Stein’s method of exchangeable pairs in multivariate functional approximations

: In this paper we develop a framework for multivariate functional approximation by a suitable Gaussian process via an exchangeable pairs coupling that satisﬁes a suitable approximate linear regression property, thereby building on work by Barbour (1990) and Kasprzak (2020). We demonstrate the applicability of our results by applying it to joint subgraph counts in an Erd˝os-Renyi random graph model on the one hand and to vectors of weighted, degenerate U -processes on the other hand. As a concrete instance of the latter class of examples, we provide a bound for the functional approximation of a vector of success runs of diﬀerent lengths by a suitable Gaussian process which, even in the situation of just a single run, would be outside the scope of the existing theory.


Introduction
In his seminal paper [64], Charles Stein introduced a method for proving normal approximations and obtained a bound on the speed of convergence to the standard normal distribution. Later, Barbour [2] and Götze [36] developed the so-called generator approach to finding Stein's equation, which made it possible to study approximations by many other probability laws. As a result, in [3], the method was adapted to approximations by the (infinite-dimensional) Wiener measure.
Moreover, the exchangeable-pair approach, first developed by Stein in his monograph [65] in the context of univariate normal approximations, has been at the heart of many results proved using Stein's method. It was extended by [56] and used in the context of non-normal approximations in [14,15,26,29,57]. The publication of [16,47,53] brought a breakthrough in the understanding of the exchangeable-pair approach and made it available for applications to a wide array of multivariate normal approximation problems. The very recent paper [25] developed a functional analytic approach that provides a substantial extension of the method of exchangeable pairs and, in particular, makes it possible to dispense with the linear regression property in finite-dimensional settings. In [42] the method was applied to the study of functional limit results and approximations by univariate Gaussian processes, using the setup of [56,65] and [3].
In this paper we combine the functional approximation of [3] and the multivariate exchangeable-pair method of [47,53]. We obtain an abstract approximation theorem, which is applied in the context of weighted degenerate U-statistics, a particularly interesting example of which are homogeneous sums. The strength of the abstract approximation result is also presented in a random-graph-theoretic application.

Motivation
We are motivated by examples of multivariate quantities whose distance from the normal distribution can be established using Stein's method of exchangeable pairs, and whose functional equivalents have not been studied yet. Functional limit results play an important role in applied fields. Scaling limits of discrete processes can be studied using stochastic analysis and are often more robust to changes in the local details than the discrete processes themselves. That is why researchers often choose to describe discrete phenomena with continuous models. The error they make by doing this is measured by rates of convergence in functional limit results. The current paper contributes to solving the problem of bounding those rates.
The two main applications motivating the paper and considered therein are a continuous Gaussian-process approximation of a rescaled weighted U-statistic and the study of an Erdős-Renyi random graph process. U-statistics are central objects in the field of mathematical statistics. Due to their appealing properties, they have found numerous applications to estimation, statistical testing and other problems. They appear in decompositions of more general statistics into sums of terms of a simpler form (see, e.g. [62,Chapter 6] or [60] and [67]) and play an important role in the study of random fields (see, e.g. [18,Chapter 4]). Moreover, functional limit theorems for rescaled U-statistics have found applications in the field of changepoint analysis (see e.g. [22,23,31,32,34,35,38,52]), where it is particularly useful to know the functional limits of the related test statistics. On the other hand, the Erdős-Renyi random graph model has found numerous applications in various fields (see [13]), including epidemic modelling [1] and modelling of evolutionary conflicts [12].
The first application discussed in the paper deals with the approximation of so-called weighted U -processes, i.e. process analogues of the class of weighted U -statistics. This class of processes is very wide, containing the so-called homogeneous sum processes as well as symmetric, degenerate (complete or incomplete) U -processes. We derive a general result and successfully apply it to the case of homogeneous sum processes in Subsection 5.5. As a concrete example, in Subsection 5.6, we provide a bound for a Gaussian approximation of a process that is defined as a vector of success runs of different lengths. For functional limit theorems involving the class of symmetric, degenerate U -processes, we refer the reader to the recent paper [27]. Moreover, we remark that, even in the univariate case of weighted U -statistics, the literature about limit theory for these random quantities is quite restricted. Indeed, apart from the abundance of references on limit theorems for homogeneous sums, the majority of articles focus on the limiting behavior of so-called reduced or incomplete U -statistics, i.e. weighted U -statistics whose weights only assume the values 0 and 1 (see e.g. [9,11,39]). Limit theorems for general weighted U -statistics can be found in references [46,51,56]. We stress, however, that the last two references focus on non-normal limiting distributions and that, in the degenerate case, [56] only considers kernels of order 2. Moreover, the literature about functional central limit theorems (FCLTs) for weighted U -statistics is even scarcer. Indeed, only for homogeneous sum processes [6,48] have we been able to find comparable results in the literature. We defer a discussion and comparison with our findings to Subsection 5.5. The second example comes originally from [40] and was studied using exchangeable pairs in a finite-dimensional context in [54]. We look at a (dynamic) Erdős-Renyi random graph with nt vertices, where t denotes the time, and study the distance from the asymptotic distribution of the joint law of the number of edges and the number of twostars. Our approach can, however, be also extended to cover the number of triangles. Those statistics are often used when approximating the clustering coefficient of a network and applied in conditionally uniform graph tests.

