High-dimensional central limit theorems for a class of particle systems *

We consider a class of particle systems that generalizes the eigenvalues of a class of matrix-valued processes, of which the empirical measures converge to deterministic measures as the dimension goes to infinity. In this paper, we obtain central limit theorems (CLTs) to characterize the fluctuations of the empirical measures around the limit measures by using stochastic calculus. As applications, CLTs for Dyson’s Brownian motion and the eigenvalues of Wishart process are recovered under slightly more general initial conditions, and a CLT for the eigenvalues of a symmetric matrixvalued Ornstein-Uhlenbeck process is obtained.


Introduction
Recently general stochastic differential equations (SDEs) on the group of symmetric matrices have attracted much interest. A prominent example is the process introduced in Graczyk and Małecki (2013) as the solution of the following matrix-valued SDE, Here, B t is a matrix-valued Brownian motion of dimension N × N , and the continuous functions g N , h N , b N : R → R act on the spectrum of X N t (a function f acts on the spectrum of a symmetric matrix X = N j=1 α j u j u j with eigenvalues (α j ) and eigenvectors (u j ) if f (X) = N j=1 f (α j )u j u j ). The matrix-valued process (1.1) extends several well-known processes such as the celebrated symmetric matrix-valued Brownian motion (Dyson, 1962), Wishart process (Bru, 1991), and the symmetric matrix-valued Ornstein-Uhlenbeck processes (Chan, 1992).
Let {λ N i (t)} 1≤i≤N be the eigenvalues of X N t . According to Theorem 3 in Graczyk and Małecki (2013), if the initial eigenvalues {λ N i (0)} 1≤i≤N are all distinct, then before the first collision time τ N = inf{t > 0 : ∃ i = j, λ i (t) = λ j (t)}, the eigenvalue processes satisfy the following system of SDEs: for 1 ≤ i ≤ N , (1.2) where {W i , 1 ≤ i ≤ N } are independent Brownian motions and G N (x, y) = g 2 N (x)h 2 N (y) + g 2 N (y)h 2 N (x). (1.3) In Małecki (2013, 2014), some other conditions on the coefficient functions were imposed to ensure that (1.2) has a unique strong solution and the collision time τ N is infinite almost surely.
Let L N (t) be the empirical measure of the eigenvalues {λ N i (t)} 1≤i≤N , that is In connection with the theory of random matrices, it is of interest to investigate possible limits of these empirical measures {L N (t), t ∈ [0, T ]} when N grows to infinity (highdimensional limits). An early result is the derivation of the Wigner semi-circle law as the only equilibrium point (with finite moments of all orders) of the equation satisfied by the limit of eigenvalue empirical measure processes in Chan (1992), where the entries of the symmetric matrix-valued processes are independent Ornstein-Uhlenbeck processes.
The results were later generalized in Rogers and Shi (1993) to the following interacting particle system Cépa and Lépingle (1997) further generalized these SDEs to (1993). High-dimensional limits for these eigenvalue SDEs appeared very recently in Song et al. (2020) and Małecki and Pérez (2019). Particularly in the former article, it was proved that under proper conditions, {L N (t), t ∈ [0, T ]} N ∈N is relatively compact in (C[0, T ], M 1 (R)) almost surely. Here M 1 (R) is the set of probability measures on R endowed with the topology induced by the weak convergence of measures.
We aim to study the fluctuations of {L N (t), t ∈ [0, T ]} of the eigenvalue SDE (1.2) around the limit {µ t , t ∈ [0, T ]} as N → ∞, up to considering a subsequence. Indeed, we shall consider the following general particle system introduced in Graczyk and Małecki (2014) which includes (1.2) as a particular case: for 1 ≤ i ≤ N , (1.6) with H N (x, y) being a symmetric function. Under proper conditions, the existence and uniqueness of the non-colliding strong solution was obtained in Graczyk and Małecki (2014), and it was shown in Song et al. (2020) that the family of empirical measures is tight almost surely, and any limit {µ t , t ∈ [0, T ]} satisfies H(x, y) (z − x)(z − y) 2 µ s (dx)µ s (dy) ds, ∀z ∈ C \ R, (1.8) where the continuous functions b(x), σ(x) and H(x, y) are the uniform limits of b N (x), σ N (x) and N H N (x, y), respectively. EJP 26 (2021), paper 87.
