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

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## Abstract

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.

## Keywords

Supercritical branching particle systems \(U\)-statistics Central limit theorem## Mathematics Subject Classification (2010)

Primary 60F05 60J80 Secondary 60G20## 1 Introduction

We consider a single particle located at time \(t=0\) at \(x \in \mathbb {R}^d\), 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 \(X = \left\{ X_t \right\} _{t\ge 0}\).

We identify the system with the empirical process, i.e., \(X\) takes values in the space of Borel measures on \(\mathbb {R}^d\) and for each Borel set \(A, X_t(A)\) 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.

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 Ornstein–Uhlenbeck process.

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 \(\left\{ X_t \right\} _{t\ge 0}\), 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 \(\mathbb {E}{}_x\) or \(\mathbb {P}_x\) to denote the fact that we calculate the expectation for the system starting from a particle located at \(x\). We use also \(\mathbb {E}{}\) and \(\mathbb {P}\) when this location is not relevant.

By \(\rightarrow ^d \), we denote the convergence in law. We use \(\lesssim \) to denote the situation when an inequality holds with a constant \(c>0\), which is irrelevant to calculations, e.g., \(f(x)\lesssim g(x)\) means that there exists a constant \(c>0\) such that \(f(x) \le c g(x)\).

By \(x \circ y = \sum _{i=1}^d x_i y_i\), we denote the standard scalar product of \(x,y\in \mathbb {R}^d\), by \(\Vert \cdot \Vert \) the corresponding Euclidean norm. By \(\otimes ^n\), we denote the \(n\)-fold tensor product.

We use also \(\left\langle f, \mu \right\rangle :=\int _{\mathbb {R}^d} f(x) \mu {({\mathrm{d}}x) }\). We will write \(X \sim \mu \) to describe the fact that a random variable \(X\) is distributed according to the measure \(\mu \), similarly \(X\sim Y\) will mean that \(X\) and \(Y\) have the same law.

For a subset \(A\) of a linear space by \(\mathrm{span}(A)\), we denote the set of finite linear combinations of elements of \(A\).

In the paper, we will use Feynman diagrams. A diagram \(\gamma \) on a set of vertices \(\left\{ 1,2,\ldots ,n \right\} \) is a graph on \(\left\{ 1,2,\ldots ,n \right\} \) consisting of a set of edges \(E_\gamma \) not having common endpoints and a set of unpaired vertices \(A_\gamma \). We will use \(r(\gamma )\) 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].

Given a function \(f\in \mathcal {P}^{}(\mathbb {R}^d)\), we will implicitly understand its derivatives (e.g., \(\frac{ \partial f}{\partial x_i}\)) in the space of tempered distributions (see, e.g., [25, p. 173]).

By \(f(a\cdot )\), we denote the function \(x\mapsto f(ax)\).

### 2.2 Basic Facts on the Galton–Watson Process

**Proposition 1**

We have \(\left\{ V_\infty = 0 \right\} = Ext\) and conditioned on non-extinction \(V_\infty \) has the exponential distribution with parameter \(\frac{2p-1}{p}\). We have \(\mathbb {E}{}(V_\infty ) =1\) and \( {{\mathrm{Var}}}(V_\infty ) = \frac{1}{2p-1}\). \(\mathbb {E}{}e^{-4\lambda _p t} |X_t|^4\) is uniformly bounded, i.e., there exists \(C>0\) such that for any \(t\ge 0\) we have \(\mathbb {E}{}e^{-4\lambda _p t}|X_t|^4 \le C\). Moreover, all moments are finite, i.e., for any \(n\in \mathbb {N}\) and \(t\ge 0\) we have \(\mathbb {E}{} |X_t|^n < +\infty \).

We will denote the variable \(V_\infty \) conditioned on non-extinction by \(W\).

### 2.3 Basic Facts on the Ornstein–Uhlenbeck Process

### 2.4 Basic Facts Concerning \(U\)-Statistics

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 \(t \rightarrow \infty \). 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.

## 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 \(\underbrace{\mathbb {R}^d\times \cdots \times \mathbb {R}^d}_{n}\) with \(\mathbb {R}^{nd}\)).