Contribution of the paper
The main achievements of the paper are the following: 1. An abstract approximation theorem (Theorem 4.1), bounding the distance between a stochastic process Y n valued in R d , for a fixed positive integer d, and a Gaussian mixture process. The estimate is derived under the assumption that that the process Y n satisfies the linear regression condition (1.1) for all f : D [0, 1], R d → R in a certain class of test functions, a random process Y n such that (Y n , Y n ) is an exchangeable pair, some Λ n ∈ R d×d and some random variable R f = R f (Y n ). In (1.1) (and in the entire paper) Df denotes the Fréchet derivative of f . The class of test functions, with respect to which the bound in non-degenerate U -processes. In order to study such examples using our theory, one may decompose the given U -process into the vector of its degenerate Hoeffding components and prove a multivariate Gaussian limit theorem for this vector. Then, by applying a linear functional, one obtains a Gaussian limit for the original process.
This strategy, in a quantified fashion, is exemplified by the application to the rruns process, discussed in Subsection 5.6. We stress that, even in the case of just one r-run process, the results about univariate functional approximations via exchangeable pairs from [42] would not be sufficient to obtain a Gaussian approximation. Thus, in this example, the multidimensionality of our approach proves to be absolutely vital. Moreover, both the kernels and the coefficients of the weighted U -processes we study in our general result may (and will in most cases) depend on the sample size n, hence yielding Gaussian limits even in degenerate situations. At the same time, our methods are flexible enough in order to yield bounds for the classical results on asymptotic Gaussianity, in non-degenerate situations, when the kernels are fixed.
3. A novel quantitative functional limit theorem for the edge counts and the number of two-stars in an Erdős-Renyi random graph G(n, p) on n vertices with fixed edge probability p. Letting I i,j , for i, j = 1, · · · , n be the indicator that edge (i, j) is present in the graph, we consider the following statistics: corresponding to the number of edges and the number of two-stars, respectively. Theorem 6.2 provides a bound on the distance between the law of the process and the law of a piecewise constant Gaussian process. Theorem 6.4 estimates the distance between the law of (1.2) and that of a continuous Gaussian process. These results extend the result of [42] bounding the distance between the distribution of the edge counts and a univariate Gaussian process. As a corollary to our results, we immediately obtain weak convergence of the law of (1.2) in the Skorokhod and uniform topologies on the Skorokhod space to that of the continuous Gaussian process.
for some λ > 0. It follows from this assumption that and so Therefore, using Taylor's theorem, it can be proved that which provides a bound on the quantity (1.3) for ν = L(W ) and A being the canonical Stein operator corresponding to the standard normal law. A multivariate version of the method was first described in [16] and then in [53]. In [53], for an exchangeable pair of d-dimensional vectors (W, W ), the following condition is used: for some invertible matrix Λ and a remainder term R. The approach of [53] was further reinterpreted and combined with the approach of [16] in [47]. Extending this multivariate version of the exchangeable-pair method to multivariate functional approximations, with the linear regression condition taking form similar to (1.5), is the subject of the current paper.
The ones belonging to the first group, containing [3,5,[41][42][43], all use, adapt and extend the setup of [3]. Therein, the author studied the rate of convergence in the celebrated functional central limit theorem, also called Donsker's theorem. Barbour considered test functions g acting on the Skorokhod space D ([0, 1], R) of càdlàg realvalued maps on [0, 1], such that g takes values in the reals, does not grow faster than a cubic, is twice Fréchet differentiable and its second derivative is Lipschitz. For each function g belonging to this class he provided a bound on the absolute difference between the expectation of g with respect to the law of a rescaled random walk and the expectation of g with respect to the Wiener measure. Crucially, he also proved that this class of functions g is so rich that his bounds imply weak convergence with respect to the Skorokhod topology of the considered rescaled random walk to Brownian Motion. This last property is vital for most applications of the limit theory for stochastic processes and may even be the main reason for the outstanding popularity of the Skorohod topology. Indeed, by means of the continuous mapping theorem, limit theorems for many natural, non-linear functionals such as the supremum over time, immediately follow from a weak limit theorem in the Skorokhod topology.
On the other hand, the results of the second group of references, containing [7,[19][20][21], develop Stein's theory on a Hilbert space using a Besov-type topology. The bounds obtained therein, however, do not imply weak convergence in the Skorokhod topology. Therefore, the continuous mapping theorem does not apply in their setting. For instance, as opposed to the results of the first group of references, one cannot study convergence of the supremum of a process using the analysis of the second group of papers.
Finally, [63] develops approximations by abstract Wiener measures on a real separable Banach space and [10] proves bounds on measure-determining distances from Gaussian random variables valued in Hilbert spaces. As for the second group, despite the elegant abstract theory used and developed in these references, the results do not imply convergence in the Skorokhod topology on D[0, 1].
In the current paper we shall follow the setup of the first group of references. We consider it more flexible than the one of the second group and more suited for applications to processes belonging to the widely-used (non-separable) Skorokhod space than the ones of the third group.
In the context of these three groups of references and the present paper, we also mention the recent paper [27] which, although not relying on functional approximation by Stein's method, provides functional limit theorems for the class of (degenerate and non-degenerate) symmetric U -processes with a kernel that may depend on the sample size n. Since it implicitly relies on a multivariate Gaussian limit theorem derived by Stein's method from [30], it is also naturally related to Stein's method.
Moreover, since one main class of applications in the present paper involves weighted U -processes, it is worthwhile to compare our results and their applicability to those of [27]. Firstly, as mentioned above, the paper [27] focuses on Gaussian limit theorems for symmetric U -processes, which constitute a narrower class than the weighted Uprocesses considered in the present work. Moreover, thanks to the finite-dimensional convergence results from [30], the conditions for convergence from [27] are phrased in term of L 2 -norms of contraction kernels and, as such, can be considered as fourth moment conditions. In contrast, as can be seen from the bounds and proofs of Section 5, the bounds and conditions in the present paper involve third moment quantities. This distinction is also clearly reflected in the respective applicability of the results proved in the present paper and those from [27]. Indeed, whereas the symmetric U -processes considered in [27] possess a global dependency structure, the results in Section 5 are most useful whenever the dependence of the weighted U -process is local in the sense that the involved array of weighting coefficients (a J ) J is sparse in some sense. The runs example in Subsection 5.6 provides an instructive showcase for this observation. Moreover, the methods used in the proofs of the main results necessitate that the quantities in the bounds involve the absolute values of both the kernels and the coefficients. Hence, no cancellation effect, typically occuring under fourth moment conditions, may be relied on in this case. We therefore consider our theorems as rather complementary to the ones in [27].