Up to considering a subsequence, we can assume, without loss of generality, the uniqueness of the limit process {µ t , t ∈ [0, T ]}. Consider the random fluctuations, (1.9) where f belongs to an appropriate space of test functions and for a measure µ on R.
The main purpose of the paper is to find, as N → ∞, a Gaussian limit for the centered process To our best knowledge, the problem of fluctuations for empirical measures of eigenvalue processes was first studied in Cabanal-Duvillard (2001) and later extended in Anderson et al. (2010). More precisely, Cabanal-Duvillard (2001) considered Dyson's Brownian motion and Wishart process with null initial condition, and established CLT for polynomial test functions; (Anderson et al., 2010, Theorem 4.3.20) extended the result for Dyson's Brownian motion by allowing bounded initial condition. Similar results were also obtained in Pérez-Abreu and Tudor (2007) and Perez-Abreu and Tudor (2009).
The rest of this paper is organized as follows.
In Section 2, we establish CLTs for the empirical measures of the general particle system (1.6). The space F of test functions is given by (2.1), which in general does not necessarily contain the set of all polynomials. In Section 3, we apply the results in Section 2 to obtain CLTs for the eigenvalues of Wishart process in Section 3.2, for Dyson's Brownian motion in Section 3.3, and for the eigenvalues of symmetric matrix-valued Ornstein-Uhlenbeck process in Section 3.4, respectively. Note that in these three cases, one key ingredient is the following uniform bound for the moments of the eigenvalue processes E sup (1.11) which enables us to obtain more precise CLTs for a larger class of test functions (in comparison with F given in (2.1)). For Dyson's Brownian motion, such bound was obtained in Anderson et al. (2010) by using the explicit joint density function of the eigenvalues (see Lemma 4.3.17 therein). This density approach is developed here in full detail for the Wishart case (Section 3.2). However, for more general particle systems that we consider in this paper, as the joint density functions are not available, some new tool is needed in order to derive the uniform moment bound (1.11). We thus establish in Section 3.1 a comparison principle for the particle system (1.6), which is also of interest EJP 26 (2021), paper 87. in itself. This comparison principle allows to obtain the uniform bound (1.11) and then extend the CLTs to a larger class of particle systems (Corollaries 3.7, 3.10 and 3.12).
Furthermore, due to the special structures of Wishart process, Dyson's Brownian motion, and matrix-valued Ornstein-Uhlenbeck process, we are able to directly characterize formulas (see Theorems 3.5, 3.8, 3.11 and the remarks thereafter). For Dyson's Brownian motion and Wishart process, our CLTs provide extensions of the existing results in Cabanal-Duvillard (2001) and Anderson et al. (2010), while the CLT obtained in Section 3.4 for the eigenvalue processes of matrix-valued Ornstein-Uhlenbeck process seems to be new. Finally, some useful lemmas are provided in Appendix A.

Central limit theorems
In this section, we derive a central limit theorem for the empirical measure (1.7) of the particle system (1.6).
Let C k b (R) be the space of bounded continuous real-valued functions with bounded continuous derivatives (up to the order k). Recall that the continuous functions b(x), σ(x) and H(x, y) are the uniform limits of b N (x), σ N (x) and N H N (x, y), respectively, and Q N t (f ) is defined in (1.10). We use the following space of test functions where the continuous functionσ(x) is the uniform limit of N (see also Section 3.2), the limits of the Section 3.4), and correspondinglyσ(x) = 1, b(x) = − 1 2 x and H(x, y) = 1 2 . In this case, 3.3 and 3.4, we will develop CLTs for a larger class of test functions which includes all polynomials for these three cases, respectively.
