U-statistics of Ornstein-Uhlenbeck branching particle system

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 $\lambda>0$. This system is known to fulfil a law of large numbers (under exponential scaling). Recently the question of the corresponding central limit theorem has been addressed. It turns out that the normalization and 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 Orstein-Uhlenbeck process. In the present paper we extend those results to $U$-statistics of the system proving a law of large numbers and a central limit theorem.


Introduction
We consider a single particle located at time t = 0 at x ∈ R d , moving according to the Orstein-Uhlenbeck process and branching after exponential time independent of the spatial movement. The branching is supercritical and given by the generating function F (s) := ps 2 + (1 − p), p > 1 2 .
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 = {X t } t≥0 .
Formally, we identify the system with the empirical process, i.e. X takes values in the space of Borel measures on R d and for each Borel set A, X t (A) is the (random) number of particles at time t in A.
It is well known (see e.g. [15]) that the system satisfies the law of large numbers, i.e. for all bounded continuous functions, conditionally on the set of non-extinction |X t | −1 X t , f → ϕ, f , a.s., (1) where |X t | is the number of particles at time t, {X t (1), X t (2), . . . , X t (|X t |)} are their positions, X t , f := |Xt| i=1 f (X t (i)) and ϕ is the invariant measure of the Orstein-Uhlenbeck process.
In a recent article [1], we investigated second order behaviour of this system and proved central limit theorems corresponding to (1). It turns out that the behaviour of the system may fall into three qualitatively different categories, depending on the relation between the branching intensity and the parameters of the Orstein-Uhlenbeck 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 (note that with this notation X t , f corresponds to U -statistics of order n = 1). Our investigation parallels the classical and well-developed 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.
The organization of the paper is as follows. After introducing the basic notation and preliminary facts in Section 2 we describe the main results of the paper in Section 3. Next (Section 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 Section 5 we provide complete proofs for arbitrary n. We conclude with some remarks concerning the so called non-degenerate case (Section 6).

Notation
For a branching system {X t } t≥0 , we denote by |X t | the number of particles at time t, and by X t (i) -the position of the i-th (in a certain ordering) particle at time t. We sometimes use E x or P x 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 equality or inequality holds with a constant c > 0, which is irrelevant for calculations. E.g. f (x) g(x) means that there exists a constant c > 0 such that f (x) = cg(x).
x i y i we denote the standard scalar product of x, y ∈ R d , by · the corresponding Euclidean norm. By ⊗ n we denote the n-fold tensor product.
In our paper we will use the Feynman diagrams. A diagram γ labeled by {1, 2, . . .} is a graph consisting of a set of edges E γ not having common endpoints, and unpaired vertices A γ . We will use r(γ) to denote the rank of the diagram i.e. the number of edges. For properties and more information we refer to [17,Definition 1.35].
In the paper we will use the space P = P (R d ) := f : R d → R : f is continuous and ∃ n such that |f (x)|/ x n → 0 as x → +∞ , (3) We endow this space with the following norm where n(x) := exp − d i=1 |x i | . We will also use to denote the space of continuous compactly supported functions.
Given a function f ∈ P (R d ) we will implicitly understand its derivatives (e.g. ∂f ∂x i ) in the space of tempered distributions (see e.g. [21, p. 173]).

Basic facts on the Galton-Watson process
The number of particles {|X t |} t≥0 is the celebrated Galton-Watson 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 λ p := (2p − 1)λ.
We will denote the extinction and non-extinction events by Ext and Ext c respectively. The process V t := e −λpt |X t | is a positive martingale. Therefore it converges (see also [2, Theorem 1.6.1]) V t → V ∞ , a.s. as t → +∞.
We have the following simple fact (we refer to [1] for the proof).
We will denote the variable V ∞ conditioned on non-extinction by W .

