\(U\) -Statistics of Ornstein–Uhlenbeck Branching Particle System

Journal of Theoretical Probability, Sep 2014

We consider a branching particle system consisting of particles moving according to the Ornstein–Uhlenbeck process in \(\mathbb {R}^d\) and undergoing a binary, supercritical branching with a constant rate \(\lambda >0\). This system is known to fulfill a law of large numbers (under exponential scaling). Recently the question of the corresponding central limit theorem (CLT) has been addressed. It turns out that the normalization and the form of the limit in the CLT fall into three qualitatively different regimes, depending on the relation between the branching intensity and the parameters of the Ornstein–Uhlenbeck process. In the present paper, we extend those results to \(U\)-statistics of the system, proving a law of large numbers and CLT.

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\(U\) -Statistics of Ornstein–Uhlenbeck Branching Particle System

Radosaw Adamczak Piotr Mios We consider a branching particle system consisting of particles moving according to the Ornstein-Uhlenbeck process in Rd and undergoing a binary, supercritical branching with a constant rate > 0. This system is known to fulfill a law of large numbers (under exponential scaling). Recently the question of the corresponding central limit theorem (CLT) has been addressed. It turns out that the normalization and the form of the limit in the CLT fall into three qualitatively different regimes, depending on the relation between the branching intensity and the parameters of the OrnsteinUhlenbeck process. In the present paper, we extend those results to U -statistics of the system, proving a law of large numbers and CLT. We consider a single particle located at time t = 0 at x Rd , moving according to the Ornstein-Uhlenbeck process and branching after an exponential time independent of the spatial movement. The branching is binary and supercritical, with probability p > 1/2 the particle is replaced by two offspring, and with probability 1 p it vanishes. - The offspring particles follow the same dynamics (independently of each other). We will refer to this system of particles as the OU branching process and denote it by X = {Xt }t0. We identify the system with the empirical process, i.e., X takes values in the space of Borel measures on Rd and for each Borel set A, Xt ( A) is the (random) number of particles at time t in A. We refer to [14] for the general construction of X as a measure-valued stochastic process. It is well known (see, e.g., [16]) that the system satisfies the law of large numbers, i.e., for any bounded continuous function f , conditionally on the set of non-extinction |Xt |1 Xt , f where |Xt | is the number of particles at time t, {Xt (1), Xt (2), . . . , Xt (|Xt |)} are their positions, Xt , f := i|X=t1| f (Xt (i )) and is the invariant measure of the Ornstein Uhlenbeck process. In a recent article [1], we investigated second-order behavior of this system and proved central limit theorems (CLT) corresponding to (1). We found three qualitatively different regimes, depending on the relation between the branching intensity and the parameters of the OrnsteinUhlenbeck process. In the present article, we extend these results on the LLN and CLT to the case of U -statistics of the system of arbitrary order n 1, i.e., to random variables of the form Utn( f ) := f (Xt (i1), Xt (i2), . . . , Xt (in)), (note that Ut1( f ) = Xt , f ). Our investigation parallels the classical and welldeveloped theory of U -statistics of independent random variables; however, we would like to point out that in our context, additional interest in this type of functionals of the process X stems from the fact that they capture average dependencies between particles of the system. This will be seen from the form of the limit, which turns out to be more complicated than in the i.i.d. case. This is one of the motivations for studying U -statistics in a more general context than for independent random variables. Another one is that while being structurally the simplest generalization of additive functionals (which are U -statistics of degree one), they may be considered building blocks for other, more complicated statistics as they appear naturally in their Taylor expansions (see [7], where in the i.i.d. context such expansions are used in the analysis of some statistical estimators). In another context, they also appear in the study of tree-based expansions and propagation of chaos in interacting particle systems (see [911,23]). The organization of the paper is as follows. After introducing the basic notation and preliminary facts in Sect. 2, we describe the main results of the paper in Sect. 3. Next (Sect. 4), we restate the results in the special case of n = 1 (as proven in [1]) to serve as a starting point for the general case. Finally, in Sect. 5, we provide proofs for arbitrary n, postponing some of the technical details (which may obscure the main ideas of the proofs) to Sect. 6. We conclude with some remarks concerning the so-called non-degenerate case (Sect. 7). 2 Preliminaries 2.1 Notation For a branching system {Xt }t0, we denote by |Xt | the number of particles at time t and by Xt (i )the position of the i th (in a certain ordering) particle at time t . We sometimes use Ex or Px to denote the fact that we calculate the expectation for the system starting from a particle located at x . We use also E and P when this location is not relevant. By d , we denote the convergence in law. We use to denote the situation when an inequality holds with a constant c > 0, which is irrelevant to calculations, e.g., f (x ) g(x ) means that there exists a constant c > 0 such that f (x ) cg(x ). By x y = id=1 xi yi , we denote the standard scalar product of x , y Rd , by the corresponding Euclidean norm. By n, we denote the n-fold tensor product. We use also f, := Rd f (x )(dx ). We will write X to describe the fact that a random variable X is distributed according to the measure , similarly X Y will mean that X and Y have the same law. For a subset A of a linear space by span( A), we denote the set of finite linear combinations of elements of A. In the paper, we will use Feynman diagrams. A diagram on a set of vertices {1, 2, . . . , n} is a graph on {1, 2, . . . , n} consisting of a set of edges E not having common endpoints and a set of unpaired vertices A . We will use r ( ) to denote the rank of the diagram, i.e., the number of edges. For properties and more information, we refer to [21, Definition 1.35]. In the paper, we will use the space P = P (Rd ) := { f : Rd R : f is continuous and k such that | f (x )|/ x k 0 as x +}. We endow this space with the following norm where n(x ) := exp Cc = Cc(Rd ), to denote the space of continuous compactly supported functions. Given a function f P (Rd ), we will implicitly understand its derivatives (e.g., xfi ) in the space of tempered distributions (see, e.g., [25, p. 173]). By f (a), we denote the function x f (ax ). Let 1, 2 be two probability measures on R, and Lip(1) be the space of 1-Lipschitz functions R [1, 1]. We define It is well known that m is a distance metrizing the weak convergence (see, e.g., [12, Theorem 11.3.3]). One easily checks that when 1, 2 correspond to two random variables X1, X2 on the same probability space then we have X1 X2 1 X1 X2 2. 