**Theorem 2**

Having formulated the law of large numbers, let us now pass to the corresponding CLTs. We recall that \(\mu \) is the drift parameter in (9) and \(\lambda _p\) is the growth rate (7). As already mentioned in the introduction, the form of the limit theorems depends on the relation between \(\lambda _p\) and \(\mu \), more specifically, we distinguish three cases: \(\lambda _p < 2\mu , \lambda _p = 2\mu \) and \(\lambda _p > 2\mu \). 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 \(\lambda _p > 2\mu \) 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 Ornstein–Uhlenbeck process.

### 3.1 Slow Branching Case: \(\lambda _p < 2\mu \)

We are now ready to formulate our main result for processes with small branching rate. Recall (8) and that \(W\) is \(V_\infty \) conditioned on \(Ext^c\).

**Theorem 3**

### 3.2 Critical Branching Case: \(\lambda _p = 2\mu \)

Before we present the main results in the critical branching case, we need to introduce some additional notation.

Consider the orthonormal Hermite basis \(\{h_i\}_{i\ge 0}\) for the measure \(\gamma = \mathcal {N}(0,\frac{\sigma ^2}{2\mu })\) (i.e, \(h_0 = 1, h_i\) is a polynomial of degree \(i\) and \(\int h_i h_j d\gamma = \delta _{ij}\)). Then for any positive integer \(n\), the set \(\{h_{i_1}\otimes \cdots \otimes h_{i_{nd}}\}_{i_1,\ldots ,i_{nd}\ge 0}\) of multivariate Hermite polynomials is an orthonormal basis in \(L_2(\mathbb {R}^{nd},\varphi ^{\otimes n})\). For a function \(f \in L_2(\mathbb {R}^{nd},\varphi ^{\otimes n})\) let \(\tilde{f}_{i_1,\ldots ,i_{nd}}\) be the sequence of coefficients of \(f\) with respect to this basis, i.e., \(f = \sum _{i_1,\ldots ,i_{nd}\ge 0} \tilde{f}_{i_1,\ldots ,i_{nd}} h_{i_1}\otimes \cdots \otimes h_{i_{nd}}\).

Let \(\mathcal {H}^n = \mathrm{span}\{h_{i_1}\otimes \cdots \otimes h_{i_{nd}}:\sum _{j=1}^d i_{kd + j} = 1, k=0,\ldots ,n-1\}\) be the subspace of \(L_2(\mathbb {R}^{nd},\varphi ^{\otimes n})\). In other words, \(\mathcal {H}^1\) is the subspace of \(L_2(\mathbb {R}^d,\varphi )\) spanned by Hermite polynomials of degree \(1\) on \(\mathbb {R}^d\) and \(\mathcal {H}^n = (\mathcal {H}^1)^{\otimes n}\).

We will identify this process with a map \(I:L_2(\mathbb {R}^d,\varphi ) \rightarrow L_2(\varOmega ,\mathcal {F},\mathbb {P})\), such that \(I(f) = G_f\). One can easily check that \(I\) is a bounded linear operator. Moreover, \(I = IP\). In fact \(I\) is the stochastic integral of \(Pf\) with respect to the random Gaussian measure on \(\mathbb {R}^d\) with intensity \(2\lambda p\varphi \) (however, we will not use this fact in the sequel).

**Theorem 4**

*Remark 5*

### 3.3 Fast Branching Case: \(\lambda _p > 2\mu \)

**Proposition 6**

\(H\) is a martingale with respect to the filtration of the OU branching system starting from \(x\in \mathbb {R}^d\). Moreover for \(\lambda _p>2\mu \), we have \(\sup _t \mathbb {E}{\left\| H_t \right\| _{ } ^{ 2 }} <+\infty \), therefore there exists \(H_\infty := \lim _{t\rightarrow +\infty } H_t\) (a.s. limit) and \(H_\infty \in L_2\). 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 \(H_\infty \) depends on the starting conditions.

**Proposition 7**

**Theorem 8**

### 3.4 Remarks on the CLT for \(U\)-Statistics of i.i.d. Random Variables

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 \(\alpha \) and \(\varphi \) 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 [9, 10, 11, 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 reader’s 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**

### 4.1 Small Branching Rate: \(\lambda _p < 2\mu \)

**Theorem 10**

### 4.2 Critical Branching Rate: \(\lambda _p = 2\mu \)

Note that the same symbol \(\sigma _f^2\) 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**

### 4.3 Fast Branching Rate: \(\lambda _p > 2\mu \)

In the following theorem, we use the notation introduced in Sect. 3.3.