Basic facts on the Orstein-Uhlenbeck process
We recall that the Ornstein-Uhlenbeck process is a time homogenous Markov process with the infinitesimal The corresponding semigroup will be denoted by T. The density of the invariant measure of the Ornstein-Uhlenbeck process is given by

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 X 1 , . . . , 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 [18,7] for more information on U -statistics of sequences of independent random variables.
In our case we will consider U -statistics based on the sequence of positions of particles from the branching system as defined by (2). We will be interested in weak convergence of properly normalised U -statistics when t → ∞. Similarly as in the classical theory, the asymptotic behaviour of U -statistics depends heavily on the so called order of degeneracy of the kernel f , which we will briefly recall in Section A function f is called completely degenerate or canonical (with respect to some measure of reference ϕ, which in our case will be the stationary measure of the Ornstein-Uhlenbeck process) if for all x 1 , . . . , x k−1 , x k+1 , . . . , x n ∈ X . The complete degeneracy may be considered a centredness condition, in the classical theory of U -statistics canonical kernels are counterparts of centred 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 Section 5.1.2).

Main results
This section is devoted to the presentation of our results. The proofs are deferred to Section 5.
We start with the following law of large numbers (throughout the article when dealing with U -statistics of order n we will identify R d × . . . × R d n with R nd ).
Moreover, when f ∈ P (R nd ), then the above convergence holds in probability.
Having formulated the law of large numbers let us now pass to the corresponding CLTs. As already mentioned in the introduction, their form depends on the relation between λ p and µ, more specifically we distinguish three cases λ p < 2µ, λ p = 2µ and λ p > 2µ. We refer the reader to [1] for a detailed discussion of this phenomenon as well as its heuristic explanation and interpretation. Here we only stress that the situation for λ p > 2µ 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 Orstein-Uhlenbeck process).

Slow branching case: λ p < 2µ
Let Z be a Gaussian stochastic measure on R d+1 with intensity µ 1 (dtdx) := δ 0 (dt) + 2λpe λpt dt ϕ(x)dx defined according to [17,Definition 7.17]. We denote the stochastic integral with respect to Z by I and the corresponding multiple stochastic integral by I n [17, Section 7.2]. We assume that Z is defined on the probability space (Ω, F, P). For f ∈ P (R nd ) we define (we recall that T is the semigroup of the Orstein-Uhlenbeck process) It will be useful to treat this function as a function of n variables of type where u i = z j,k if (j, k) ∈ E γ and i = j or i = k and u i = z i if i ∈ A γ . Less formally, for each pair (j, k) we integrate over diagonal of coordinates j and k with respect to µ 2 . The function obtained in this way is integrated using the multiple stochastic integral I |Aγ| . We define where the sum spans over all Feynman diagrams labeled by {1, 2, . . . , n}.
For any canonical f ∈ P (R nd ) we have EL 1 (f ) 2 < +∞. Moreover L 1 is a continuous function where Can = f ∈ P (R nd ) : f is a canonical kernel and Can is endowed with the norm · P .
We are now ready to formulate our main result for processes with the small branching rate.
be the OU branching system starting from x ∈ R d . Let us assume that f ∈ P (R nd ) is a canonical kernel and λ p < 2µ. Then conditionally on the set of non-extinction Ext c there is the convergence where where G 1 ∼ N (0, 1/(2p − 1)) and W, G 1 , L 1 (f ) are independent random variables.

Critical branching case: λ p = 2µ
Consider the space L : ∂ϕ(x) ∂x l and a centred Gaussian process (G f ) f ∈L defined on some probability space (Ω, F, P), with the covariance structure given by We will identify the process with a map I : L → L 2 (Ω, F, P), such that I(f ) = G f . One can easily check that I is a bounded linear operator.
To formulate the central limit theorem in this case we will need the following where Can = f ∈ P (R nd ) : f is a canonical kernel endowed with the norm · P , such that for every The above lemma will be proved together with the following theorem, which describes the asymptotic behaviour of U -statistics in the critical case.
Theorem 3.5. Let {X t } t≥0 be the OU branching system starting from x ∈ R d . Let us assume that f ∈ P (R nd ) is a canonical kernel and λ p = 2µ. Then conditionally on the set of non-extinction Ext c there is the convergence where G 1 ∼ N (0, 1/(2p − 1)) and W, G, L 2 (f ) are independent random variables.