Structure of the paper
Section 2 includes some introductory remarks about notation and the spaces of test functions with respect to which bounds on distances between probability laws in this paper will be derived. Section 3 gives a general form of the pre-limiting process to which all the processes of interest will be compared using Stein's method. It also presents the corresponding Stein equation, its solution and the smoothness properties of the solution. Section 4 contains the main abstract result of this paper providing a bound on the distance between a process valued in the Skorokhod space D([0, 1], R d ) and the prelimiting process described in the previous section. Section 5 discusses the application of the abstract theorem to weighted, degenerate U-statistics and presents a bound on their distance from a continuous Gaussian process. It furthermore explains how the bound simplifies in the context of homogeneous sums and applies it to the example of r-runs on the line. Section 6 discusses the example concerning an Erdős-Renyi random graph process and the bound on the distance between the number of its edges and two-stars and a continuous Gaussian process. Technical details of some of the proofs in this paper are postponed to Section 7.
2 Notation and spaces M and M 0 The following notation, similar to the one of [3] and [43], is used throughout the paper. For a fixed positive integer d, let D([0, 1], R d ) be the Skorokhod space of càdlàg R d -valued functions on [0, 1]. For i = 1, · · · , d, by e i we denote the ith unit vector of the canonical basis of R d . The ith component of any x ∈ R d will be denoted by x (i) , so that x = x (1) , · · · , x (d) . For a function w defined on [0, 1] and taking values in a Euclidean space, we will also write w = sup where | · | denotes the Euclidean norm. Moreover, the notation E W [ · ] will be used to represent E[ · |W ].
Furthermore, we define and let L be the Banach space of continuous functions f : By D k f we will always mean the k-th Fréchet derivative of f . The norm · of a k-linear form B on L will be taken to be As in [3], we define M ⊂ L as a subspace of L consisting of the twice Fréchet differentiable functions f , such that: for some constant k f , uniformly in w, h ∈ D([0, 1], R d ). We have following lemma (whose proof we omit), which may be proved in an analogous way to that used to show (2.6) and Then, for all f ∈ M , we have f M < ∞.
We, furthermore, let M 0 be the class of functionals g ∈ M such that: for each g ∈ M 0 and if T n log 2 (1/r n ) n→∞ − −−− → 0, then the law of Y n converges weakly to that of Z in D([0, 1], R d ), in both the uniform and the Skorokhod topologies.