Also assume that (1.6) has a non-exploding and non-colliding strong solution, such that Then, for any k ∈ N and any f 1 , .
Recall that the limit measure µ t satisfies (1.8). For f ∈ F, under condition (2.2), one may apply the approach used in the proof of Theorem 1 in Małecki and Pérez (2019) (see also Theorem 2.2 in Song et al. (2020)) to get, noting that σ(x) ≡ 0, Thus, (2.4) and (2.7) yield (2.8) The third term on the right-hand side of (2.8) can be written as Thus, by (2.2), we have For the fourth term on the right-hand side of (2.8), Noting that both convergences √ N σ N →σ and N H N (x, x) → H(x, x) are uniform as EJP 26 (2021), paper 87.
we have that the term converges to 0 almost surely as N → ∞, uniformly in t ∈ [0, T ]. Note that in (2.9), (2.10) and (2.12), the integrand function is bounded, and hence the convergence is also in Therefore, to prove the desired result, it suffices to show that, for any k ∈ N and with covariance given by (2.3). To this end, by Lemma A.1 it suffices to prove that By the uniform convergence of The term , L N (s) ds converges to 0 a.s. and in L p for all p ≥ 1 due to the boundedness of f 1 (x) and f 2 (x) and the uniform convergence of Furthermore, the following convergence converges to t 0 f 1 f 2σ 2 , µ s ds a.s. and in L p for all p ≥ 1.
The proof is concluded.
If the particles in (1.6) (2.14) almost surely for some constant C(T ) depending on T . Then Theorem 2.2 still holds if the set F of test function is replaced by C 2 (R).
Proof. Note that by (2.14), all but finitely many terms in T ]} also has the same support. By (Rudin, 1991, 1.46 a.s. using the same argument as in the proof of Theorem 2.2. Then following the rest part of the proof, it is easy to get the result of Theorem 2.2. Remark 2.4. Under the conditions in Theorem 2.2, (2.14) yields almost sure conver- However, it is not clear whether the convergence also holds in L p for p ≥ 1.
The following corollary provides a sufficient condition for L p -convergence for p ≥ 1.
Corollary 2.5. Assume the same conditions as in Theorem 2.2. For T < ∞ and all p ≥ 1 and all N ≥ cp for some positive constant c, assume that E sup for some positive constant C(T ) which depends only on T . Furthermore, assume that (σ(x) 2 − H(x, x))f (x) and its derivative has at most polynomial growth. Then for f ∈ C 3 (R) of which the derivatives have at most polynomial growth, Proof. By the analysis in the proof of Theorem 2.2, it suffices to show for p ≥ 1 and g ∈ C 1 (R) of which the derivative has at most polynomial growth. More precisely, one can check that under conditions (2.15) and (2.16), the convergences to 0 in (2.9), (2.10) and (2.12) are uniform in L p , and hence By Markov inequality and (2.15), EJP 26 (2021), paper 87.
Remark 2.6. The proof of Corollary 2.5 clearly shows that condition (2.15) implies condition (2.14). Therefore, under (2.15), for the test function f satisfying conditions in Corollary 2.5, Q N t (f ) − N M N f (t) converges to 0 a.s. and in L p for all p ≥ 1 uniformly in t ∈ [0, T ]. As a consequence, Theorem 2.2 holds for such test functions.
High-dimensional CLTs for a class of particle systems To end this section, we provide the following linear property for the Gaussian family is actually a zero process. Thus, as the limit of the convergence in distribution, is also a zero process, which implies (2.20).

Applications
In this section, we first provide a comparison principle in Section 3.1, and then we apply our main results obtained in Section 2 to the eigenvalues of Wishart process (Section 3.2), Dyson's Brownian motion (Section 3.3) and the eigenvalues of symmetric matrix-valued Ornstein-Uhlenbeck process (Section 3.4).
