π◦:= max π t 1≤j≤ν˜t| j| involved, for example, with the proof of Theorem 2. The starting point for this discussion is given by the following: Lemma 1. For every x in (0,1) and p>2, one has 1 P π◦>x exp t(1 2α ) . t{ t }≤ xp {− − p } In particular, P π◦>x 0, as t + , even if t{ t }→ → ∞ (12) x=x =e−tc, t provided that 0<c<(1 2α )/p. p − For the proof of Lemma 1, see the Appendix. CENTRAL LIMIT THEOREM FOR THE SOLUTIONOFTHE KACEQUATION 9 3. Proofs of Theorems 1 and 2. It is useful to premise a remark about the real (Re) and imaginary (Im) parts of the solution of (2). In fact, it is easy to prove that Reφ(,t) is the unique solution of the same problem as · (2) with initial data Reφ , while Imφ(,t) has an explicit form, that is, 0 · Imφ(ξ,t)=(Imφ (ξ))e−t. 0 Then, one can prove Theorems 1 and 2 by assuming, temporarily, that φ is 0 a real-valued characteristic function, which is tantamount to admitting that µ is symmetric, that is, 0 µ (( , x])=µ ([x,+ )) for every x>0. 0 0 −∞ − ∞ 3.1. Proof of Theorem 2 and of sufficiency in Theorem 1. Webeginwith Theorem 2 which, among other things, entails the sufficiency part of Theo- rem 1. The starting point is an estimate of ∆˜(ξ) , where | | ∆˜(ξ):=φ˜ (ξ) e−ξ2/2 t − and φ˜ denotes the conditional characteristic function of t 1 ν˜t π x˜ , j j σ · j=1 X given (ν˜,γ˜,θ˜): t For every ε>0 and ξ in R, ν˜t π2x˜2 ∆˜(ξ) e−ξ2/2 E ξ2 j j (π x˜ >σε) | |≤ 0 σ2 1 | j j| j=1 (cid:20) X π2x˜2 +εξ 3 j j [π x˜ σε]+ξ4π2(π◦)2 , | | σ2 1| j j|≤ j t (cid:21) where E indicates expectation with respect to µ∞ and 0 0 ε:=(π◦)a t for some a in (0,1), π◦ being the same as in Lemma 1. The above inequality t follows from a well-known “sharp” estimate of the remainder in the Tay- lor expansion of exp(it). A complete proof can be found in Section 9.1 of Chow and Teicher (1997). Now, ν˜t π2x˜2 ν˜t x˜2 E ξ2 j j (π x˜ >σε) ξ2π2E (π◦x˜ >σε) j 0 σ2 1 | j j| ≤ j 0 1 | t j| σ2 j=1 (cid:20) (cid:21) j=1 (cid:20) (cid:21) X X ξ 2 = x2µ (dx) 0 (cid:18)σ(cid:19) Z|πt◦x|>σε 10 E. GABETTA AND E. REGAZZINI ξ 2 x2µ (dx); 0 ≤(cid:18)σ(cid:19) Z|x|>σ(πt◦)a−1 ν˜t π2x˜2 ν˜t E εξ 3 j j (π x˜ σε) εξ 3π2= ξ 3(π◦)a 0" | | σ2 1 | j j|≤ #≤ | | j | | t j=1 j=1 X X and ν˜t E ξ4 π2(π◦)2 =ξ4(π◦)2. 0 j t t " # j=1 X Hence, E (eiξVt/σ) e−ξ2/2 E ∆˜(ξ) t t | − |≤ | | ξ 2 (13) E x2µ (dx) t 0 ≤(cid:18)σ(cid:19) (cid:18)Z|x|>σ(πt◦)a−1 (cid:19) + ξ 3E ((π◦)a)+ξ4E ((π◦)2). | | t t t t Next, E u2µ (du) =E u2µ (du) π◦ x t(cid:18)Z|u|>σ(πt◦)a−1 0 (cid:19) t(cid:18)Z|u|>σ(πt◦)a−1 0 ·1{ t ≤ } + u2µ (du) π◦>x Z|u|>σ(πt◦)a−1 0 ·1{ t }(cid:19) (x>0) u2µ (du)+σ2P π◦>x , ≤Z|u|>σxa−1 0 { t } which, for x=x :=e−cpt like in (12), gives t (14) E u2µ (du) u2µ (du)+σ2e−t(1−2αp−cp). t 0 0 (cid:18)Z|u|>σ(πt◦)a−1 (cid:19)≤Z|u|>σ(xt)a−1 Moreover, E ((π◦)a)=E ((π◦)a π◦ x )+E ((π◦)a π◦>x ) (15) t t t t 1{ t ≤ t} t t 1{ t t} xa+e−t(1−2αp−cp) ≤ t and (16) E ((π◦)2) x2+e−t(1−2αp−cp). t t ≤ t Then, by (13), (14), (15) and (16), E (eiξVt/σ) e−ξ2/2 t | − |