Setting up Stein's method for the pre-limiting approximation
We set up Stein's method in a fashion similar to [3] and [43]. First, we define the process D n whose distribution will be treated as the target measure. We then construct a process (W n (·, u) : u ≥ 0) for which the target measure is stationary. We subsequently calculate its infinitesimal generator A n and take it as our Stein operator. Next, we solve the Stein equation A n f = g, using the analysis of [44], and prove several smoothness properties of the solution f n = φ n (g).

Stein equation
The following result follows immediately from [44, Propositions 4.1 and 4.4]: Proposition 3.2. The infinitesimal generator of the process (W n (·, u)) u≥0 acts on any f ∈ M (for M defined in Section 2) in the following way: Moreover, for any g ∈ M such that Eg(D n ) = 0, the Stein equation A n f n = g is solved by: for any constant function c : Remark 3.3. The fact that the process (W n (·, u)) u≥0 is built using Ornstein-Uhlenbeck processes and that the corresponding semigroup T n,u takes the convenient form, coming from Mehler's formula, plays an important role in the proof of Proposition 3.2. It is not clear to us whether this result can easily be extended beyond this context.

An abstract approximation theorem
The following result provides an expression for a bound on the distance between a process Y n and D n , defined by (3.1). It assumes that we can find some Y n such that (Y n , Y n ) is an exchangeable pair satisfying an appropriate condition. We explain in , for all f ∈ M , some Λ n ∈ R d×d and some random variable . Let D n be defined by (3.1). Then, for any g ∈ M : Proof of Theorem 4.1. We will bound |Eg(Y n ) − Eg(D n )| by bounding |EA n f (Y n )|, where f is the solution to the Stein equation: for A n defined in Proposition 3.2. Note that, by exchangeability of (Y n , Y n ) and (4.1): and so: Therefore: where the last inequality follows by Taylor's theorem and Proposition 3.2.

Weighted, degenerate U -statistics
In this Section we will apply Theorem 4.1 in order to prove bounds for the approximation of a vector of weighted, degenerate U -processes by suitable Gaussian processes.

Introduction
The setup will be the following. We fix positive integers d, p 1 , . . . , p d and consider a sequence (X i ) i∈N of i.i.d. random variables with distribution µ on some measurable We denote by D p (n) the collection of p-subsets of the set [n] := {1, . . . , n} (if p > n, we set D p (n) = ∅).
Furthermore, we fix an integer n ≥ max(p 1 , . . . , p d ) and let {a J (i) : 1 ≤ i ≤ d, J ∈ D pi (n)}, be a (given) set of real numbers (weights). We further let {σ n (i) : 1 ≤ i ≤ d} be a set of positive real numbers and, for t ∈ [0, 1], define i.e. equal to the variance of the sum in the definition of Y (i) n (1). This is, however, not necessary for our results. For fixed t (in particular for t = 1), the quantity Y (i) n (t) is customarily referred to as a degenerate, weighted U -statistic based on X 1 , . . . , X nt and, thus, we coin the whole random function Y (i) n a degenerate, weighted U -process. Limit theorems (not necessarily central) for such weighted U -statistics have been derived in [46,51,55,56] and in the (somehow) more special case of incomplete U -statistics in [9,11,39]. However, we have not been able to find FCLTs for degenerate, weighted U -process in the literature.
With the above definitions, we let , which is, as one can easily observe, an element of D([0, 1], R d ). We will write X := (X 1 , . . . , X n ) and construct an X := (X 1 , . . . , X n ) such that the pair (X, X ) is exchangeable. Specifically, we let X 0 be another random variable with distribution µ and let I be uniformly distributed on [n] in such a way that I, X 0 , (X j ) j∈N are jointly independent. For 1 ≤ j ≤ n, we let Therefore condition (4.1) is satisfied for Λ n of (5.1) and R f = 0. In what follows we will assume that 1 ≤ p 1 ≤ p 2 ≤ · · · ≤ p d .