Fast branching case: λ p > 2µ
In order to describe the limit we introduce an R d -valued process The following two facts have been proved in [1].
Fact 3.6. H is a martingale with respect to the filtration of the OU-branching system starting from x ∈ R d .
Moreover for λ p > 2µ we have sup t EH 2 t < +∞, therefore there exists H ∞ := lim t→+∞ H t (a.s. limit) and H ∞ ∈ L 2 . When the OU branching system starts from 0 then martingales V t and H t are orthogonal.
It is worthwhile to note that the distribution of H ∞ depends on the starting conditions. Fact 3.7. Let {X t } t≥0 and {X t } t≥0 be two OU branching processes, the first one starting from 0 and the second one from x. Let us denote the limit of corresponding martingales by H ∞ ,H ∞ respectively. Theñ where V ∞ is the limit given by (6) for the system X.
H ∞ is R d -valued, we denote its coordinates by H i ∞ . Let f ∈ P (R nd ). We definẽ where we adopted convention that x j,l is the l-th coordinate of the j-th variable. By L 3 (f ) we will denotẽ Theorem 3.8. Let {X t } t≥0 be the OU branching system starting from x ∈ R d . Let us assume that f ∈ P (R nd ) is a canonical kernel and λ p > 2µ. Then conditionally on the set of non-extinction Ext c there is the convergence where G 1 ∼ N (0, 1/(2p − 1)) and (W, L 3 (f )), G 1 are independent. Moreover 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 central limit theorem for Ustatistics of independent random variables. U -statistics were introduced in the 1940's in the context of unbiased estimation by Halmos [14] and Hoeffding who obtained the central limit theorem for nondegenerate (degenerate of order 0) kernels [16]. The full description of the central limit theorem was obtained in [20,11] (see also the article [13] where the CLT is proven for a related class of V -statistics).
Similarly as in our case, the asymptotic behaviour of U -statistics based on a function f : X n → R and an i.i.d. X -valued sequence X 1 , X 2 , . . . is governed by the order of degeneracy of the function f (see Section 5.1.2) with respect to the law of X 1 (call it P ). The case of general f can be reduced to the canonical one, for which one has the weak convergence where J n 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 .
Let us note that in the i.i.d. case the limiting distribution is simpler than for U -statistics of the OU branching processes. For small branching rate however, the behaviour of U -statistics in our case still resembles the classical one as it is a sum of multiple stochastic integrals of different orders. In the remaining two cases the behaviour differs substantially. This can be seen as a result of the lack of independence. Although asymptotically the particles' positions become less and less dependent, in short time scale 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 analysed the behaviour 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 α and ϕ 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 is [19], where the authors consider U -statistics of interacting particle systems.

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 form 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 reader's convenience, we will now restate this case in a simpler language of [1], not involving multiple stochastic integrals.
We will start with the law of large numbers Theorem 4.1. Let {X t } t≥0 be the OU branching system starting from x ∈ R d . Let us assume that or equivalently on the set of non-extinction, Ext c , we have Moreover, if f is bounded then the almost sure convergence holds.

Small branching rate: λ p < 2µ
We denotef (x) := f (x) − f, ϕ and Let us also recall (6) and that W is V ∞ conditioned on Ext c . In this case, the behaviour of X is given by the following be the OU branching system starting from x ∈ R d . Let us assume that Then σ 2 f < +∞ and conditionally on the set of non-extinction Ext c there is the convergence where G 1 ∼ N (0, 1/(2p − 1)), G 2 ∼ N (0, σ 2 f ) and W, G 1 , G 2 are independent random variables.

Critical branching rate: λ p = 2µ
We denote Note that the same symbol σ 2 f 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.
be the OU branching system starting from x ∈ R d . Let us assume that λ p = 2µ and f ∈ P (R d ). Then σ 2 f < +∞ and conditionally on the set of non-extinction Ext c there is the convergence where G 1 ∼ N (0, 1/(2p − 1)), G 2 ∼ N (0, σ 2 f ) and W, G 1 , G 2 are independent random variables.