For these three cases, we are able to obtain the boundedness of the moments for the empirical measures assuming proper initial conditions (see Lemma 3.4 for Wishart process, Eq. (3.30) for Dyson's Brownian motion, and Eq. (3.45) for Ornstein-Uhlenbeck process). This enables us to apply Corollary 2.5 and Remark 2.6. As a consequence, the CLT in Theorem 2.2 holds for a larger space of test functions consisting of all the functions in C 3 (R) of which the derivatives have at most polynomial growth. In particular, if we choose the space R[x] of polynomial functions as the space of test functions, we are able to obtain recursive formulas for the basis . Note that these results are more precise than the general results in Section 2, where we study the centered processes

Comparison principle
In this subsection, we provide comparison principles for SDE (1.2) and particle system (1.6), which allow us to obtain the boundedness of the eigenvalues/particles under more general initial conditions in Sections 3.2, 3.3 and 3.4.
Throughout this subsection, the dimension N is fixed and thus subscripts/superscripts are removed. Precisely, consider the following two particle systems: EJP 26 (2021), paper 87.
Note that conditions for the existence and uniqueness of a non-colliding and nonexploding strong solution to (3.1) (or (3.2)) were obtained in Graczyk and Małecki (2014). In particular, under conditions (A2) -(A5) therein, the particles will separate from each other immediately after starting from a colliding initial state, and will not collide forever.
Assume that there exists a strictly increasing function ρ : If we further assume that b i (u) ≤b i (u) for all u ∈ R, and x i (0) ≤ y i (0) a.s., 1 ≤ i ≤ N , Proof. The continuity of the functions H ij and the condition (3.3) imply that for all Hence, the drift functions , satisfy the quasi-monotonously increasing condition in Lemma A.2. In order to apply Lemma A.2 to get the desired result, we use an approximation argument to remove the singularities of the drift functions F and F . For > 0, let One can find continuous quasi-monotonously increasing functions F and F , such that they coincide with F and F in ∆ , respectively. Before time τ , both x-particles and y-particles stay in ∆ and thus satisfy (3.1) and (3.2) with drift functions F and F , respectively.
Applying Lemma A.2 to the processes x and y , we have The desired result now follows from the non-colliding property lim →0 + τ = ∞.
As a corollary of Theorem 3.1, we have the following comparison principle for SDE (1.2) of eigenvalue processes. Note that the existence and uniqueness of the non-colliding and non-exploding strong solution was obtained under proper conditions in Graczyk and Małecki (2013).

Application to eigenvalues of Wishart process
In this subsection, we discuss the limit theorem for Wishart process. As illustrated in Graczyk and Małecki (2013) and Song et al. (2020), the scaled Wishart process is the solution to (1.1) with the coefficient functions The eigenvalue processes now satisfy Hence, the eigenvalue processes are the particles in (1.6) with By (Graczyk and Małecki, 2019, Theorem 3), all the components of the solution to (3.5) are non-negative if all the components of the initial value are non-negative. Let where C N,P > 0 is a normalization constant. Then we have the following estimation on the eigenvalues. Lemma 3.3. Let ξ N = (ξ N 1 , . . . , ξ N N ) be a random vector independent of (W 1 , . . . , W N ) with (3.7) as its joint probability density function. Assume that (λ N 1 (0), . . . , λ N N (0)) is independent of (W 1 , . . . , W N ) and that there exists a constant a > 0, such that λ N i (0) ≤ aξ N i for 1 ≤ i ≤ N almost surely. Then there exists a stationary stochastic process u N (t) with initial value u N (0) = ξ N satisfying, for 1 ≤ i ≤ N and t ≥ 0, Proof. Consider the following system of SDEs, Note that the pathwise uniqueness proved in (Graczyk and Małecki, 2013, Theorem 2) is still valid if the coefficient functions depend on the time t and the corresponding conditions therein hold uniformly in t. Furthermore, the boundedness estimation and the McKean's argument in (Graczyk and Małecki, 2013, Theorem 5) is also valid when t ≥ 0. Therefore, the system of SDEs (3.8) has a unique non-colliding strong solution.