2.2 Basic Facts on the GaltonWatson Process The number of particles {|Xt |}t0 is the celebrated GaltonWatson process. We present basic properties of this process used in the paper. The main reference in this section is [2]. In our case, the expected total number of particles grows exponentially at the rate The process becomes extinct with probability (see [2, Theorem I.5.1]) pe = We will denote the extinction and non-extinction events by E x t and E x t c, respectively. The process Vt := ept |Xt | is a positive martingale. Therefore, it converges (see also [2, Theorem 1.6.1]) Vt V, a.s. as t +. We have the following simple fact (we refer to [1] for the proof). Proposition 1 We have {V = 0} = E x t and conditioned on non-extinction V has the exponential distribution with parameter 2 pp1 . We have E(V) = 1 and 1 . Ee4pt |Xt |4 is uniformly bounded, i.e., there exists C > 0 such Var(V) = 2 p1 that for any t 0 we have Ee4pt |Xt |4 C . Moreover, all moments are finite, i.e., for any n N and t 0 we have E|Xt |n < +. 2.3 Basic Facts on the OrnsteinUhlenbeck Process We recall that the OrnsteinUhlenbeck process with parameters , > 0 is a timehomogeneous Markov process with the infinitesimal operator The corresponding semigroup will be denoted by T. The density of the invariant measure of the OrnsteinUhlenbeck process is given by (x ) := 2.4 Basic Facts Concerning U -Statistics We will now briefly recall basic notation and facts concerning U -statistics. A U statistic of degree n based on an X -valued sample X1, . . . , X N and a function f : X n R is a random variable of the form The function f is usually referred to as the kernel of the U -statistic. Without loss of generality, it can be assumed that f is symmetric, i.e., invariant under permutation of its arguments. We refer the reader to [7,22] for more information on U -statistics of sequences of independent random variables. In our case, we will consider U -statistics based on positions of particles from the branching system as defined by (2). We will be interested in weak convergence of properly normalized U -statistics when t . Similarly as in the classical theory, the asymptotic behavior of U -statistics depends heavily on the so-called order of degeneracy of the kernel f , which we will briefly recall in Sect. 5.2. A function f is called completely degenerate or canonical (with respect to some measure of reference , which in our case will be the stationary measure of the OrnsteinUhlenbeck process) if f (x1, . . . , xn )(dxk ) = 0, for all x1, . . . , xk1, xk+1, . . . , xn X . The complete degeneracy may be considered a centeredness condition, in the classical theory of U -statistics canonical kernels are counterparts of centered random variables from the theory of sums of independent random variables. Their importance stems from the fact that each U -statistic can be decomposed into a sum of canonical U -statistics of different degrees, a fact known as the Hoeffding decomposition (see Sect. 5.2). Thus, in the main part of the article, we prove results only for canonical U -statistics. Their counterparts for general U statistics, which can be easily obtained via the Hoeffding decomposition, are stated in Sect. 7. 3 Main Results This section is devoted to the presentation of our results. The proofs are deferred to Sect. 5. We start with the following law of large numbers (throughout the article when dealing with U -statistics of order n we will identify Rd Rd with Rnd ). aTshseuomree mtha2t Lfet:{RXnt}dt0 be the OU branching system starting from x Rd . Let us R is a bounded continuous function. Then, on the set of non-extinction E x t c there is the convergence Having formulated the law of large numbers, let us now pass to the corresponding CLTs. We recall that is the drift parameter in (9) and p is the growth rate (7). As already mentioned in the introduction, the form of the limit theorems depends on the relation between p and , more specifically, we distinguish three cases: p < 2, p = 2 and p > 2. We refer the reader to [1] (Introduction and Section 3) for a detailed discussion of this phenomenon as well as its heuristic explanation and interpretation. Here, we only stress that the situation for p > 2 differs substantially from the remaining two cases, as we obtain convergence in probability and the limit is not Gaussian even for n = 1. Intuitively, this is caused by large branching intensity which lets local correlations between particles prevail over the ergodic properties of the OrnsteinUhlenbeck process. 3.1 Slow Branching Case: p < 2 Let Z be a Gaussian stochastic measure on Rd+1 with intensity 1(dt dx ) := 0(dt ) + 2pept dt (x )dx , defined according to [21, Definition 7.17], where 0 is the Dirac measure concentrated at 0. We denote the stochastic integral with respect to Z by I and the corresponding multiple stochastic integral by In [21, Section 7.2]. We assume that Z is defined on some probability space (, F , P). For f P (Rnd ) we define (we recall that T is the semigroup of the Ornstein Uhlenbeck process) si R+, xi Rd . n H ( f )(s1, x1, s2, x2, . . . , sn, xn) := i=1Tsi f (x1, x2, . . . , xn), 2(dsdx ) := 2peps (x )dsdx , 2(dz j,k ) where ui = z j,k if ( j, k) E and (i = j or i = k) and ui = zi if i A . Less formally, for each pair ( j, k), we integrate over diagonal of coordinates j and k with respect to 2. The function obtained in this way is integrated using the multiple stochastic integral I|A |. We define L1( f ) := where the sum spans over all Feynman diagrams labeled by {1, 2, . . . , n}. We are now ready to formulate our main result for processes with small branching rate. Recall (8) and that W is V conditioned on E x t c. Theorem 3 Let {Xt }t0 be the OU branching system starting from x Rd . Let us assume that f P (Rnd ) is a canonical kernel and p < 2. Then conditionally on the set of non-extinction E x t c there is the convergence |Xt | 3.2 Critical Branching Case: p = 2 Before we present the main results in the critical branching case, we need to introduce some additional notation. 2 Consider the orthonormal Hermite basis {hi }i0 for the measure = N (0, 2 ) (i.e, h0 = 1, hi is a polynomial of degree i and hi h j d = i j ). Then for any positive integer n, the set {hi1 hind }i1,...,ind 0 of multivariate Hermite polynomials is an orthonormal basis in L2(Rnd , n ). For a function f L2(Rnd , n ) let fi1,...,ind be the sequence of coefficients of f with respect to this basis, i.e., Cov(G f , Gg) = 2p( f1,0,...,0g1,0,...,0 + f0,1,...,0g0,1,...,0 + + f0,...,0,1g0,...,0,1) (16) We will identify this process with a map I : L2(Rd , ) L2(, F , P), such that I ( f ) = G f . One can easily check that I is a bounded linear operator. Moreover, I = I P. In fact I is the stochastic integral of P f with respect to the random Gaussian measure on Rd with intensity 2p (however, we will not use this fact in the sequel). Since Hn = (H1)n, there exists a unique linear operator L 2 : Hn L2(, F , P) 1 such that for any functions f1, . . . , fn H , L 2( f1 fn) = I ( f1) I ( fn) (we used here the fact that Gaussian variables have all moments finite). Let now Pn : L2(Rnd , n) Hn be the orthogonal projection onto Hn. We have Pn = Pn and using the fact that Hn are finite dimensional we obtain that the linear operator L2 : L2(Rnd , n ) L2(, F , P) defined as L2( f ) = L 2( Pn f ) L2( f1 fn) = G f1 G fn . We are now ready to formulate the theorem, which describes the asymptotic behavior of U -statistics in the critical case. Theorem 4 Let {Xt }t0 be the OU branching system starting from x Rd . Let us assume that f P (Rnd ) is a canonical kernel and p = 2. Then conditionally on the set of non-extinction E x t c there is the convergence |Xt | , Remark 5 One can express L2( f ) in terms of the Hermite expansion of the function f , which might give more insight into the structure of the limiting law. Indeed, define hi : Rd R, i = 1, . . . , d, by hi (x1, . . . , xd ) = h1(xi ) (thus, hi = h0(i1) h1 h(di)). Then for f L2(Rdn, n), 0 f hi1 . . . hin , n Gi1 Gin , i1,...,ind where Gi = (2p)1/2Ghi . In particular (G1, . . . , Gd ) is a vector of independent standard Gaussian variables (this follows easily from the covariance structure of the process (G f ) and the fact that the functions hi form an orthonormal system). 