Fast branching rate: λ p > 2µ
In the following theorem we use the notation introduced in Section 3.3.
Theorem 4.4. Let {X t } t≥0 be the OU branching system starting from x ∈ R d . Let us assume that λ p > 2µ and f ∈ P (R d ). Then conditionally on the set of non-extinction Ext c there is the convergence where G ∼ N (0, 1/(2p − 1)), (W, J), G are independent and J is H ∞ conditioned on Ext c . Moreover

Proofs
We will now pass to the proofs of the results announced in Section 3. Their general structure is similar as in 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 proper function spaces.
In the next section we will recall some additional (standard) facts concerning the Orstein-Uhlenbeck process and U -statitistics. Next, in Section 5.2 we will develop general tools needed for the approximation, which will be the most technical part of the proof. Section 5.3 will be devoted to rather short proofs of the main results.
From now on we will often work conditionally on the set of non-extinction Ext c , which will not be explicitly mentioned in the proofs (however should be clear from the context).

The Orstein-Uhlenbeck process
The semigroup of the Ornstein-Uhlenbeck process can be represented by where Let us recall (8). We denote ou(t) := √ 1 − e −2µt and let G ∼ ϕ. The semigroup T has the following useful representations We also denote T λ s := e λs T s .

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 corresponding to I as the function given by the formula Once can easily see that for Note that if f is symmetric (i.e. invariant with respect to permutations of arguments), Π I f depends only on the cardinality of f . In this case we speak about the k-th Hoeffding projection (k = 0, . . . , n), given by The order of degeneracy is responsible for the normalisation and the form of the limit in the central limit theorem for U -statistics, e.g. if the kernel is non-degenerate, i.e. Π 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 Section 6).
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 where we use the convention U 0 t (a) = a for any constant a. For technical reasons we will also consider the notion of a V -statistic which is closely related to Ustatistics, and is defined as The corresponding Hoeffding decomposition is where again we set V 0 t (a) = a for any constant a. In the proof 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 ). An easy application of the inclusion-exlusion formula yields that

Approximation of functions
First we will show that any function in P (R nd ) can be approximated by tensor functions. For a subset A of a linear space by span(A) we denote the set of finite linear combinations of elements of A.
f i : f i bounded continuous} and f ∈ P (R nd ) be a canonical kernel then there exists a sequence {f k } ⊂ span(A) such that each f k is canonical and Proof. First we prove that span(A) is dense in P (R nd ). Let us notice that given a function f ∈ P (R nd ) it The box is a compact set and an approximation exists due to the Stone-Weierstrass theorem.
Now, let f ∈ P (R nd ). We may find a sequence {h k } ⊂ span(A) such that h k → f in P . Let us recall the Hoeffding projection (24) and denote I = {1, 2, . . . , n}. Now direct calculation (using the exponential integrability of Gaussian variables) reveals that the sequence f k := Π I h k fulfils the conditions of the lemma.
Let f ∈ P (R nd ) and I ⊂ {1, 2, . . . , n} with I = k. We definê where z i = y i if i ∈ I, z i = x i otherwise and g 1 is given by (22). We have where C > 0 depends only on σ, µ, d, l, n. Moreover, when f is canonical, so isf I .
Proof. Let us fix some I and Λ. Using (28) we get Therefore by the properties of Gaussian density g 1 and easy calculations we arrive at for some constant C Λ .
To conclude it is enough to take the maximum over all admissible pairs I, Λ.
Let us now assume that f is canonical. We would like to check that for any j ∈ {1, 2, . . . , n} we have There are two cases, the first when j / ∈ I. Then we have The second case is when j ∈ I. Then where the second equality holds by the fact that ϕ is the invariant measure of the Ornstein-Uhlenbeck process. Now the proof reduces to the first case.