**Theorem 12**

## 5 Proofs of Main Results

### 5.1 Outline of the Proofs

- 1.
Using the one-dimensional versions of the results, presented in Sect. 4, and the Cramér–Wold device, one proves convergence for functions \(f\), which are linear combinations of tensor products. This class is shown to be dense in \(\mathcal {P}^{}\).

- 2.
Using algebraic properties of the covariance, one obtains explicit formulas for the limit, which are well defined for any function \(f\in \mathcal {P}^{}\). 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 \(U^n_t(f)\) and of \(U^n_t(g)\) in terms of the distance between \(f\) and \(g\) in \(\mathcal {P}^{}\). 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 \(f_n\) whose laws converge by step 1 as \(t\rightarrow \infty \) to some limiting measure \(\mu _n\). By step 2, when the approximation becomes finer and finer (\(n\rightarrow \infty )\), one has \(\mu _n \rightarrow \mu \) for some probability measure \(\mu \). Finally, it is easy to see that \(\mu \) is the limiting measure for the original \(U\)-statistic.

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 \(Ext^c\), 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.

*degenerate of order*\(k-1\) (\(1 \le k \le n\)) iff \(k = \min \{i> 0 :\varPi _i f \not \equiv 0 \}\). 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., \(\varPi _1 f \not \equiv 0\), 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).

### 5.3 Proof of the Law of Large Numbers

*Proof of Theorem 2*

Consider the random probability measure \(\mu _t = |X_t|^{-1} X_t\) (recall that formally we identify \(X_t\) with the corresponding counting measure). By Theorem 9 with probability one (conditionally on \(Ext^c\)), \(\mu _t\) converges weakly to \(\varphi \). Thus, by Theorem 3.2 in [3], \(\mu _t^{\otimes n}\) converges weakly to \(\varphi ^{\otimes n}\).

Let \(f\) be bounded and continuous. We notice that \(\langle f,\mu _t^{\otimes n}\rangle = |X_t|^{-n}V_t^n(f)\), which gives the almost sure convergence \(|X_t|^{-n}V_t^n(f) \rightarrow \langle f,\varphi \rangle \). Now it is enough to note that the number of “off-diagonal” terms in the sum (25) defining \(V_t^n(f)\) is of order \(|X_t|^{n-1}\) and use the fact that \(|X_t| \rightarrow \infty \) a.s. on \(Ext^c\).

We note that in the proofs below we will use this fact only in the version for \(f\in \mathcal {C}(\mathbb {R}^{nd})\) which we have just proven. The proof of convergence in probability for \(f\in \mathcal {P}^{}(\mathbb {R}^{nd})\) follows directly from the CLT presented in Sect. 7. \(\square \)

### 5.4 Approximation

Before we proceed to the proofs of CLTs, we will demonstrate the simple fact that any function in \(\mathcal {P}^{}(\mathbb {R}^{nd})\) can be approximated by tensor functions.

**Lemma 13**

*Proof*

First, we prove that \(\mathrm{span}(A)\) is dense in \(\mathcal {P}^{}(\mathbb {R}^{nd})\). Let us notice that given a function \(f\in \mathcal {P}^{}(\mathbb {R}^{nd})\) 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 Stone–Weierstrass theorem.

Now, let \(f\in \mathcal {P}^{}(\mathbb {R}^{nd})\). We may find a sequence \(\left\{ h_k \right\} \subset \mathrm{span}(A)\) such that \(h_k(m\cdot ) \rightarrow f(m\cdot )\) in \(\mathcal {P}^{}\). Let us recall the Hoeffding projection (24) and denote \(I=\left\{ 1,2,\ldots ,n \right\} \). Now direct calculation (using exponential integrability of Gaussian variables) reveals that the sequence \(f_k := \varPi _{I} h_k\) fulfills the conditions of the lemma. \(\square \)

### 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 \(L_1\) given in (14).

**Proposition 14**

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**

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 \(Ext^c\).