3.3 Fast Branching Case: p > 2 i=1 The following two facts have been proved in [1, Propositions 3.9,3.10]. Proposition 6 H is a martingale with respect to the filtration of the OU branching system starting from x Rd . Moreover for p > 2, we have supt E Ht 2 < +, therefore there exists H := limt+ Ht (a.s. limit) and H L2. When the OU branching system starts from 0, then martingales Vt and Ht are orthogonal. It is worthwhile to note that the distribution of H depends on the starting conditions. Proposition 7 Let {Xt }t0 and {X t }t0 be two OU branching processes, the first one starting from 0 and the second one from x . Let us denote the limit of the corresponding martingales by H, H, respectively. Then H H + x V, where V is the limit given by (8) for the system X . H is Rd -valued, we denote its coordinates by H i . Let f P (Rnd ). We define L 3( f ) := n f i1,i2,...,in=1 x1,i1 x2,i2 , . . . , xn,in where we adopted the convention that x j,l is the lth coordinate of the j th variable. By L3( f ), we will denote L 3( f ) conditioned on E x t c. Theorem 8 Let {Xt }t0 be the OU branching system starting from x Rd . Let us assume that f P (Rnd ) is a canonical kernel and p > 2. Then conditionally on the set of non-extinction E x t c there is the convergence |Xt | |Xt | ept V , en(p)t Utn( f ) ept |Xt |, en(p)t Utn( f ) V, L 3( f ) i n pr obabili t y. 3.4 Remarks on the CLT for U -Statistics of i.i.d. Random Variables For comparison purposes, we will now briefly recall known results on the CLT for U statistics of independent random variables. U -statistics were introduced in the 1940s in the context of unbiased estimation by Halmos [19] and Hoeffding who obtained the CLT for non-degenerate (degenerate of order 0, see Sect. 5.2) kernels [20]. The full description of the CLT was obtained in [15,24] (see also the article [18] where the CLT is proven for a related class of V -statistics). Similarly as in our case, the asymptotic behavior of U -statistics based on a function f : X n R and an i.i.d. X -valued sequence X1, X2, . . . is governed by the order of degeneracy of the function f (see Sect. 5.2) with respect to the law of X1 (call it P). The case of general f can be reduced to the canonical one, for which one has the weak convergence N n/2 where Jn is the n-fold stochastic integral with respect to the so-called isonormal process on X , i.e., the stochastic Gaussian measure with intensity P. For the small branching rate case, the behavior of U -statistics in our case resembles the classical one as the limit is a sum of multiple stochastic integrals of different orders. In the remaining two cases, the behavior differs substantially. This can be regarded as a result of the lack of independence. Although asymptotically the particles positions become less and less dependent, in short timescale, offspring of the same particle stay close one to another. Let us finally mention some results for U -statistics in dependent situations, which have been obtained in the last years. In [6], the authors analyzed the behavior of U -statistics of stationary absolutely regular sequences and obtained the CLT in the non-degenerate case (with Gaussian limit). In [5], the authors considered and mixing sequences and obtained a general CLT for canonical kernels. Interesting results for long-range dependent sequences have been also obtained in [8]. A more recent interesting work (already mentioned in the introduction) is [911,23], where the authors consider U -statistics of interacting particle systems. 4 The Case of n = 1 In the special case of n = 1, the results presented in the previous section were proven in [1]. Although this case obviously follows immediately from the results for general n, it is actually a starting point in the proof of the general result (similarly as in the case of U -statistics of i.i.d. random variables). Therefore, for the readers convenience, we will now restate this case in a simpler language of [1], not involving multiple stochastic integrals. We start with the law of large numbers Theorem 9 Let {Xt }t0 be the OU branching system starting from x Rd . Let us assume that f P (Rd ). Then tli+m ept Xt , f = f, V i n pr obabili t y, or equivalently on the set of non-extinction, E x t c, we have Moreover, if f is bounded then the almost sure convergence holds. 4.1 Small Branching Rate: p < 2 We denote f(x ) := f (x ) f, and Let us also recall (8) and that W is V conditioned on E x t c. In this case, the behavior of X is given by the following Theorem 10 Let {Xt }t0 be the OU branching system starting from x Rd . Let us assume that p < 2 and f P (Rd ). Then 2f < + and conditionally on the set of non-extinction E x t c, there is the convergence |Xt | where G1 N (0, 1/(2 p 1)), G2 N (0, 2f) and W, G1, G2 are independent random variables. 4.2 Critical Branching Rate: p = 2 Recall the notation related to Hermite polynomials introduced in Sect. 3.2 and denote 2 xi i=1 2 2 2 = 2p( f1,0,...,0 + f0,1,...,0 + + f0,...,0,1) (where the first equality follows from the form of the Gaussian density and its relation to Hermite polynomials, whereas the second one from the definition of P). Note that the same symbol 2f has already been used to denote the asymptotic variance in the small branching case. However, since these cases will always be treated separately, this should not lead to ambiguity. Theorem 11 Let {Xt }t0 be the OU branching system starting from x Rd . Let us assume that p = 2 and f P (Rd ). Then 2f < + and conditionally on the set of non-extinction E x t c there is the convergence |Xt | where G1 N (0, 1/(2 p 1)), G2 N (0, 2f) and W, G1, G2 are independent random variables. 4.3 Fast Branching Rate: p > 2 In the following theorem, we use the notation introduced in Sect. 3.3. Theorem 12 Let {Xt }t0 be the OU branching system starting from x Rd . Let us assume that p > 2 and f P (Rd ). Then conditionally on the set of non-extinction E x t c, there is the convergence |Xt | d (W, G, f, J ), where G N (0, 1/(2 p 1)), (W, J ), G are independent and J is H conditioned on E x t c. Moreover We will now pass to the proofs of the results announced in Sect. 3. Their general structure is similar to the case of U -statistics of independent random variables, i.e., all the theorems will be proved first for linear combinations of tensor products and then via suitable approximations extended to the function space P (Rnd ). Below we provide a brief outline of the proofs, common for all the cases considered in the paper. 1. Using the one-dimensional versions of the results, presented in Sect. 4, and the CramrWold device, one proves convergence for functions f , which are linear combinations of tensor products. This class is shown to be dense in P . 