Approximation of U -statistics
Bookkeeping of trees So far we have shown that one can approximate functions in P by linear combinations of tensor products. Our next goal is to show that two functions which are close in P generate U -statistics which (after proper normalization, specific for each regime) are close in distribution. To this end we will use the "bookkeeping of trees" technique (see e.g. [4] or [10, Section 2]), 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.
We recall (25). Let f 1 , f 2 , . . . , f n ∈ C c (R d ) and f i ≥ 0. We would like to calculate Let Λ ⊂ {1, 2, . . . , n}, slightly abusing notation we denote Note that this differentiation is valid by Fact 2.1 and properties of the Laplace transform (e.g. [12, Chapter XIII.2]). By the calculations from Section 4.2. in [1] we know that It is easy to check that Assume that |Λ| > 0. We denote by P 1 (Λ) all pairs (Λ 1 , Λ 2 ) such that Λ 1 ∪ Λ 2 = Λ and Λ 1 ∩ Λ 2 = ∅, and by P 2 (Λ) ⊂ P 1 (Λ) pairs with an additional restriction that Λ 1 = ∅ and Λ 2 = ∅. We easily check that We evaluate it at α = 0, (let us notice that ∂ |Λ 1 | s)), multiply both sides by (−1) |Λ| and use the definition This can be easily transformed to (recall that T 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 T n 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 τ ∈ T n , by l(τ ) we denote the set of its leaves. Each leaf l ∈ l(τ ) is assigned a label, denoted by lab(l), which is a non-empty subset of {1, 2, . . . , n}, the labels fulfil two conditions: leaves with single and multiple labels respectively.
Let τ ∈ T n , we consider an Ornstein-Uhlenbeck branching walk on τ as follows (in this part we ignore the labels). Let us fix t ∈ R + and {t i } i∈i(τ ) . The initial particle is placed at time 0 at location x, it evolves up to the time t − t 1 and splits into two offspring, the first one is associated with the left branch of vertex . The construction makes sense provided that t i ≤ t and t i ≤ t p(i) for all i ∈ i(τ ). We where j(a) = l ∈ l(τ ) is the unique leaf such that a ∈ lab(l). We also define Similarly as before the first term of (31) corresponds to τ s . By induction the second term can be written as We have where p is the transition density of the Ornstein-Uhlenbeck process. Now we create a new tree τ by setting τ 1 and τ 2 to be descendants of the vertex born at time t − t 1 . We keep labels and split times unchanged and assume that the first particles of OU processes on τ 1 and τ 2 are put at x. Thus by the Markov property of the Ornstein-Uhlenbeck process we can identify the branching random walk on τ 1 and τ 2 with the branching random walk on τ . It is also easy to check that the described correspondence is a bijection from set of pairs (τ 1 , τ 2 ) (as in the sum above) to T n \ {τ s } and therefore the expression above is equal to the sum Tn\{τs} S(τ, t, x).
The calculations will be more tractable when we derive an explicit formula for {Y i } i∈l(τ ) . Let us recall the notation introduced in (23) and consider a family of independent random variables {G i } i∈τ , such that G i ∼ ϕ for i = 0 and G 0 ∼ δ x . Recall also that ou(t) = √ 1 − e −2µt . The following fact follows easily from the construction of the branching walk on τ and (23).
Fact 5.5. Let {Y i } i∈l(τ ) be positions of particles at time t of the Ornstein-Uhlenbeck process on tree τ with labels {t i } t∈i(τ ) . We have where for any i ∈ l(τ ) we put where P (i) := {predecessors of i}, by convention we set t 0 = t and ou(t p(0) − t 0 ) = 1.
We are now ready to prove an extended version of Fact 5.4.