Let us now consider a general canonical function \(f \in \mathcal {P}^{}\). We put \(h(x) := f(2n x)\). By Lemma 13 we may find a sequence of canonical functions \(\left\{ f_k \right\} _k \subset \mathrm{span}(A)\) such that \(f_k(2n\cdot ) \rightarrow h\) in \(\mathcal {P}^{}\). Now by Proposition 15, we may approximate \(|X_t|^{-(n/2)} U^n_t(f)\) with \(|X_t|^{-(n/2)} U^n_t(f_k)\) uniformly in \(t>1\). This together with Proposition 14 and standard metric-theoretic considerations concludes the proof.\(\square \)

### 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**

*Proof of Theorem 4*

As in the subcritical case, we will focus on the third coordinate.

Now we pass to general canonical functions \(f\in \mathcal {P}^{}\). By Lemma 13, we can approximate \(f\) by canonical \(f_k\) from \(\mathrm{span}(A)\) in such a way that \(\Vert f - f_k\Vert _{\mathcal {P}^{}} \le \Vert f(2n\cdot ) - f_k(2n\cdot )\Vert _{\mathcal {P}^{}} \rightarrow 0\). Thus, by Proposition 16, the law of \((t|X_t|)^{-1/2}U_t^n(f_k)\) converges to the one of \((t|X_t|)^{-1/2}U_t^n(f)\) as \(k \rightarrow \infty \) uniformly in \(t>1\). Moreover by the fact that \(L_2\) is bounded on \(L_2(\mathbb {R}^{nd},\varphi ^{\otimes n})\) and there exists \(C<\infty \) such that \(\Vert \cdot \Vert _{L_2(\mathbb {R}^{dn},\varphi ^{\otimes n})} \le C\Vert \cdot \Vert _{\mathcal {P}^{}}\) (which follows easily from exponential integrability of Gaussian variables), we obtain \(L_2(f_k) \rightarrow L_2(f)\) in the space \(L_2(\varOmega ,\mathcal {F},\mathbb {P})\). The proof may now be concluded by standard metric-theoretic arguments.\(\square \)

### 5.7 Large Branching Rate: Proof of Theorem 8

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**

*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 \(n=1\)).

## 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 Ornstein–Uhlenbeck process, and then we introduce certain auxiliary combinatorial structures which will play a prominent role in the proofs.

### 6.1 Auxiliary Facts on the Ornstein–Uhlenbeck Process

### 6.2 Smoothing Things Out

It is well known that the Ornstein–Uhlenbeck semigroup increases the smoothness of a function. We will now introduce some simple auxiliary lemmas which quantify this statement and give bounds on the \(\Vert \cdot \Vert _{\mathcal {P}^{}}\) norms of derivatives of certain functions obtained from \(f\) by applying the Ornstein–Uhlenbeck semigroup on a subset of coordinates. Such bounds will be useful, since they will allow us to pass in the analysis to smooth functions.

**Lemma 18**

*Proof*

To prove (34), it is enough to take the maximum over all admissible pairs \(I,\varLambda \).

We will also need the following simple identity. We consider \(\left\{ x_i \right\} _{i=1,2,\ldots ,n}, \left\{ \tilde{x}_i \right\} _{i=1,2,\ldots ,n}\). By induction one easily checks that the following lemma holds (we slightly abuse the notation here, e.g., \(\frac{\partial }{\partial y_i}\) denotes the derivative in direction \(\tilde{x}_i - x_i\) and \(\int _a^b\) the integral over the segment \([a,b] \subset \mathbb {R}^d\)).

**Lemma 19**

### 6.3 Bookkeeping of Trees

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.

**Proposition 20**

*Proof*

The calculations will be more tractable when we derive an explicit formula for \(\left\{ Y_i \right\} _{i\in l(\tau )}\). Let us recall the notation introduced in (32) and consider a family of independent random variables \(\left\{ G_i \right\} _{i\in \tau }\), such that \(G_i \sim \varphi \) for \(i\ne 0\) and \(G_0 \sim \delta _x\). Recall also that \(\mathrm{ou}(t) = \sqrt{1 - e^{-2\mu t}}\). The following proposition follows easily from the construction of the branching walk on \(\tau \) and (32).