A pre-limiting process
We will construct a pre-limiting Gaussian process D n of the form (3.1) which has the same covariance structure as Y n . We take D n = D where, for i = 1, . . . , d and J ∈ D pi (n), Z J (i) are jointly Gaussian random variables that are independent of X and satisfy Multivariate functional Stein's method of exchangeable pairs

Distance from the pre-limiting process
Having established the setup and defined the pre-limiting process above, we prove the following result: Theorem 5.1. Let Y n be defined as in Section 5.1 and D n be defined as in Section 5.2. Then, for any g ∈ M , Proof.
Step 1. First note that, for 1 in Theorem 4.1, which follows directly from the definition of Λ n in (5.1) and our assumption that Furthermore, for max(J) := max{j : j ∈ J} and for all i = 1, . . . , d: Step 2. We will now bound 2 of Theorem 4.1. Denoting by e i the ith element of the canonical basis of R d , for i = 1, . . . , d, for any f ∈ M , we have We now let f = φ n (g), as defined by (3.3), and fix some i, j ∈ {1, . . . , d}. We have that Then, using independence, from (5.9) we obtain that Finally, (5.8) and (5.11) imply that

Distance from a continuous process
We now prove the following theorem, which bounds the distance between the law of Y n and that of a continuous Gaussian process. Let us introduce some notation first.
Let Σ (m) n ∈ R d×d be given by . . , n}, and and, for any g ∈ M 0 , Brownian motions in R.
Step 1. Consider process D n defined in Section 5.2. Note that, for i = 1, . . . , d, } is a jointly Gaussian collection of centred random variables with the following covariance structure: Using this observation, note that D n has the same distribution asZ n given bỹ Step 2. By Doob's L 2 inequality and Itô's isometry, we note that E sup Similarly, by Doob's L 3 inequality, the formula for Gaussian moments and Itô's isometry, (5.14) Step 3. We now apply an argument similar to that of [33,Theorem 1]. Note that is a martingale vanishing at zero. In particular, so are the coordinate processes Step 3. Using the calculations above, we note that We furthermore note that, using Doob's L 3 inequality, the formula for Gaussian moments and Itô's isometry, Therefore, using the mean value theorem The result now follows by Theorem 5.1 and the triangle inequality.
Remark 5.3. The approximation results in this Section are merely stated for vectors of degenerate weighted U -processes. In many applications, however, the given weighted U -process might involve non-degenerate kernels. If is such a non-degenerate, weighted U -process, then it can be written in its Hoeffding decompoition as a sum of degenerate, weighted U -processes as follows: where the kernels ψ q , 1 ≤ q ≤ p, are degenerate kernels which are expressible in terms of ψ. Hence, the results of this Section for the vector (U n ) together with the application of a linear functional immediately yield bounds on the approximation of U n by a suitable Gaussian process. For simplicity we do not state the resulting bounds explicitly but leave their derivation to the interested reader. In the very particular example of d-runs on the line, however, we will work out this procedure in full detail.

Homogeneous sum processes
In this subsection we consider an important subclass of weighted, degenerate Uprocessess, namely the processes given as so-called homogeneous sums or homogeneous sum processes. In this case, the random variables X i , i ∈ N, are real-valued such that E|X 1 | 3 < ∞, E[X 1 ] = 0 and E[X 2 1 ] = 1. Moreover, for each 1 ≤ i ≤ d, the kernel ψ(i) is given by In particular, ψ(i) does not depend on n. Hence, for 1 ≤ i ≤ d and t ∈ [0, 1] we have that where the σ n (i) are positive reals and, in this special case, the random variables Z J (i) making up the processes D (i) n , defined in Subsection 5.2, are standard normally distributed. In this situation we have the following results, which are direct consequences of Theorems 5.1 and 5.2, respectively. Then, for any g ∈ M , |Eg(Y n ) − Eg(Z)| ≤ g M (γ 1 + γ 2 + γ 3 + γ 4 + γ 5 ) and for any g ∈ M 0 ,