5 Proofs of Main Results 5.1 Outline of the Proofs (V, f, H), i n pr obabili t y. 2. Using algebraic properties of the covariance, one obtains explicit formulas for the limit, which are well defined for any function f P . Further, one shows that they depend on f in a continuous way. 3. One obtains a uniform in t bound on the distance between the laws of Utn( f ) and of Utn(g) in terms of the distance between f and g in P . This is the most involved and technical step as it relies on the analysis of moments of U -statistics. It turns out that the formulas for moments can be expressed in terms of auxiliary branching processes indexed by combinatorial structures, more specifically by labeled trees of a special type (introduced in Sect. 6.3). Having this representation, one can then obtain moment bounds via combinatorial arguments. 4. Combining the above three steps, one can easily conclude the proofs by standard metric-theoretic arguments. By step 3, a general U-statistic based on a function f can be approximated (uniformly in t ) by a U-statistics based on special functions fn whose laws converge by step 1 as t to some limiting measure n. By step 2, when the approximation becomes finer and finer (n ), one has n for some probability measure . Finally, it is easy to see that is the limiting measure for the original U -statistic. The organization of the rest of the paper is as follows. First, we recall some basic facts about U - and V -statistics and Hoeffding projections, which we will need already at step 1. Then we present the proof of the law of large numbers and CLTs. In the latter proofs, we formulate and use the estimates related to step 3 without proving them. Only later in Sect. 6 do we introduce the necessary notation and combinatorial arguments which give those estimates. We choose this way of presentation since it allows the readers to see the structure of the proofs without being distracted by rather heavy notation and quite lengthy technical arguments related to step 3. From now on, we will often work conditionally on the set of non-extinction E x t c, which will not be explicitly mentioned in the proofs (however, should be clear from the context). 5.2 Basic Facts on U - and V -Statistics We will now briefly recall one of the standard tools of the theory of U -statistics, which we will use in the sequel, namely the Hoeffding decomposition. Let us introduce for I {1, . . . , n} the Hoeffding projection of f : Rnd R corresponding to I as the function I f : R|I |d R, given by the formula iI ORnned cfa(nx1e,a.s.il.y, xsene) thina=t 1for(d|Ix|i ). 1, I f is a canonical kernel. Moreover f = Note that if f is symmetric (i.e., invariant with respect to permutations of arguments), I f depends only on the cardinality of f . In this case, we speak about the kth Hoeffding projection (k = 0, . . . , n), given by k f (x1, . . . , xk ) = (x1 ) (xk ) (nk), f . A symmetric kernel in n variables is called degenerate of order k 1 (1 k n) iff k = min{i > 0 : i f 0}. The order of degeneracy is responsible for the normalization and the form of the limit in the CLT for U -statistics, e.g., if the kernel is non-degenerate, i.e., 1 f 0, then the corresponding U -statistic of an i.i.d. sequence behaves like a sum of independent random variables and converges to a Gaussian limit. The same phenomenon will be present also in our situation (see Sect. 7). In the particular case k = n, the definition of the Hoeffding projection reads as One easily checks that which gives us the aforementioned Hoeffding decomposition of U -statistics which in the case of symmetric kernels simplifies to f (x1, . . . , xn ) = I {1,...,n} Utn( f ) = (|Xt | |I |)! Ut|I |(I f ), I {1,...,n} (|Xt | n)! Utn( f ) = k=0 where we use the convention Ut0(a) = a for any constant a. For technical reasons, we will also consider the notion of a V -statistic which is closely related to U -statistics, and is defined as Vtn( f ) := i1,i2,...,in=1 f (Xt (i1), Xt (i2), . . . , Xt (in)). The corresponding Hoeffding decomposition is Vtn( f ) = I {1,...,n} where again we set Vt0(a) = a for any constant a. In the proofs of our results, we will use a standard observation that a U -statistic can be written as a sum of V -statistics. More precisely, let J be the collection of partitions of {1, . . . , n} i.e., of all sets J = { J1, . . . , Jk }, where Ji s are non-empty, pairwise disjoint and i Ji = {1, . . . , n}. For J as above let f J be a function of | J | variables x1, . . . , x|J |, obtained by substituting xi for all the arguments of f corresponding to the set Ji , e.g., for n = 3 and J = {{1, 2}, {3}}, f J (x1, x2) = f (x1, x1, x2). An easy application of the inclusionexclusion formula yields that Utn( f ) = J J where aJ are some integers depending only on the partition J . Moreover one can easily check that if J = {{1} , . . . , {n}}, then aJ = 1, whereas if J consists of sets with at most two elements then aJ = (1)k where k is the number of two-element sets in J . Let us also note that partitions consisting only of one- and two-element sets can be in a natural way identified with Feynman diagrams (defined in Sect. 2.1). 5.3 Proof of the Law of Large Numbers Proof of Theorem 2 Consider the random probability measure t = |Xt |1 Xt (recall that formally we identify Xt with the corresponding counting measure). By Theorem 9 with probability one (conditionally on E x t c), t converges weakly to . Thus, by Theorem 3.2 in [3], n converges weakly to n. t Let f be bounded and continuous. We notice that f, tn = | Xt |n Vtn( f ), which gives the almost sure convergence |Xt |n Vtn( f ) f, . Now it is enough to note that the number of off-diagonal terms in the sum (25) defining Vtn( f ) is of order |Xt |n1 and use the fact that |Xt | a.s. on E x t c. We note that in the proofs below we will use this fact only in the version for f C(Rnd ) which we have just proven. The proof of convergence in probability for f P (Rnd ) follows directly from the CLT presented in Sect. 7. 5.4 Approximation Before we proceed to the proofs of CLTs, we will demonstrate the simple fact that any function in P (Rnd ) can be approximated by tensor functions. Lemma 13 Let A := in=1gi : gi bounded continuous! and f P (Rnd ) be a canonical kernel. For every m > 0 there exists a sequence { fk } span( A) such that each fk is canonical and fk (m) f (m) P 0, as k +. Proof First, we prove that span( A) is dense in P (Rnd ). Let us notice that given a function f P (Rnd ) it suffices to approximate it uniformly on some box [M, M ]d , M > 0. The box is a compact set and an approximation exists due to the StoneWeierstrass theorem. Now, let f P (Rnd ). We may find a sequence {hk } span( A) such that hk (m) f (m) in P . Let us recall the Hoeffding projection (24) and denote I = {1, 2, . . . , n}. Now direct calculation (using exponential integrability of Gaussian variables) reveals that the sequence fk := I hk fulfills the conditions of the lemma. 