Fact 5.6. Let {X t } t≥0 be the OU branching system starting from x ∈ R d and f ∈ P (R nd ) then where in (34) we extend the definition of OU in (33) by putting Moreover all the quantities above are finite.
Proof. . Let now f ∈ P (R nd ), f ≥ 0. We notice that for any τ ∈ T n the expression OU (f, τ, t, {t i } i∈i(τ ) , x) is finite, which follows easily from Fact 5.5. Further, one can find a sequence {f k } such that f k ∈ C c (R d ), f k ≥ 0 and f k ր f (pointwise). Appealing to the monotone Lebesgue theorem yields that (36) still holds (and is finite). To conclude, once more we remove the positivity condition.
As a simple corollary we obtain Corollary 5.7. Let {X t } t≥0 be the OU branching system, then for any n ≥ 1 there exists C n such that Proof. We apply the above fact with f = 1. Using definition (34) and the inequality |i(τ )| ≤ n − 1 for τ ∈ T n , it is easy to check that for any t ∈ T n we have S(τ, t, x) ≤ C τ e nλpt , for a certain constant depending only on τ .
By the fact thatf is smooth with respect to coordinates from I and canonical and Lemma 5.3 we obtain . . .
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 5.2 and applying the Schwarz inequality multiple times we have variable with the mean bounded by C x i e −µt p(i) and the standard deviation of order e −µt p(i) . In particular 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 fulfil t p(i) ≥ 1 we repeat the above proof with s(τ ) replaced by the set s ′ of indices for which additionally t p(i) ≥ 1. In this way we obtain (37) with i∈s ′ t p(i) . In our setting hence (37) still holds (with a worse constant C).
Bounds for U -and V -statistics Using Fact 5.8 we will now be able to estimate moments of V -and U -statistics of the branching particle system. The inequalities will be stated separately for each regime discussed in Section 3.
We will first develop L 2 bounds for U -statistics with deterministic normalization. Since in the slow and critical branching case the normalization in our theorems is random, related to the size of the process, we will later transform these bounds to L 0 -bounds for U -statistics with a random normalization.
Fact 5.9 (Small branching rate). Let {X t } t≥0 be the OU branching particle system with λ p < 2µ. There exist C, c > 0 such that for any canonical kernel f ∈ P (R nd ) we have Proof. We need to estimate E x e −nλpt |Xt| Obviously the function f ⊗ f is canonical. Moreover, it is easy to check, that f ⊗ f P ≤ f 2 P . By Fact 5.6 it suffices to show that for each τ ∈ T 2n there exist C, c > 0 such that for any t > 0 we have e −nλpt S(τ, t, x) ≤ C exp {c x } f 2 P . Let us fix τ ∈ T 2n and denote by P 1 (τ ) and P 2 (τ ) the sets of inner vertices of τ with respectively one and two children in s(τ ). Set also P 3 (τ ) := i(τ ) \ (P 1 (τ ) ∪ P 2 (τ )).
We will now pass to an analogous estimate in the critical case.
Fact 5.10 (Critical branching rate). Let {X t } t≥0 be the OU branching particle system with λ p = 2µ.
There exist C, c > 0 such that for any canonical kernel f ∈ P (R nd ) we have Proof. We will use similar ideas as in the proof of Fact 5.9 as well as the notation introduced therein.
Fact 5.11 (Fast branching rate). Let {X t } t≥0 be the OU branching particle system with λ p > 2µ. There exist C, c > 0 such that for any canonical kernel f ∈ P (R nd ) we have Proof. As in the previous cases, consider any τ ∈ T 2n . We have e −2n(λp−µ)t S(τ, t, x) Thus it is enough to prove that where for simplicity we write P i instead of P i (τ ) (in the rest of the proof we will use the same convention with other characteristics of τ ). Using the equality |s| = |P 1 | + 2|P 2 |, we may rewrite (38) as so by the inequalities 2n ≥ |s| and λ p − µ > λ p /2 it is enough to prove that But |P 3 | = |i| − |P 2 | − |P 1 | and so which ends the proof.