**Proposition 21**

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**

*Proof*

(Sketch) Using Proposition 20, Proposition 1, (35) and Lebesgue’s monotone convergence theorem one may prove that (45) is valid for \(f \equiv C, C>0\). Using standard methods, we may drop the positivity assumption in (35) and (42). Therefore, by the Stone–Weierstrass theorem, linearity and Lebesgue’s dominated convergence theorem, (45) is valid for any \(f\in \mathcal {C}^{}_c(\mathbb {R}^{nd})\). Let now \(f\in \mathcal {P}^{}(\mathbb {R}^{nd}), f\ge 0\). We notice that for any \(\tau \in \mathbb {T}_{n}\) the expression \(\mathrm{OU}(f, \tau , t, \left\{ t_i \right\} _{i \in i(\tau )},x)\) is finite, which follows easily from Proposition 21. Further, one can find a sequence \(\left\{ f_k \right\} \) such that \(f_k \in \mathcal {C}^{}_c(\mathbb {R}^{nd}), f_k\ge 0\) and \(f_k\nearrow f\) (pointwise). Appealing to Lebesgue’s monotone convergence theorem yields that (45) still holds (and both sides are finite). To conclude, once more we remove the positivity condition.\(\square \)

As a simple corollary we obtain

**Corollary 23**

*Proof*

We apply the above proposition with \(f=1\). Using the definition (41) and the inequality \(|i(\tau )| \le n-1\) for \(\tau \in \mathbb {T}_{n}\), it is easy to check that for any \(t\in \mathbb {T}_{n}\) we have \(S(\tau , t, x)\le C_\tau e^{n \lambda _p t}\), for a certain constant depending only on \(\tau \) and \(p,\lambda \). \(\square \)

Let us recall the notation of (39). The following proposition will be crucial in proving moment estimates for \(V\)- and \(U\)-statistics.

**Proposition 24**

*Proof*

### 6.4 Small Branching Rate: Proofs of Propositions 14 and 15

*Proof of Proposition 14*

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 \(L_2\) bounds on \(V\)-statistics with deterministic normalization (Proposition 25), then we will pass to \(L_1\) bound of \(U\)-statistics with random normalization, restricted to the subset of the probability space, where \(|X_t|\) 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 \(\mathcal {P}^{}\) of the generating kernels (proof of Proposition 15).

**Proposition 25**

*Proof*

By Proposition 22, it suffices to show that for each \(\tau \in \mathbb {T}_{2n}\) there exist \(C,c>0\) such that for the function \(f\otimes f\) and any \(t>0\) we have \(e^{-n\lambda _p t}S(\tau , t,x) \le C\exp \left\{ c\left\| x \right\| _{ } ^{ } \right\} \left\| f \right\| _{ \mathcal {P}^{} } ^{ 2 }\).

Let us fix \(\tau \in \mathbb {T}_{2n}\) and denote by \(P_1(\tau )\) and \(P_2(\tau )\) the sets of inner vertices of \(\tau \) with, respectively, one and two children in \(s(\tau )\) [we recall (39)]. Set also \(P_3(\tau ) := i(\tau )\setminus (P_1(\tau )\cup P_2(\tau ))\).

Note that \(|P_1(\tau )| + 2|P_2(\tau )| = |s(\tau )|\) and \(\sum _{i=1}^3|P_i(\tau )| = |i(\tau )| = |l(\tau )|-1\). Thus, \(|P_1(\tau )| + 2|P_3(\tau )| = 2|l(\tau )| - 2 - |s(\tau )|=|l(\tau )| + |m(\tau )| - 2 \le 2n -2\) (recall that \(m(\tau )\) denotes the set of leaves with multiple labels). This ends the proof. \(\square \)

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 \(|X_t|^{n/2}\), we need to restrict the range of integration in the moment bound to the set on which \(|X_t|\) is relatively large. It will not be an obstacle in the proof of Proposition 15, since the probability that \(|X_t|\) is small will be negligible (on the set of non-extinction), which will allow us to pass from restricted \(L_1\) estimates to bounds on the distance between distributions.