5.5 Small Branching rate: Proof of Theorem 3 Let us first formulate two crucial facts, whose rather technical proofs we defer to Sect. 6.4. The first one corresponds to Step 2 in the outline of the proof presented in Sect. 5.1. Recall the definition of L1 given in (14). Proposition 14 For any canonical f P (Rnd ) we have EL1( f )2 < +. Moreover L1 is a continuous function L1 : Can L2(, F , P), where Can = " f P (Rnd ) : f is a canonical kernel# , and Can is endowed with the norm P . The other fact we will use allows for a uniform in t approximation of general canonical U -statistics by those, whose kernels are sums of tensor products. This corresponds to step 3 of the outline. Recall the distance m given by (5). Proposition 15 Let {Xt }t0 be the OU branching system starting from x Rd and p < 2. For any n 2 there exists a function ln : R+ R+, fulfilling lims 0 ln(s) = 0 and such that for any f1, f2 Can and any t > 1 we have m(1, 2) ln( f1(2n) f2(2n) P ), where 1 |Xt |n/2Utn( f1), 2 |Xt |n/2Utn( f2) (the U -statistics are considered here conditionally on E x t c). We can now proceed with the proof of Theorem 3. Proof of Theorem 3 For simplicity, we concentrate on the third coordinate. The joint convergence can be easily obtained by a straightforward modification of the arguments below (using the joint convergence in Theorem 10 for n = 1). In the whole proof, we work conditionally on the set of non-extinction E x t c. Let us consider bounded continuous functions fil : Rd R, l = 1, . . . , m, i = 1, . . . , n, which are centered with respect to and set fl := in=1 fil and f := lm=1 fl . In this case the U -statistic (2) writes as Utn( f ) = l=1 iij1=,ii2k,,..f.,oirn =j=1,k f1l (Xt (i1)) f2l (Xt (i2)) . . . fnl (Xt (in)). Let be a Feynman diagram labeled by {1, 2, . . . , n}, with edges E and unpaired vertices A . Let Decomposition (27) writes here as f (Xt (i1), Xt (i2), . . . , Xt (in)). Utn( f ) = where the sum spans over all Feynman diagrams labeled by {1, 2, . . . , n} (note that when has no edges, then St ( ) = Vtn( f )), and the remainder R is the sum of V -statistics corresponding to partitions of {1, . . . , n} containing at least one set with more than two elements. First, we will prove that |Xt |(n/2) Rt 0. To this end, let us consider a partition J = { Ar }1rm1 {Br }1rm2 {Cr }1rm3 of the set {1, 2, . . . , n}, in which | Ar | 3, |Br | = 2 and |Cr | = 1. Assume that m1 1 and recall the definition of f J used in (27). For any l = 1, . . . , m, we have rm1 rm3 kAr kCr rm2 kBr By the first part of Theorem 9, the first product on the right-hand side converges almost surely to 0 and the second one converges to a finite limit. Each factor of the third product, by Theorem 10, converges (in law) to a Gaussian random variable. We conclude that |Xt |(n/2)Vt|J |( f Jl ) d 0. Thus, only the first summand of (28) is relevant for the asymptotics of |Xt |(n/2)Utn( f ). Consider now i=1 i=1 Let us denote Z f l (t ) := |Xt |1/2 j Wold device, we get that i|X=t1| f jl (Xt (i )). By Theorem 10 and the Cramr(Z f jl (t ))1 jn,1lm d (G f jl )1 jn,1lm , Let D := R+ Rd and note that = Cov(I1(H ( f jl1 )), I1(H ( fkl2 ))). i=1 We conclude that (recall that I1 is the Gaussian stochastic integral with respect to the random Gaussian measure with intensity 1). Thus, without loss of generality, we can assume that G f jl = I1(H ( f jl )) for all l, j . On the other hand by Theorem 9, one easily obtains |Xt |1 G frl Thus, by decomposition (28) and the considerations above, we obtain |Xt |(n/2)t Utn( f ) d We will now show that L is equal to L1( f ) given by (14). By linearity of L1( f ) it is enough to consider the case of m = 1. We will therefore drop the superscript and write fi instead of fil . We recall (12) and denote P( fi , f j ) := D H ( fi f j )(z, z)2(dz), where D = R+ Rd . By (29) and the definition of 2 given in (13) L = (EG f j G fk P( f j , fk )) L = Let us notice that the inner sum can be written as where runs over all Feynman diagrams on the set of vertices A. Thus, by [21, Theorem 3.4 and Theorem 7.26] this equals I|A| H (iA fi ) and in consequence L = P( f j , fk )I|A| H (iA fi ) . n It is easy to see that in the case of f = i=1 fi , the expression above is equivalent to (14). Thus, for each f , which is a finite sum of tensors, we have L = L1( f ) and in consequence |Xt |n/2Utn( f ) d L1( f ). Let us now consider a general canonical function f P . We put h(x ) := f (2nx ). By Lemma 13 we may find a sequence of canonical functions { fk }k span( A) such that fk (2n) h in P . Now by Proposition 15, we may approximate |Xt |(n/2)Utn( f ) with |Xt |(n/2)Utn( fk ) uniformly in t > 1. This together with Proposition 14 and standard metric-theoretic considerations concludes the proof. 5.6 Critical Branching Rate: Proof of Theorem 4 For the critical case, we will need the following counterpart of Proposition 15, which will be proved in Sect. 6.5. Proposition 16 Let {Xt }t0 be the OU branching system starting from x Rd and p = 2. For any n 2 there exists a function ln : R+ R+, fulfilling lims0 ln(s) = 0 and such that for any canonical f1, f2 P (Rnd ) and any t > 1 we have where 1 (t |Xt |)n/2Utn( f1), 2 (t |Xt |)n/2Utn( f2) (the U -statistics are considered here conditionally on E x t c). Proof of Theorem 4 As in the subcritical case, we will focus on the third coordinate. The proof is slightly easier than the one of Theorem 3 as, because of larger normalization, the notion of U -statistics and V -statistics coincide in the limit. Indeed, let us consider bounded continuous functions f1l , f2l , . . . , fnl , l = 1, . . . , m, which are centered with respect to and denote f := lm=1 in=1 fil . By (28) we have Utn( f ) Vtn( f ) = simply by the fact that the Feynman diagram without edges corresponds to Vtn( f ). Analogously as in the proof of Theorem 3, we have (t |Xt |)n/2 Rt d 0. Let us now fix some diagram with at least one edge. Without loss of generality, we assume that E = {(1, 2), (2, 3), . . . , (2k 1, 2k)} for k 1. We have l=1 ik (|Xt |t )1/2 Xt , fil Let us denote Z f l (t ) := (t |Xt |)1/2 j CramrWold device, we get that By Theorem 11 each of the factors in the second product converges in distribution, whereas by the first part of Theorem 2 each factor in the first product converges almost surely to 0, in consequence (t |Xt |)n/2 S( ) converges in probability to 0, which shows that (t |Xt |)n/2Utn( f ) and (t |Xt |)n/2Vtn( f ) are asymptotically equivalent. i|X=t1| f jl (Xt (i )). By Theorem 11 and the (Z f jl (t ))1in,lm d (G f jl )1in,lm , where (G f l )in,lm is centered Gaussian with the covariances given by (16). Thus, i (t |Xt |)n/2Vtn( f ) d G f jl = L2( f ), l=1 j=1 where L2 is defined by (17). Now we pass to general canonical functions f P . By Lemma 13, we can approximate f by canonical fk from span( A) in such a way that f fk P f (2n) fk (2n) P 0. Thus, by Proposition 16, the law of (t |Xt |)1/2Utn( fk ) converges to the one of (t |Xt |)1/2Utn( f ) as k uniformly in t > 1. Moreover by the fact that L2 is bounded on L2(Rnd , n ) and there exists C < such that L2(Rdn,n) C P (which follows easily from exponential integrability of Gaussian variables), we obtain L2( fk ) L2( f ) in the space L2(, F , P). The proof may now be concluded by standard metric-theoretic arguments. As in the previous two cases, we start with a fact, which allows to approximate general U -statistics, by those with simpler kernels. It is slightly different than the corresponding statements in the small and critical branching case, which is related to a different type of convergence and a deterministic normalization which we have for large branching. The proof is deferred to Sect. 6.6. Proposition 17 Let {Xt }t0 be the OU branching system with p > 2. There exist constants C, c > 0 such that for any canonical f P (Rnd ) we have Ex en(p)t |Utn( f )| C exp c x ! f (2n) P . Proof of Theorem 8 Again we concentrate on the third coordinate. The joint convergence can be easily obtained by a modification of the arguments below (using the joint convergence in Theorem 12 for n = 1). First, note that U -statistics and V -statistics are asymptotically equivalent. The argument is analogous to the one presented in the proof of Theorem 4, since under the assumption p > 2 we have as t and consequently we can disregard the sum over all multi-indices (i1, . . . , in ) in which the coordinates are not pairwise distinct. 