Random normalization Using the facts obtained above we will now prove estimates for U -statistics normalized by a proper power of |X t |, which will be relevant in the proofs of Theorems 3.3 and 3.5. Since asymptotically |X t | exp(−λ p t) behaves (conditionally on Ext c ) as an exponential random variable W and EW −1 does not exist, we will have to introduce a truncation, cutting out the set where |X t | is small.
Corollary 5.12. Let {X t } t≥0 be the OU branching system with λ p < 2µ. There exist constants C, c > such that for any canonical f ∈ P (R nd ) and r ∈ (0, 1) we have Proof. Let J be the collection of partitions of {1, . . . , n} i.e. of all sets J = {J 1 , . . . , J k }, where J i 's are nonempty, pairwise disjoint and i J i = {1, . . . , n}. Using (27) and notation introduced there we have where a J 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 Let us thus consider J = {J 1 , . . . , J k } and let us assume that among the sets J i there are exactly l sets of cardinality 1, say J 1 , . . . , J l . We would like to use Fact 5.9. To this end we have to express V k t (f J ) as a sum of V -statistics with canonical kernels. This can be easily done by means of Hoeffding's decomposition (26). Since f J is already degenerate with respect to variables x 1 , . . . , x l , we get where I c := {1, . . . , k} \ I. Let us notice that n ≥ 2k − l, so n − k ≥ k − l ≥ |I|, which gives Thus we have for I = {1, . . . , k}, where in the third inequality we used Fact 5.9. One can check that for any n ≥ 2 there exists C > 0 such that for any I, J ⊂ {1, 2, . . . , n} we have Π I f P ≤ C f P and f J P ≤ f (2n·) P .
Therefore it remains to bound the contribution from I = {1, . . . , k} (in the case l = 0). But in this case In an analogous way, replacing Fact 5.9 by Fact 5.10 one proves Corollary 5.13. Let {X t } t≥0 be the OU branching system with λ p = 2µ. There exist constants C, c such that for any canonical f ∈ P (R nd ) and r ∈ (0, 1) we have for t ≥ 1, We also have an analogous statement in the supercritical case.
Corollary 5.14. Let {X t } t≥0 be the OU branching system with λ p > 2µ. There exist constants C, c such that for any canonical f ∈ P (R nd ) we have Proof. Using the notation from the proof of Fact 5.12, we get Let us note that by λ p > 2µ and (41) we get n( Thus the summands on the right hand side above for I = {1, . . . , k} can be bounded using Corollary 5.7 and Fact 5.11 by (the last inequality is analogous as in the proof of Fact 5.12).
The contribution from I = {1, . . . , k} (in the case l = 0) also can be bounded like in Fact 5.12.
It is well known that m is a distance metrizing the weak convergence (see e.g. [9,Theorem 11.3.3]). One easily checks that when µ 1 , µ 2 correspond to two random variables X 1 , X 2 on the same probability space then we have Fact 5.15. Let {X t } t≥0 be the OU branching system starting from x and λ p < 2µ. For any n ≥ 2 there exists a function l n : R + → R + , fulfilling lim s→0 l n (s) = 0 and such that for any canonical f 1 , f 2 ∈ P (R nd ) and any t > 1 we have where µ 1 ∼ |X t | −n/2 U n t (f 1 ), µ 2 ∼ |X t | −n/2 U n t (f 2 ) (the U -statistics are considered here conditionally on Ext c ).
Proof. Let us fix g ∈ Lip(1). We consider Let h(x) := f 1 (2nx) − f 2 (2nx), take r := h 1/n P and assume that r < 1. Then by Corollary 5.12 we get On the other hand, Since on Ext c we have |X t | ≥ 1 and |X t |e −λpt converges to an absolutely continuous random variable, which ends the proof.
An analogous proof using Corollary 5.13 gives a counterpart of the above fact in the critical case. and any t > 0 we have (the U -statistics are considered here conditionally on Ext c ).