**Corollary 26**Let \(\left\{ X_t \right\} _{t\ge 0}\) be the OU branching system with \(\lambda _p<2\mu \). There exist constants \(C, c>\) such that for any canonical \(f \in \mathcal {P}^{}(\mathbb {R}^{nd})\) and \(r \in (0,1)\) we have

*Proof*

*Proof of Proposition 15*

### 6.5 Critical Branching Rate: Proof of Proposition 16

As the proofs in this section follow closely the line of those in the subcritical case, we present only outlines, emphasizing differences.

**Proposition 27**

*Proof*

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**

Using the above corollary, we can now obtain Proposition 16 in an analogous way as we derived Proposition 15 from Corollary 26.

### 6.6 Supercritical Branching Case: Proof of Proposition 17

The proofs in this section diverge slightly from those in the critical and subcritical cases, and hence, we present more details.

**Proposition 29**

*Proof*

**Proof of Proposition 17**

The contribution from \(I = \{1,\ldots ,k\}\) (in the case \(l=0\)) also can be bounded like in Corollary 26. Namely, \(I^c = \emptyset \), so \(|V_t^{|I^c|}(\varPi _{I^c} f_J)| = |\varPi _{I^c} f_J| =| \langle \varphi ^{\otimes k},f_J\rangle | \le C\Vert f_J\Vert _\mathcal {P}^{} \le C^2 \left\| f(2n\cdot ) \right\| _{ \mathcal {P}^{} } ^{ }\) and \(\exp (-n(\lambda _p-\mu )t + \lambda _p |I| t) \le \exp (-(n-2|I|)(\lambda _p-\mu ))\le 1\), which easily gives the desired estimate.\(\square \)

## 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 \(\varPi _k :\mathcal {P}^{}(\mathbb {R}^{nd}) \rightarrow \mathcal {P}^{}(\mathbb {R}^{kd})\) is continuous, give the following

**Corollary 30**

Let \(\{X_t\}_{t\ge 0}\) be the OU branching system starting from \(x\in \mathbb {R}^d\). Assume that \(\lambda _p <2\mu \) and let \(f \in \mathcal {P}^{}(\mathbb {R}^{nd})\) be symmetric and degenerate of order \(k-1\). Then conditionally on \(Ext^c, |X_t|^{-(n-k/2)}U_t^n(f - \langle f, \varphi ^{\otimes n}\rangle )\) converges in distribution to \(\left( {\begin{array}{c}n\\ k\end{array}}\right) L_1(\varPi _k f)\).

Similar results can be derived in the remaining two cases. Using the fact that on the set of non-extinction \(|X_t|\) grows exponentially in \(t\), we obtain

**Corollary 31**

Let \(\{X_t\}_{t\ge 0}\) be the OU branching system starting from \(x\in \mathbb {R}^d\). Assume that \(\lambda _p =2\mu \) and let \(f \in \mathcal {P}^{}(\mathbb {R}^{nd})\) be symmetric and degenerate of order \(k-1\). Then conditionally on \(Ext^c, t^{-k/2}|X_t|^{-(n-k/2)}U_t^n(f - \langle f, \varphi ^{\otimes n}\rangle )\) converges in distribution to \(\left( {\begin{array}{c}n\\ k\end{array}}\right) L_2(\varPi _k f)\).

Similarly, using (8) and the definition of \(W\) we obtain

**Corollary 32**

Let \(\{X_t\}_{t\ge 0}\) be the OU branching system starting from \(x\in \mathbb {R}^d\). Assume that \(\lambda _p >2\mu \) and let \(f \in \mathcal {P}^{}(\mathbb {R}^{nd})\) be symmetric and degenerate of order \(k-1\). Then conditionally on \(Ext^c, \exp (-(\lambda _pn - \mu k)t)U_t^n(f - \langle f, \varphi ^{\otimes n}\rangle )\) converges in probability to \(\left( {\begin{array}{c}n\\ k\end{array}}\right) W^{n-k}L_3(\varPi _k f)\).

Since in all the corollaries above the normalization is strictly smaller than \(|X_t|^n\), they in particular imply that \(|X_t|^{-n} U_t^n(f - \langle f,\varphi ^{\otimes n}\rangle ) \rightarrow 0\) in probability, which proves the second part of Theorem 9 (as announced in Sect. 5.3).

## Notes

### 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.

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