Corollary 5.4. With the above definitions and notation we have that
Remark 5.6.
1. In the case p = 2 the array (a J ) J∈D2(n) := (a J (1)) J∈D2(n) may be identified with the (symmetric) matrix A = (a i,j ) 1≤i,j≤n , where a i,i = 0 and a i,j = a j,i for all 1 ≤ i, j ≤ n. Many papers [24,37,48,50,59] have established sufficient conditions for the (univariate) CLT to hold for Y n := Y n (1) in this case (with the choice of σ 2 n = 1≤i =j≤n a 2 i,j ). Remarkably, in [50] the authors prove a universality principle for homogeneous sums of any order p ≥ 1. In other words, they find necessary and sufficient conditions on the coefficient functions for the asymptotic normality of Y n to hold in the case when the X j 's are i.i.d. standard Gaussian. They also show that these conditions imply asymptotic normality of Y n for any possible choice of the distribution of the X j 's, as long as the X j 's are independent and the usual moment assumptions hold. Now concentrating on p = 2 and letting λ * n := max{|λ| : λ eigenvalue of A} , for the matrix A introduced above, a well-known sufficent condition (see, e.g. [48, Theorem 1.1]) for Y n , n ∈ N, to be asymptotically normal is that lim n→∞ λ * n /σ n = 0 (under our standing assumption that E|X 1 | 3 < ∞). The well-known inequalities (see e.g. [37]) imply that this condition in particular implies the Lindeberg type condition lim n→∞ ρ 2 n /σ 2 n = 0, which roughly says that the asymptotic influence of every individual X i vanishes. On the other hand, it is implied by the stronger (and maybe easier to verify) condition that lim n→∞ Γ n /σ n = 0. We remark that the sufficient condition provided by [50] for d = 2 reduces to lim n→∞ Tr(A 4 )/σ 4 n = 0, which is easily seen to be equivalent to lim n→∞ λ * n /σ n = 0. Here Tr(B) = n i=1 b i,i denotes the trace of a matrix B = (b i,j ) 1≤i,j≤n .
From the easy to derive inequality we conclude that, in the univariate case, the condition γ 1 → 0 as n → ∞, which follows from our bound in Corollary 5.5, is also stronger than the Lindeberg condition. The Lindeberg condition is, however, neither necessary (consider e.g. Y n = (n − 1) −1/2 X 1 n j=2 X j where the X j are i.i.d symmetric Rademacher random variables) nor sufficient for the asymptotic normality of the Y n . Hence, by the above inequality, also the sufficient condition lim n→∞ λ * n /σ n = 0 is not necessary for asymptotic normality to hold. We now provide upper bounds on the quantities γ 1 and γ 2 from our bound in this special case. First note that where [n] p = denotes the collection of all (i 1 , . . . , i p ) ∈ [n] p such that i k = i l whenever k = l. We have Hence, there is an absolute constant C 1 such that The second term γ 2 in our bound in this case is of the same order as where, for positive sequences, we write b n d n if there are 0 < c < C < ∞ such that cb n < d n < Cb n for all sufficiently large n. Note that we have Thus, there is another absolute constant C 2 such that In particular, we obtain the asymptotic normality of Y n = Y n (1) under the assumption that which is a stronger condition than λ * n = o(σ n ). However, if additionally the terms γ 3 , γ 4 and γ 5 in Corollary 5.5 converge to zero, we can conclude the much stronger result that the whole process Y n converges to a continuous Gaussian process on [0, 1].

The literature around FCLTs for homogeneous sum processes is non-void but
nevertheless extremely scarce. Indeed, the only references we have found, whose results might compare to ours (in the one-dimensional case) are [48] and [6], of which [48] only considers quadratic forms, i.e. the case p = 2. It turns out that comparing our results to those in [48] (for p = 2) and to those in [6] is complicated. Indeed, [48,Theorem 1.6] states the FCLT for the quadratic from Y n under the (additional) assumption that Ã −2 Ã TÃ → 0 as n → ∞, where · denotes the Frobenius norm of a matrix and whereÃ = (ã i,j ) 1≤i,j≤n has entriesã i,j = a i,j 1 {i>j} . Thus, the matrix C :=Ã TÃ has entries c i,j = n k=(i∨j)+1 a i,k a k,j and, hence, its only specifies the one-dimensional distributions of ξ k but not its covariance function.
A similar setup was considered in [53], where the authors studied the rate of the (finite-dimensional) weak convergence of the law of V (r) n (1) to the normal distribution.
The authors of [53] note that the standard exchangeable-pair construction of [56] does not lead to a bound going to zero as n → ∞. In order to solve this problem, they apply their embedding method and study the joint convergence of V (1) n (1), . . . , V (r) n (1) to a multivariate normal law, using a slightly unusual construction of the exchangeable pair. Our propositions in this subsection provide bounds on the rate of the functional to a Gaussian process for any collection {r 1 , . . . , r d }.
They implicitly use the standard exchangeable-pair construction of Subsection 5.1. Our bounds are of the same order as the bound on the rate of the (finite-dimensional) convergence provided in [53]. We start with the following result on the pre-limiting approximation: Proposition 5.7. Adopt the notation from above. Let d ≥ 1 and n 2 > r 1 ≥ r 2 ≥ · · · ≥ r d ≥ 1. Let Proof.
Next, we deal with the continuous process approximation as given in Corollary 5.5.
For this, we need to either compute or estimate the quantities δ A simple computation shows that, for q > 1 and r i ∧ r l ≤ m ≤ n + 1 − r i ∧ r l , Note that, for ϕ(s) ≡ Σ 1/2 and ϕ n (s) = n m=1 Σ Moreover, with obvious notation, Furthermore, for q > 1, Therefore, for all q = 1, . . . , r 1 , Thus, taking (5.19) into account, we note that Hence, using Corollary 5.5 and Proposition 5.7 (and noting that reordering the arguments of function f does not change the bound on g • f M 0 obtained in Lemma 7.1), we obtain the following result: Proposition 5.8. Adopt the notation form above. In particular, let N be as in (5.21), V n be defined as in Proposition 5.7 and Σ be the block diagonal matrix with blocks Σ(1) ∈ R N (1)×N (1) , . . . , Σ(r 1 ) ∈ R N (r1)×N (r1) defined by (5.22) and (5.23).
Then, for any g ∈ M 0 , we have where γ 1 and γ 2 are as in Proposition 5.7 and for the larger class of test functions M . It would, however, require some more involved computations, which would make the discussion of this example rather long.