The law of large numbers
Proof of Theorem 3.1. Consider the random probability measure µ t = |X t | −1 X t (recall that formally we identify X t with the corresponding counting measure). By Theorem 4.1 with probability one (conditionally on Ext c ), µ t converges weakly to ϕ. Thus, by Theorem 3.2 in [3], µ ⊗n t converges weakly to ϕ ⊗n . But f, µ ⊗n t = |X t | −n V n t (f ), which gives the almost sure convergence |X t | −n V n t (f ) → f, ϕ . Now it is enough to note that the number of "off-diagonal" terms in the sum (39) defining U n t (f ) is of order |X t | n−1 and use the fact that |X t | → ∞ a.s. on Ext c .
The proof for f ∈ P (R nd ) follows directly from the central limit theorems from Section 6 (we will not use Theorem 3.1 in their proof).
where D := R + × R d . We know that L(f, γ) = I n−2k (J(x 2k+1 , . . . , x n )). By the properties of the multiple stochastic integral [17,Theorem 7.26] we know that EL(f, γ) 2 i∈{2k+1,...,n} D µ 1 (dz i ) |J(z 2k+1 , . . . , z n )| 2 . Therefore we need to estimate Using (23), Lemma 5.3, the assumption that f is canonical and the semigroup property, we can rewrite (10) as By Lemma 5.2 we have (in order to simplify the notation we calculate for d = 1, the general case is an easy modification) Obviously for any x, y ∈ R we have max (exp(x), exp(y)) ≤ exp(x) + exp(y). Therefore by the mean value theorem we get Using the Schwarz inequality and performing easy calculations we get Since we also have E exp(|Y i (s i )|) exp(|x i |), we have thus proved that We use the above inequality to estimate (44). To this end let us denote To simplify the notation let us introduce the following convention. For subsets I 1 , I 2 ⊂ {1, . . . , k}, . . , n} and i ∈ {1, . . . , n} we will write I j (i) = 1 if i ∈ I j and I j (i) = 0 otherwise. Let us also denote for z = (z 1 1 , . . . , z 1 k , z 2 1 , . . . , z 2 k , z 2k+1 , . . . , z n ). With this notation we can estimate (44) as follows ( * ) = I 1 ,I 2 ⊂{1,...,k} I 3 ⊂{2k+1,...,n} Now, using (45) in combination with the Fubini theorem, the definition of the measures µ i (given in Section 3.1) and our assumption λ p < 2µ, we get To conclude the proof we use the fact that f → L(f, γ) is linear and · P is a norm.
On the other hand by Theorem 4.1 one could easily obtain f l j (X t (i))f l k (X t (i)) → f l j f l k , ϕ , a.s.
We will now show that L is equal to L 1 (f ) given by (12). By linearity of L 1 (f ) it is enough to consider the case of m = 1. We will therefore drop the superscript and write f i instead of f l i . We denote by P (f i , f j ) := D H(f i ⊗ f j )(z, z)µ 2 (dz). By (47) and definition of µ 2 given in (11) We now adopt the notation that η ⊂ γ when E η ⊂ E γ and write It is easy to notice that in the case of f = ⊗ n i=1 f i the expression above is equivalent to (12). Let us now consider a function f ∈ P . We put h(x) := f (2nx). By Lemma 5.1 we may find a sequence of functions {h k } k ⊂ span(A) such that h k → h in P . Next we define f k (x) := h k (x/2n). Now by Fact 5.16 we may approximate |X t | −(n/2) U n This is a well known fact in the theory of polynomial chaos and follows e.g. from a combination of Theorem 3.2.5, Theorem 3.2.10 and Proposition 3.3.1. (i.e. the Paley-Zygmund inequality) in [7]. Note that c does not depend on N , which is crucial in our application. Let f ∈ A ∩ Can. Now, for sufficiently large D, by Markov's inequality and the fact that W is with probability one strictly positive, P(|L 2 (f )| ≥ D 2 f P ) ≤ P(|W n/2 L 2 (f )| ≥ D f P ) + P(W −n/2 > D) ≤ C D + P(W −n/2 > D) < c, which, together with (48), implies that L 2 (f ) 2 ≤ c −1 D 2 f P and in consequence the operator L 2 is bounded on A ∩ Can. In particular it admits a unique extension to a bounded operator L 2 : Can → L 2 (Ω, F, P), which proves Fact 3.4.
Theorem 3.5 follows now by the already established case of f ∈ span(A), Fact 5.15 and standard approximation arguments.

CLT -supercritical branching rate
Proof of Theorem 3.8. 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 4.4 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 3.5, since under assumption λ p > 2µ we have Before our final step we recall that the convergence in probability can be metrised by d(X, Y ) := Let us now consider a function f ∈ P . By Lemma 5.1 we may find a sequence of functions {f k } ⊂ span(A) such that f k → f in P . Now by Corollary 5.14 we may approximate e −n(λp−µ)t U n t (f ) with e −n(λp−µ)t U n t (f k ) uniformly in t in the sense of metric d. Moreover, one can easily show that lim k→+∞ d(L 3 (f k ),L 3 (f )) = 0. This concludes the proof.

Remarks on the non-degenerate case
Let us remark that 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