If at any time t, u N (t) has the distribution P N , then Lemma A.3 yields that d dt E[f (u N (t))] vanishes for f ∈ C 2 b (R). Since u N (0) is distributed according to P N , we can conclude that (u N (t)) t≥0 is a stationary process with marginal distribution P N .
. Then the Itô formula shows that v N (t) is a solution to (3.5) with initial value v N (0) = au N (0) = aξ N . Noting that the solution of (3.5) is non-negative and that G N (x, y) = (x + y)/N with non-negative variables satisfies condition (3.3), we can apply the comparison principle in Theorem 3.1 to obtain The proof is concluded. Proof. Noting that the probability density of u N (t) considered in Lemma 3.3 is (3.7) for all t, we can obtain the following tail probability estimation with α being a positive constant independent of N , (3.9) By Lemma 3.3 and (3.9), we have for t ≥ 0, x k y n−k (x + y)L N (s)(dx)L N (s)(dy)ds.
The limit [x] are characterized by the following properties.

The basis {L
and for n ≥ 0, t∧s 0 x n+m−1 , µ u du, n, m ≥ 1. (3.16) Proof. First, note that by Lemma 3.4 and Corollary 2.5, Q N t (x n ) defined by (1.10) converges in distribution to a centered Gaussian family {G t (x n ), t ∈ [0, T ]} n∈N with covariance given by (3.16). Furthermore, by (1.9), (1.10) and (3.6), for n ≥ −1, we have for q ≥ 1 and n ∈ N by using an induction argument on n.
To estimate the last term on the right-hand side of (3.17), we apply the Cauchy-EJP 26 (2021), paper 87.
Schwarz inequality to obtain, for 0 ≤ k ≤ n, for some constant C(n, T, q). Thus, the last term on the right-hand side of (3.17) converges to 0 in L q for q > 1, as N tends to infinity. By Markov inequality and Borel-Cantelli Lemma, one can also obtain the almost sure convergence. If we definẽ where " d =" means equality in distribution. The proof is concluded.
). With these identities and the linearity of L t (·), (3.15) can be simplified as, for n ≥ 0, Note that the case t = 1 corresponds to the classical Wishart matrix, and µ 1 is the Marchenko-Pastur law. More precisely, recalling that L 1 (1) = 0 and L 1 ( are determined recursively by (3.19).
We now study a more general particle system: (3.20) Compared to (3.5), the constant P/N is replaced by a function b N (x) that will be assumed to converge to a constant c in Corollary 3.7 below. Despite the extension being small, the system (3.20) may not have an explicit joint density function or/and stationary distribution, and hence cannot be treated in the same way as for the eigenvalues of Wishart process.
Corollary 3.7. Consider the SDEs (3.20), where b N (x) satisfies, for some constant c ≥ 1, (3.21) Assume the same initial conditions as in Theorem 3.5. Then the conclusion of Theorem 3.5 still holds.
with the initial conditions x N i (0) = y N i (0) = λ N i (0). By the comparison principle in Corollary 3.2, we have Thus, almost surely, are the empirical measures of the two particle systems (x N i (t)) 1≤i≤N and (y N i (t)) 1≤i≤N , respectively. Noting that p 1 /N and p 2 /N converge to c as N → ∞ by (3.21), we have that Lemma 3.4 holds for the two systems (3.22) and (3.23), and thus also holds for (3.20) by (3.24).
Furthermore, condition (3.21) also yields that b N (x) → c uniformly as N → ∞, and hence (3.17) still holds. Then the rest of the proof follows that of Theorem 3.5.

Application to Dyson's Brownian motion
In this subsection, we discuss the CLT for Dyson's Brownian motion. It was shown in Anderson et al. (2010); Graczyk and Małecki (2014); Song et al. (2020), the scaled symmetric matrix-valued Brownian motion X N t = (B (t) +B(t))/ √ 2N , whereB(t) is a N × N matrix-valued Brownian motion, is the solution of the matrix SDE (1.1) with the coefficient functions The system of SDEs of the eigenvalue processes, that is, Dyson's Brownian motion, is (3.25) Hence, the eigenvalue processes are the particles in (1.6) with (3.26) Here, we consider the distribution Q N on ∆ N = {x = (x 1 , x 2 , . . . , x N ) ∈ R N : x 1 < . . . < x N } with the density function (3.27) where C N is a normalization constant.
Similar to Wishart process, we can obtain the following central limit theorem.
Theorem 3.8. Let ξ N = (ξ N 1 , . . . , ξ N N ) be a random vector independent of (W 1 , . . . , W N ) with (3.27) as its joint probability density function. Assume that (λ N 1 (0), . . . , λ N N (0)) is independent of (W 1 , . . . , W N ) and that there exist constants a, b ≥ 0, such that √ for 1 ≤ i ≤ N almost surely. Besides, assume that for any polynomial f (x) ∈ R[x], the initial value L N 0 (f ) converges in probability to a random variable L 0 (f ). Furthermore, assume that for all n ∈ N, Then for any 0 < T < ∞, there exists a family of , such that for any n ∈ N and any polynomials P 1 , . . . , P n ∈ R[x], the vector-valued process [x] are characterized by the following properties.
Proof. The proof is similar to the proofs of the Wishart case (Lemma 3.3, Lemma 3.4 and Theorem 3.5), which is sketched below.
Consider the following SDE, for 1 ≤ i ≤ N , ] vanishes for any f ∈ C 2 b (R) if u N (t) has the distribution Q N given in (3.27), and hence the process u N (t) with initial value u N (0) = ξ N is stationary (see (Anderson et al., 2010, Lemma 4.3.17 ) Then v N (t) and λ N (t) solve the same SDEs (3.25), and by the comparison principle in Corollary 3.2, we have EJP 26 (2021), paper 87.
A similar argument leads to Using the tail probability estimation based on the density function (3.27) of u N i (t), Then applying Corollary 2.5 and following the approach in the proof of Theorem 3.5, we may get the desired result.
Remark 3.9. A CLT was obtained in (Anderson et al., 2010, Theorem 4.3.20) for Dyson's Brownian motion with bounded initial values. (We would like to point out that there should be a constant factor 2/β in the covariance function which equals to 2 in the real case and equals to 1 in the complex case in Anderson et al. (2010).) Thanks to the comparison principle Corollary 3.2, the CLT also applies to Dyson's Brownian motion with possibly unbounded initial values satisfying (3.28), as stated in Theorem 3.8. For example, the CLT is valid for Dyson's Brownian motion with unbounded initial value ξ N . Note that ξ N has the joint density (3.27), which is also the joint density of {λ i (1)} 1≤i≤N , assuming {λ i (t)} 1≤i≤N is a Dyson's Brownian motion starting at 0.
Similar to the Wishart case, the self-similarity of the Brownian motion implies , µ 1 when the initial value X N 0 = 0. Thus, (3.29) can be simplified as, for n ≥ 0, m+n 2 x m+n−2 , µ 1 , n, m ≥ 1.
The following Corollary extends the result of Theorem 3.8 by allowing asymptotical constant drift coefficient functions.
Corollary 3.10. Consider the following SDEs (3.33) Furthermore, assume the same initial conditions as in Theorem 3.8. Then the conclusion of Theorem 3.8 still holds with (3.29) replaced by x n , µ s ds Proof. Set c 1 = c − 1 and c 2 = c + 1. Then by (3.33), there exists N 0 ∈ N such that for N ≥ N 0 , c 1 ≤ b N (x) L ∞ (R) ≤ c 2 . Without loss of generality, we assume c 1 ≤ b N (x) L ∞ (R) ≤ c 2 for all N ≥ 1.
Consider the following two systems of SDEs:       E sup for some positive constant C(a, b, T ) depending only on (a, b, T ) and all p ≥ 1, N ≥ αp for some positive constant α. Note that (3.33) also implies that b N (x) converges to the constant c uniformly as N → ∞. Then applying Corollary 2.5 and following the approach in the proof of Theorem 3.5, we get the desired result.

Application to eigenvalues of symmetric matrix-valued OU matrix
In this subsection, we discuss the CLT for the eigenvalues of a symmetric matrixvalued Ornstein-Uhlenbeck process. It was shown in Chan (1992), the symmetric N × N matrix X N (t), whose entries {X N ij (t), i ≤ j} are independent Ornstein-Uhlenbeck processes with invariant distribution N (0, (1 + δ ij )/(2N )), where δ ij is the Kronecker delta function, is the solution of the matrix SDE (1.1) with the coefficient functions The SDEs of the eigenvalue processes are Hence, the eigenvalue processes are the particles in (1.6) with and thus, EJP 26 (2021), paper 87.
Similar to the eigenvalues of Wishart process and Dyson's Brownian motion, we have the following CLT.
By Knight's Theorem, there exists a family of independent standard one-dimensional Let Y N t be a symmetric matrix-valued stochastic process whose entries {Y N ij (t), i ≤ j} are given by where {λ N i (t)} 1≤i≤N and {λ N i (t)} 1≤i≤N are the eigenvalues of X N (t) and Y N (t), respectively. Thus, almost surely, we have where R t (n) is given in (3.40). Without loss of generality, we may replace " d =" by "=" in the above equation. Thus we have whose solution is given by (3.39). The proof is concluded.
Now we extend the result of Theorem 3.11 to a generalized system of (3.38).
Corollary 3.12. Consider the following SDEs (3.47) Furthermore, assume the same initial conditions as in Theorem 3.11. Then the conclusion of Theorem 3.11 still holds with R t (n) in (3.40) replaced by R t (n) = c(n + 2) t 0 L N s (x n+1 )ds + (n + 2)(n + 1) 4 t 0 x n , µ s ds + n + 2 2 n k=0 t 0 L s (x n−k ) x k , µ s ds.
Proof. The proof is similar to the proof of Corollary 3.10, which is sketched below. By (3.47), without loss of generality, we assume where the processes (x N i (t)) 1≤i≤N and (y N i (t)) 1≤i≤N are the solutions of the following systems of SDEs respectively: with the initial conditions x N i (0) = y N i (0) = λ N i (0) for 1 ≤ i ≤ N . Noting that (x N i (t) − 2c + 2) 1≤i≤N and (y N i (t) − 2c − 2) 1≤i≤N solve the SDEs (3.38), by (3.45) and (3.48), we get that the uniform L p bound (2.15) holds for system (3.46).
Then applying Corollary 2.5 and following the approach in the proof of Theorem 3.5, we get the desired result.

A Some lemmas
In this section, we provide some results that were used in the preceding sections. The following CLT for martingales was used in the proof of Theorem 2.2.
Lemma A.1 (Rebolledo's Theorem). Let n ∈ N, and let {M N } N ∈N be a sequence of continuous centered martingales with values in R n . If the quadratic variation M N t converges in L 1 (Ω) to a continuous deterministic function φ(t) for all t > 0, then for any T > 0, as a continuous process from [0, T ] to R n , (M N (t), t ∈ [0, T ]) converges in law to a Gaussian process G with mean 0 and covariance E[G(s)G(t) ] = φ(t ∧ s).
The following lemma was employed in the proof of Lemma 3.3. Lemma A.3. Let u N (t) be the strong solution to (3.8). If u N (t) is distributed according to P N in (3.7), then for Proof. For f ∈ C 2 b (R N ), applying Itô's formula to (3.8), we have ds.
Here, ∂ i is the partial derivative with respect to the i-th component x i . Therefore, for t ≥ 0, Thus it suffices to show, with the density function p(x) in (3.7), where ∆ N = {x ∈ R N : 0 < x 1 < . . . < x N } is the support of P N . Noting that p(x) vanishes on ∂∆ N , we have by the integration by parts formula, Hence, to show (A.2), it is sufficient to verify (p(x) + x i ∂ i p(x)) = 0.
By the chain rule, which gives the desired result.