Edge and two-star counts in Erdős-Renyi random graphs
In this section we study an Erdős-Renyi random graph with a fixed edge probability p and nt edges for t ∈ [0, 1]. We analyse the asymptotic behaviour of the joint law of its (rescaled) number of edges and its (rescaled) number of two-stars (i.e. subgraphs which are trees with one internal node and 2 leaves). Hence, we extend the result of [42], where the univariate process convergence of the rescaled number of edges is studied. We also extend the analysis of [54], whose authors provide a bound on the distance between the (three-dimensional) joint law of the (rescaled) number of edges, two-stars and triangles in a G(n, p) graph and a Gaussian vector. In Theorem 6.2, we establish a bound on the distance between our process and a pre-limiting Gaussian processes with paths in D([0, 1], R 2 ). Then, in Theorem 6.4, a bound on the quality of a continuous Gaussian process approximation is provided. It is worth noting that the analysis of a three-dimensional process representing the number of edges, triangles and two-stars in a G( nt , p) graph does not pose any additional challenges except that it makes the algebraic computations more involved. The only reason we do not do it here is that it would make this section rather lengthy.

Introduction
Consider an Erdős-Renyi random graph G( nt , p) on nt vertices, for t ∈ [0, 1], with a fixed edge probability p. Let I i,j = I j,i 's be i.i.d. Bernoulli (p) random variables indicating that edge (i, j) is present in this graph. We consider the following process, representing the re-scaled total number of edges and a re-scaled statistic related to the number of two-stars V n (t) = 1 6n 2 1≤i,j,k≤ nt i,j,k distinct The scaling therefore ensures that the covariances are of the same order in n.

A pre-limiting process
Suppose that the collection {Z (1) i,j : i, j ∈ [n], i < j} ∪ {Z (2) i,j,k : i, j, k ∈ [n], i < j < k} is jointly centred Gaussian with the following covariance structure: , if (i, j, k) = (r, s, t), n ) be defined in the following way: Note that the covariance structure of the collection {Z i,j,k : i, j, k ∈ [n], i < j < k} is the same as the covariance structure of the summands in the formulas (6.1) and (6.2).

Distance from the pre-limiting process
We provide an estimate of the distance between Y n and the pre-limiting piecewise constant Gaussian process. Theorem 6.2. Let Y n be defined as in Section 6.1 and D n be defined as in Section 6.3. Then, for any g ∈ M , |Eg(Y n ) − Eg(D n )| ≤ 23 g M n −1 . Remark 6.3. Our bound in Theorem 6.2 is of the same order as the analogous bound obtained in [54] on the distance between the (finite-dimensional) distributions of Y n (1) and D n (1).
The proof is based on Theorem 4.1. In Step 1 we estimate term 1 , which involves bounding Λ n 2 of (6.3) and the third moment of Y n − Y n . In Step 2 we treat 2 , using involved computations, which are, to a large extent, postponed to the appendix.
Term 3 is equal to zero as R f of Section 6.2 is equal to zero.
Proof of Theorem 6.2. We adopt the notation of sections 6.1, 6.2, 6.3 and apply Theorem 4.1.
Step 1. First note that, for 1 in Theorem 4.1, where | · | denotes the Euclidean norm in R 2 and · 2 is the induced operator 2-norm.