1, .L.e.t, mus, wcohniscihdearrebcoeunntdeereddcwonitthinrueosupsecftutnoctioannsd fd1le,nfo2lt,e. .f. ,:=fnl : Rlm=d1 in=1Rf,il .l B=y Theorem 12 for n = 1 we have fil , H l=1 i=1 in probability. Before our final step, we recall that the convergence in probability can be metrized by d(X, Y ) := E |X|XY Y|+| 1 X Y 1. Let us now consider a function f P . By Lemma 13 we may find a sequence of canonical functions { fk } span( A) such that fk (2n) f (2n) in P . Now by Proposition 17, we may approximate en(p)t Utn( f ) with en(p)t Utn( fk ) uniformly in t in the sense of the metric d. Moreover, one can easily show that limk+ d(L 3( fk ), L 3( f )) = 0. This concludes the proof. 6 Proofs of Technical Lemmas We will now provide the proofs of the technical facts formulated in Sect. 5. The proofs are quite technical and require several preparatory steps. In what follows, we first recall some additional properties of the OrnsteinUhlenbeck process, and then we introduce certain auxiliary combinatorial structures which will play a prominent role in the proofs. The semigroup of the OrnsteinUhlenbeck process can be represented by Tt f (x ) = (gt f )(xt ), xt := et x , gt (x ) = x 2- , t2 := 2(1 e2t ). Let us recall (10). We denote ou(t ) := 1 e2t and let G . Then (30) can be written as We also denote Tt f (x ) = f (xt y)gt (y)dy f x et + ou(t )y (y)dy It is well known that the OrnsteinUhlenbeck semigroup increases the smoothness of a function. We will now introduce some simple auxiliary lemmas which quantify this statement and give bounds on the P norms of derivatives of certain functions obtained from f by applying the OrnsteinUhlenbeck semigroup on a subset of coordinates. Such bounds will be useful, since they will allow us to pass in the analysis to smooth functions. Let f P (Rnd ) and I {1, 2, . . . , n} with |I | = k. We define fI (x1, x2, . . . , xn ) := f (z1, z2, . . . , zn) iI Lemma 18 Let f P (Rnd ) and l N. Then for any I {1, 2, . . . , n} the function fI is smooth with respect to coordinates in I . For any multi-index = (i1, . . . , il ) {1, . . . , nd}l such that { i j /d : j = 1, . . . , l} I we have C f where C > 0 does not depend on f and depends only on the parameters of the system (that is , , d, l, n). Moreover, when f is canonical, so is fI . K := || f (z1, z2, . . . , zn) x iI g1(xi e yi )dyi . Therefore, by (4), the properties of the Gaussian density g1 and easy calculations we arrive at fI (x1, x2, . . . , xn)(x j )dx j = 0. There are two cases, the first when j / I . Then we have fI (x1, x2, . . . , xn)(x j )dx j f (z1, z2, . . . , zn) iI The second case is when j I . Then fI (x1, x2, . . . , xn)(x j )dx j f (z1, z2, . . . , zn) f (z1, z2, . . . , zn) iI\{ j} iI\{ j} where the second equality holds by the fact that is the invariant measure of the OrnsteinUhlenbeck process. Now the proof reduces to the first case. and ab the integral over the segment [a, b] Rd ). We will also need the following simple identity. We consider {xi }i=1,2,...,n , {xi }i=1,2,...,n. By induction one easily checks that the following lemma holds (we slightly abuse the notation here, e.g., yi denotes the derivative in direction xi xi Lemma 19 Let f be a smooth function, then n (1) i=1 i f (x1 + 1(x1 x1), x2 + 2(x2 x2), . . . , xn + n(xn xn)) ( 1, 2,..., n){0,1}n n y1 y2 . . . yn 6.3 Bookkeeping of Trees f (y1, y2, . . . , yn)dyndyn1 . . . dy1. We will now introduce the bookkeeping of trees technique (for similar considerations see, e.g., [13, Section 2] or [4]), which via some combinatorics and introduction of auxiliary branching processes will allow us to pass from equations on the Laplace transform in the case of n = 1 to estimates of moments of V -statistics and consequently U -statistics, which will be crucial for proving Propositions 15, 16 and 17. Our starting point is classical. We will use the equation on the Laplace transform of the branching process to obtain, via integration, recursive formulas for moments of V -statistics generated by tensors. Recall thus (25). Let f1, f2, . . . , fn Cc(Rd ) and fi 0. We would like to calculate i=1 Ex Vtn(in=1 fi ) = Ex Note that this differentiation is valid by Proposition 1 and properties of the Laplace transform (e.g., [17, Chapter XIII.2]). By the calculations from Section 4.1. in [1] we know that It is easy to check that Tts 4 pw2(, s, ) w(, s, ) + (1 p)5 (x )ds. The last formula is much easier to handle if written in terms of auxiliary branching processes. Firstly, we introduce the following notation. For n N \ {0} we denote by Tn the set of rooted trees described below. The root has a single offspring. All inner vertices (we exclude the root and the leaves) have exactly two offspring. For Tn, = Tt = Tt We evaluate it at = 0, (let us notice that |11| w(x , s, 0) = 1 |1| w1 (x , s, 0) = Assume that || > 0. We denote by P1() all pairs (1, 2) such that 1 2 = and 1 2 = , and by P2() P1() pairs with an additional restriction that 1 = and 2 = . Using (36) we easily check that Tts || w(, s, ) w(, s, ) (x )ds. by l( ) we denote the set of its leaves. Each leaf l l( ) is assigned a label, denoted by lab(l), which is a non-empty subset of {1, 2, . . . , n}. The labels fulfill two conditions: : lab(l) = {1, 2, . . . , n} , l1,l2l( ) (l1 = l2 # lab(l1) lab(l2) = ) . ll( ) In other words, the labels form a partition of {1, 2, . . . , n}. For a given Tn let i ( ) denote the set of inner vertices (we exclude the root and the leaves), clearly |i ( )| = |l( )| 1 (as usual | | denotes the cardinality). Let us identify the vertices of with {0, 1, 2, . . . , | | 1} in such a way, that for any vertex i its parent, denoted by p(i ), is smaller. Obviously, this implies that 0 is the root and that the inner vertices have numbers in the set {1, 2, . . . , | | 1}. We denote also leaves with singleton and non-singleton label sets, respectively. Given Tn and t R+ and {ti }ii( ), we consider an OrnsteinUhlenbeck branching walk on as follows. The initial particle is placed at time 0 at location x , it evolves up to the time t t1 according to the OrnsteinUhlenbeck process and splits into two offspring, the first one is associated with the left branch of vertex 1 in tree and the second one with the right branch. Further each of them evolves independently until time t ti , where i is the first vertex in the corresponding subtree, when it splits and so on. At time t , the particles are stopped and their positions are denoted by {Yi }il( ) (the number of particles at the end is equal to the number of leaves). The construction makes sense provided that ti t and ti t p(i) for all i i ( ) (which we implicitly assume). We define OU an=1 fa , , t, {ti }ii( ) , x := Ex a=1 where we set t0 = t . The reason to study the above objects becomes apparent by the following statement. Proof The claim is a consequence of the identity where {1, 2, . . . , n} and T() is the set of trees, as T||, with the exception that the labels are in the set . This identity in turn will follow by induction with respect to the cardinality of . For = {i } Eq. (38) reads as v(x , t ) = ept Tt fi (x ) (note that P2() = ). The space T() contains only one tree, denoted by s , consisting of the root and a single leaf labeled by {i }. We have i ( ) = and S(s , t, x ) = ept Tt fi (x ) so (43) follows. Let now || = k > 1 and assume that (43) holds for all sets of cardinality at most k 1. Apply again (38). Similarly as before, the first term corresponds to s . Let p be the transition density of the OrnsteinUhlenbeck semigroup. By induction the second term of (38) can be written as ept1 Ttt1 By (41) the above expression equals (1,2)P2() 1T(1) 2T(2) t1 OU i2 fi , 2, t1, {ti }ii(2) , y dydt1. Now we create a new tree by setting 1 and 2 to be descendants of the vertex born from the root at time t t1 (thus this vertex is assigned the split time t1). We keep labels and the remaining split times unchanged. Consider the branching random walk on with the initial position of the first particle equal to x . Note that by the branching property the evolution of this process on subtrees 1 and 2 is conditionally independent given the evolution of the first particle up to time t t1. Thus, by the Markov property of the OrnsteinUhlenbeck process, we can identify the branching random walk on 1 and 2 with the branching random walk on . More precisely we have p(t t1, x , y)OU i1 fi , 1, t1, {ti }ii(1) , y Using the Fubini theorem together with the equality |i ( )| = |i (1)| + |i (2)| + 1 we see that the summand corresponding to 1, 2 in (44) equals S(, t, x ). It is also easy to check that the described correspondence is a bijection from the set of pairs (1, 2) [as in (44)] to Tn \ {s } and therefore the expression (44) is equal to Tn\{s } S(, t, x ), which ends the proof. The calculations will be more tractable when we derive an explicit formula for {Yi }il( ). Let us recall the notation introduced in (32) and consider a family of independent random variables {Gi }i , such that Gi for i = 0 and G0 x . Recall also that ou(t ) = 1 e2t . The following proposition follows easily from the construction of the branching walk on and (32). Proposition 21 Let {Yi }il( ) be positions of particles at time t of the Ornstein Uhlenbeck process on tree with labels {ti }ti( ). We have Zi := ou(t p(l) tl )Gl etl + ou(t p(i))Gi , lP(i) We are now ready to prove an extended version of Proposition 20. This result will be instrumental in proving bounds needed to implement step 3 of the outline presented in Sect. 5.1. Proposition 22 Let {Xt }t0 be the OU branching system starting from x Rd and f P (Rnd ). Then where in (41) we extend the definition of OU in (40) by putting OU( f, , t, {ti }ii( ) , x ) := Ex f (Y j (1), Y j (2), . . . , Y j (n)). Moreover all the quantities above are finite. Proof (Sketch) Using Proposition 20, Proposition 1, (35) and Lebesgues monotone convergence theorem one may prove that (45) is valid for f C, C > 0. Using standard methods, we may drop the positivity assumption in (35) and (42). Therefore, by the StoneWeierstrass theorem, linearity and Lebesgues dominated convergence theorem, (45) is valid for any f Cc(Rnd ). Let now f P (Rnd ), f 0. We notice that for any Tn the expression OU( f, , t, {ti }ii( ) , x ) is finite, which follows easily from Proposition 21. Further, one can find a sequence { fk } such that fk Cc(Rnd ), fk 0 and fk % f (pointwise). Appealing to Lebesgues monotone convergence theorem yields that (45) still holds (and both sides are finite). To conclude, once more we remove the positivity condition. As a simple corollary we obtain Corollary 23 Let {Xt }t0 be the OU branching system, then for any n 1 there exists Cn such that Proof We apply the above proposition with f = 1. Using the definition (41) and the inequality |i ( )| n 1 for Tn, it is easy to check that for any t Tn we have S(, t, x ) C enpt , for a certain constant depending only on and p, . Let us recall the notation of (39). The following proposition will be crucial in proving moment estimates for V - and U -statistics. Proposition 24 For any n > 0 there exist C, c > 0, such that for any Tn, any split times {ti }ii( ) and any canonical f P (Rnd ) we have exp c x ! exp Proof Let k n. Without loss of generality we may assume that I := {1, 2, . . . , k} are single numbers (i.e., j (i ) s( )) and {k + 1, . . . , n} are multiple ones. Let us also assume for a moment that for i {1, 2, . . . , k} we have t p( j (i)) 1. Let Zi and Gi be as in Proposition 21. We have E f (Y j (1), Y j (2), . . . , Y j (n)) = E f (Z j (1), Z j (2), . . . , Z j (n)). For i k we define Z i := ou(t p(l) tl )Gl e(tl 1) + ou(t p(i) 1)Gi . lP(i) A = E n (1) i=1 i f(Z j (1) + 1(G j (1) Z j (1)), . . . , Z j (k) ( 1,..., k ){0,1}k + k (G j (k) Z j (k)), Z j (k+1), . . . , Z j (n)) = E f(y1, . . . , yk , Z j (k+1), . . . , Z j (n))dyk . . . dy1. From now on, we restrict to the case d = 1. The proof for general d proceeds along the same lines but it is notationally more cumbersome. Using Lemma 18 and applying the Schwarz inequality multiple times we have | A| C f P C f / / / exp |Z j (i)|!/ / / / Note that by the definition of Z i we have Z i Gi = Hi e(tp(i)1) + ou(t p(i) 1) 1 Gi , where Hi is independent of Gi and Hi N (xi , i2) with i /2 and xi x . Thus, Z i Gi is a Gaussian variable with the mean bounded by C xi etp(i) and the standard deviation of order etp(i) . In particular Z i Gi l Cl exp(Cl x t p(i)). Since exp {|yi |} dyi e|G j(i)| + e|Z j(i)| |Z j (i) G j (i)|, the proof can be concluded by yet another application of the Schwarz inequality and standard facts on exponential integrability of Gaussian variables. Finally, if some i s do not fulfill t p(i) 1, we repeat the above proof with s( ) replaced by the set s& of indices from s( ) for which additionally t p(i) 1. In this way, we obtain (46) with is& t p(i) instead of is( ) t p(i). In our setting t p(i) t p(i) = is& hence (46) still holds (with a worse constant C ). Proof of Proposition 14 The sum (14) is finite hence it is enough to prove our claim for one L( f, ). Without loss of generality let us assume that E = {(1, 2), (3, 4), (2k 1, 2k)} and A = {2k + 1, . . . , n} (we recall notation in Sect. 2.1). Using the same notation as in (13) we write J (z2k+1, . . . , zn) := 2(dz j,k ) H ( f )(u1, u2, . . . , un) H ( f )(z1, z1, z2, z2, . . . , zk , zk , z2k+1 . . . , zn) where D := R+Rd . We have L( f, ) = In2k ( J (x2k+1, . . . , xn)). By the properties of the multiple stochastic integral [21, Theorem 7.26] we know that EL( f, )2 iI c By Lemma 18, we have (in order to simplify the notation, we calculate for d = 1, the general case is an easy modification) f (yi )iI , (Yi (si ))iI c . / / e|yi |dyi // // iI c / (1) iI i f Yi (si 1) + i (Yi Yi (si 1)) iI iI c For any x , y R we have max (exp(x ), exp(y)) exp(x ) + exp(y). Therefore, by the mean value theorem, we get + E|Yi (si 1) Yi | exp(Yi ). Using the Schwarz inequality and performing easy calculations, we get E| exp(Yi (si 1)) exp(Yi )| F (z) = H ( f )(z11, z11, z21, z21, . . . , zk1, zk1, z2k+1 . . . , zn) H ( f )(z12, z12, z22, z22, . . . , zk2, zk2, z2k+1 . . . , zn) () Now, using (48) in combination with the Fubini theorem, the definition of the measures i (given in Sect. 3.1) and our assumption p < 2, we get I1,I2{1,...,k} I3{2k+1,...,n} A(I1, I2, I3). A(I1, I2, I3) To conclude the proof, we use the fact that f L( f, ) is linear and P is a norm. Our next goal is to prove Proposition 15, which is the last remaining ingredient used in the proof of Theorem 3. This is where we will use for the first time the bookkeeping of trees technique introduced in Sect. 6.3. We will proceed in three steps. First we will obtain L2 bounds on V -statistics with deterministic normalization (Proposition 25), then we will pass to L1 bound of U -statistics with random normalization, restricted to the subset of the probability space, where |Xt | is large (Corollary 26). Finally, we will obtain bounds on the distance between the distribution of two U -statistics (with random normalization) in terms of the distance in P of the generating kernels (proof of Proposition 15). Proposition 25 Let {Xt }t0 be the OU branching particle system with p < 2. There exist C, c > 0 such that for any canonical kernel f P (Rnd ) we have Proof We need to estimate i1,i2,...i2n=1 f (Xt (i1), . . . , Xt (in)) f (Xt (in+1), . . . , Xt (i2n)). f 2 exp(c x ) exp ((n 1) + |P1( )|/2 + |P3( )|)pt P f 2 exp(c x )e(n1)pt P etp(i) f 2 exp(c x ) exp ((n 1) + |P1( )|/2 + |P3( )|)pt . P Note that | P1( )| + 2| P2( )| = |s( )| and i3=1 | Pi ( )| = |i ( )| = |l( )| 1. Thus, | P1( )| + 2| P3( )| = 2|l( )| 2 |s( )| = |l( )| + |m( )| 2 2n 2 (recall that m( ) denotes the set of leaves with multiple labels). This ends the proof. The next corollary is a technical step toward the proof of Proposition 15. Since we would like to normalize the U -statistic by the random quantity |Xt |n/2, we need to restrict the range of integration in the moment bound to the set on which |Xt | is relatively large. It will not be an obstacle in the proof of Proposition 15, since the probability that |Xt | is small will be negligible (on the set of non-extinction), which will allow us to pass from restricted L1 estimates to bounds on the distance between distributions. Corollary 26 Let {Xt }t0 be the OU branching system with p < 2. There exist constants C, c > such that for any canonical f P (Rnd ) and r (0, 1) we have Ex |Xt |n/2|Utn( f )|1 |Xt |rept ! C exp c x ! r n/2 f (2n) P . Proof Let J be the collection of partitions of {1, . . . , n}, i.e., of all sets J = { J1, . . . , Jk }, where Ji s are non-empty, pairwise disjoint and i Ji = {1, . . . , n}. Using (27) and the notation introduced there, we have n 2|I | k |I | = |I c|. |Xt |n/2Utn( f ) = |Xt |n/2 J J where aJ are some integers depending only on the partition J . Since the cardinality of J depends only on n, it is enough to show that for each J J and some constants C, c > 0 we have where I c := {1, . . . , k} \ I . Let us notice that n 2k l, so n k k l |I |, which gives We consider a single term of (50). We have Ex |Xt ||I |n/2|Vt|I c|(I c f J )|1 |Xt |rept ! r n/2+|I | Ex ept (n2|I |)/2|Vt|I c|(I c f J )| r n/2+|I | for I = {1, . . . , k}, where in the third inequality we used Proposition 25. One can check that for any n 2 there exists C > 0 such that for any I, J we have I f J P (R|I|d ) C f J P (R|J|d ) and f J P (R|J|d ) f (2n) P (Rnd ). Therefore, it remains to bound the ccontribution from I = {1, . . . , k} (in the case l = 0). But in this case I c = , so |Vt|I |(I c f J )| = |I c f J | = | k , f J | C f J P and exp( pt (n 2|I |)) 1, which easily gives the desired estimate. Let h(x ) := f1(2nx ) f2(2nx ), take r := Corollary 26 we get On the other hand, 2 g P E x t c |Xt |ept < r ! . Since on E x t c, we have |Xt | 1 and |Xt |ept converges to an absolutely continuous random variable, which ends the proof. As the proofs in this section follow closely the line of those in the subcritical case, we present only outlines, emphasizing differences. Proposition 27 Let {Xt }t0 be the OU branching particle system with p = 2. There exist C, c > 0 such that for any canonical kernel f P (Rnd ) and t > 1 we have Proof We will use similar ideas as in the proof of Proposition 25 as well as the notation introduced therein. Consider any T2n. By the definition of S(, t, x ), Proposition 24 and the assumption p = 2, we obtain C2 f 2 exp(c x )t n exp ((n 1) + | P1( )|/2 + | P3( )|) pt P where we used the fact that | P2( )| n and the estimate | P1( )| + 2| P3( )| 2n 2 obtained in the proof of Proposition 25. Now we can repeat the proof of Corollary 26 using Proposition 27 instead of Proposition 25 and obtain the following corollary, whose role is analogous to the one played by Corollary 26 in the slow branching case. Corollary 28 Let {Xt }t0 be the OU branching system with p = 2. There exist constants C, c such that for any canonical f P (Rnd ) and r (0, 1) we have for t 1, The proofs in this section diverge slightly from those in the critical and subcritical cases, and hence, we present more details. Proposition 29 Let {Xt }t0 be the OU branching particle system with p > 2. There exist C, c > 0 such that for any canonical kernel f P (Rnd ) we have Proof As in the previous cases, consider any T2n. We use the same notation as in the proof of Proposition 25. We have e2n(p)t S(, t, x ) C1e2n(p)t+pt Thus, it is enough to prove that 2n( p )t + pt + | P1( )|( p )t where for simplicity we write Pi instead of Pi ( ) (in the rest of the proof we will use the same convention with other characteristics of ). Using the equality |s| = | P1|+2| P2|, we may rewrite (52) as p + |s|( p ) | P2| p + | P3| p 2n( p ), 2 + |s| 2| P2| + 2| P3| 2n. But | P3| = |i | | P2| | P1| and so which ends the proof. Ex Proof of Proposition 17 Using the notation from the proof of Corollary 26, we get en(p)t+p|I |t |Vt|I c|(I c f J )| 7 Remarks on the Non-degenerate Case As in the case of U -statistics of i.i.d. random variables, by combining the results for completely degenerate U -statistics with the Hoeffding decomposition, we can obtain limit theorems for general U -statistics, with normalization, which depends on the order of degeneracy of the kernel. For instance, in the slow branching case Theorem 3, the Hoeffding decomposition and the fact that k : P (Rnd ) P (Rkd ) is continuous, give the following Corollary 30 Let {Xt }t0 be the OU branching system starting from x Rd . Assume that p < 2 and let f P (Rnd ) be symmetric and degenerate of order k 1. Then conditionally on E x t c, |Xt |(nk/2)Utn( f f, n ) converges in distribution to nk L1(k f ). Similar results can be derived in the remaining two cases. Using the fact that on the set of non-extinction |Xt | grows exponentially in t , we obtain Corollary 31 Let { Xt }t0 be the OU branching system starting from x Rd . Assume that p = 2 and let f P (Rnd ) be symmetric and degenerate of order k 1. Then conditionally on E x t c, t k/2| Xt |(nk/2)Utn ( f f, n ) converges in distribution to nk L 2(k f ). Similarly, using (8) and the definition of W we obtain Corollary 32 Let { Xt }t0 be the OU branching system starting from x Rd . Assume that p > 2 and let f P (Rnd ) be symmetric and degenerate of order k 1. Then conditionally on E x t c, exp(( pn k)t )Utn ( f f, n ) converges in probability to nk W nk L 3(k f ). Since in all the corollaries above the normalization is strictly smaller than | Xt |n , they in particular imply that | Xt |nUtn ( f f, n ) 0 in probability, which proves the second part of Theorem 9 (as announced in Sect. 5.3). Acknowledgments Research of R.A. was partially supported by the MNiSW grant N N201 397437 and by the Foundation for Polish Science. Research of P.M. was partially supported by the MNiSW grant N N201 397537. The authors would like to thank the referees for their constructive remarks. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


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Radosław Adamczak, Piotr Miłoś. \(U\) -Statistics of Ornstein–Uhlenbeck Branching Particle System, Journal of Theoretical Probability, 2014, 1071-1111, DOI: 10.1007/s10959-013-0503-2