Distance from the continuous process
We now study the approximation of Y n by a continuous Gaussian process with covariance equal to the limit of the covariance of D n . We obtain a bound on the quality of this approximation. This is achieved by applying Theorem 6.2 and by bounding the distance between D n and the continuous process via the Brownian modulus of continuity. Theorem 6.4. Let Y n be defined as in Subsection 6.1 and let Z = (Z (1) , Z (2) ) be defined by: , where B 1 , B 2 are independent standard Brownian Motions. Then, for any g ∈ M : |Eg(Y n ) − Eg(Z)| ≤ g M 16422n −1/2 log n + 138n −1/2 . Remark 6.5.
Theorem 6.4, together with Proposition 2.2, implies that Y n converges to Z in distribution with respect to the Skorokhod and uniform topologies.
Remark 6.6. Theorem 6.4 can be adapted to situations in which p = p n varies with n.
More precisely, as indicated by the necessary and sufficient conditions for approximate normality of the marginal distributions given in [61], Theorem 6.4 can be modified to yield a quantitative functional CLT in the case that n 3 p 2 n → ∞ and n 2 (1 − p n ) → ∞.
In Step 1 of the proof of Theorem 6.4, we use i.i.d standard Brownian Motions to construct a process Z n having the same distribution as D n . In Step 2 we couple Z n and Z and use the Brownian modulus of continuity to bound moments of the supremum distance between them. In Step 3 we combine those bounds with the mean value theorem to obtain the desired final estimate.
Proof of Theorem 6.4.
Step 1. Let B 3 be another standard Brownian Motion, mutually independent with B 1 and B 2 . Let Z n = Z (1) n , Z (2) n be defined by: . To see this, observe that for all u, t ∈ [0, 1], n (t)Z (2) n (u). (6.9) Step 2. We now let Z be constructed as in Theorem 6.4, using the same Brownian Motions B 1 , B 2 , as the ones used in the construction of Z n . In Lemma 7.3, proved in the appendix, we obtain the following bounds: (6.10) Step 3. We note that, by (6.10): 1] Dg(Z + c(Z n − Z)) Z − Z n By noting that the mean zero Z (1) i,k and Z (1) i ,j are independent for i = i , we obtain: and so, by (7.3), where the last inequality holds because |I ij − 2pI ij + p| ≤ 1, |I ij − I ij | ≤ 1 and I jk + I ik ≤ 2 for all k ∈ {1, · · · , n}. For S 2 , let Y ijk n equal to Y n except that I ij , I jk , I ik are replaced by I ij , I jk , I ik , i.e. for all t ∈ [0, 1] let Now, by (7.5), we note that: where the second inequality follows from the fact that for all a, b, c ∈ {1, · · · , n}, |I ab − I ab | ≤ 1, (I ab + I bc ) ≤ 2 and |I ab I bc − I ab I bc | ≤ 1. Therefore, by (7.6): Similarly, for S 3 , ≤ √ 178 g M 6n . (7.9) Now, for S 4 , let Y ijkl n be equal to Y n except that I ij , I ik , I il , I jk , I jl , I kl are replaced with independent copies I ij , I ik , I il , I jk , I jl , I kl , i.e. for all t ∈ [0, 1] and for all t ∈ [0, 1] let Y ijkl n (t) = T ijkl n (t) − ET n , V ijkl n (t) − EV n (t) . Note that: + (I ij − 2pI ij + p) (I ik I il + I ik I jl + I jk I il + I jk I jl ) − 16p 3 (1 − p)  · (2pI jk + 2pI ik + I ik I il + I ik I jk + I jk I il + I jk I jl ) · Y n − Y ijkl n + g M 12n 4 1≤i<j≤n 1≤k≤n k ∈{i,j} E |(I ij − 2pI ij + p) (2pI jk + 2pI ik + I ik + 2I ik I jk + I jk )| 1≤i<j≤n 1≤k≤n k ∈{i,j} E Y n − Y ijk n . (7.11) Now, by (7.10), note that: Y n − Y ijkl n ≤ 1 n 2 (n − 2) 2 |I ij − I ij | + |I ik − I ik | + |I il − I i | + |I jk − I jk | + |I jl − I jl | + |I kl − I kl | 2 + m:m =i,j,k,l I ij − I ij (I im + I jm ) + |I ik − I ik | (I im + I km ) + |I il − I il | (I im + I lm ) + I jk − I jk (I jm + I km ) + I jl − I jl (I jm + I lm ) + |I kl − I ll | (I km + I lm ) + |I ij I jk − I ij I jk | + |I ij I ik − I ij I ik | + |I ik I jk − I ij I jk | + |I ij I jl − I ij I jl | + |I ij I il − I ij I il | + |I il I jl − I ij I jl | + |I ik I kl − I ik I kl | + |I ik I il − I ik I il | + |I il I kl − I ik I kl | + |I jk I jl − I jk I jl | + |I jl I kl − I jl I kl | + |I kl I jk − I kl I jk | Therefore, by (7.11) and (7.7), S 4 ≤ g M · √ 180n 2 − 1008n + 1440 + √ 73n 2 − 372n + 477 3n 2 ≤ √ 612 + √ 178 g M 3n . (7.12) The result now follows by (7.6), (7.8), (7.9), (7.12).

Technical details of the proof of Theorem 6.4
Lemma 7.3. Using the notation of Step 2 of the proof of Theorem 6.4, Proof. Note the following: