High-dimensional central limit theorems for eigenvalue distributions of generalized Wishart processes

TL;DR

This paper establishes central limit theorems for eigenvalue distributions of generalized Wishart processes and related particle systems, revealing their fluctuations around deterministic limits in high dimensions.

Contribution

It introduces new CLTs for eigenvalue fluctuations of generalized Wishart processes, extending results to Dyson's Brownian motion and Ornstein-Uhlenbeck matrix processes.

Findings

CLTs for eigenvalue fluctuations of generalized Wishart processes

Extension of CLTs to Dyson's Brownian motion and Ornstein-Uhlenbeck processes

Eigenvalue empirical measures converge with quantifiable fluctuations

Abstract

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

Equations

$$d{X}_t^N=g_N(X_t^N)d{B}_t{h}_N(X_t^N)+h_N(X_t^N)d{B}_t^{⊺}{g}_N(X_t^N)+b_N(X_t^N)dt,t\ge 0.$$

$${\tau }_N=\inf \{t>0:\exists i\ne j,{\lambda }_i(t)={\lambda }_j(t)\},$$

$$d{\lambda }_i^N(t)=2g_N({\lambda }_i^N(t))h_N({\lambda }_i^N(t))d{W}_i(t)+\left(b_N({\lambda }_i^N(t))+\sum _{j:j\ne i}\frac{G_N({\lambda }_i^N(t),{\lambda }_j^N(t))}{{\lambda }_i^N(t)-{\lambda }_j^N(t)}\right)dt,$$

$$G_N(x,y)=g_N^2(x)h_N^2(y)+g_N^2(y)h_N^2(x).$$

$$L_N(t)=\frac{1}{N}\sum _{i=1}^N{\delta }_{{\lambda }_i^N(t)}.$$

$$d{X}_i=\sqrt{\frac{2\alpha }{N}}d{B}_i+\left(-\theta X_i+\frac{\alpha }{N}\sum _{j:j\ne i}\frac{1}{X_i-X_j}\right)dt,1\le i\le N,t\ge 0.$$

$$d{X}_i=\sigma (X_i)d{B}_i+\left(b(X_i)+\sum _{j:j\ne i}\frac{\gamma }{X_i-X_j}\right)dt,1\le i\le N,t\ge 0,$$

$$\int \frac{{\mu }_t(dx)}{z-x}=$$

$$+{\int }_0^t\left[\iint \frac{G(x,y)}{(z-x)(z-y{)}^2}{\mu }_s(dx){\mu }_s(dy)\right]ds,\forall z\in \mathbb{C}\setminus \mathbb{R},$$

$$b(x)=\underset{N\to \infty }{\lim }{b}_N(x)\text{ and }G(x,y)=\underset{N\to \infty }{\lim }N{G}_N(x,y),$$

$${\mathcal{L}}_t^N(f)=N⟨f,L_N(t)-{\mu }_t⟩=\sum _{i=1}^{N}f({\lambda }_i^N(t))-N⟨f,{\mu }_t⟩,$$

$$Q_t^N(f)$$

$$-{\int }_0^t{\mathcal{L}}_s^N\left(\int \frac{f^{\prime }(x)-f^{\prime }(y)}{x-y}G(x,y){\mu }_s(dx)\right)ds$$

$$-\frac{N}{2}{\int }_0^t\iint \frac{f^{\prime }(x)-f^{\prime }(y)}{x-y}G(x,y)[L_N(s)(dx)-{\mu }_s(dx)][L_N(s)(dy)-{\mu }_s(dy)]ds,$$

$$\mathbb{F}=$$

$$\displaystyle\qquad\left\|\dfrac{f^{\prime}(x)-f^{\prime}(y)}{x-y}G(x,y)\right\|_{L^{\infty}(\mathbb{R}^{2})}<\infty,~{}~{}\|(f^{\prime}(x))^{2}G(x,x)\|_{L^{\infty}(\mathbb{R})}<\infty\Bigg{\}}.$$

$$\begin{array}{r}\underset{N\to \infty }{\lim }N\Vert b_N(x)-b(x){\Vert }_{L^{\infty }(\mathbb{R})}=0,\\ \underset{N\to \infty }{\lim }N\Vert N{G}_N(x,y)-G(x,y){\Vert }_{L^{\infty }({\mathbb{R}}^2)}=0.\end{array}$$

$$\mathbb{E}\left[G_t(f_i)G_s(f_j)\right]=2{\int }_0^{t\wedge s}⟨f_i^{\prime }(x)f_j^{\prime }(x)G(x,x),{\mu }_u⟩du,1\le i,j\le k.$$

$$⟨f,L_N(t)⟩$$

$$+\frac{N}{2}{\int }_0^t\iint \frac{f^{\prime }(x)-f^{\prime }(y)}{x-y}{G}_N(x,y)L_N(s)(dx)L_N(s)(dy)ds,$$

$$M_f^N(t)=\frac{2}{N}\sum _{i=1}^N{\int }_0^t{f}^{\prime }({\lambda }_i^N(s))g_N({\lambda }_i^N(s))h_N({\lambda }_i^N(s))d{W}_i(s),$$

$$⟨M_f^N{⟩}_t=\frac{4}{N}{\int }_0^t⟨|f^{\prime }{g}_N{h}_N{|}^2,L_N(s)⟩ds=\frac{2}{N}{\int }_0^t⟨|f^{\prime }(x){|}^2{G}_N(x,x),L_N(s)⟩ds.$$

$$⟨f,{\mu }_t⟩$$

$${\mathcal{L}}_t^N(f)=N⟨f,L_N(t)-{\mu }_t⟩$$

$$=N⟨f,L_N(0)-{\mu }_0⟩+N{M}_f^N(t)$$

$$+N{\int }_0^t⟨f^{\prime }{b}_N,L_N(s)⟩-⟨f^{\prime }b,{\mu }_s⟩ds+N{\int }_0^t⟨f^{\prime \prime }{g}_N^2{h}_N^2,L_N(s)⟩ds$$

$$+\frac{N}{2}{\int }_0^t\iint \frac{f^{\prime }(x)-f^{\prime }(y)}{x-y}[N{G}_N(x,y)L_N(s)(dx)L_N(s)(dy)-G(x,y){\mu }_s(dx){\mu }_s(dy)]ds.$$

$$N{\int }_0^t⟨f^{\prime }{b}_N,L_N(s)⟩-⟨f^{\prime }b,{\mu }_s⟩ds$$

$$=N{\int }_0^t⟨f^{\prime }{b}_N-f^{\prime }b,L_N(s)⟩ds+N{\int }_0^t⟨f^{\prime }b,L_N(s)-{\mu }_s⟩ds$$

$$=N{\int }_0^t⟨f^{\prime }{b}_N-f^{\prime }b,L_N(s)⟩ds+{\int }_0^t{\mathcal{L}}_s^N(f^{\prime }b)ds.$$

$$\underset{N\to \infty }{\lim }\left|N{\int }_0^t⟨f^{\prime }{b}_N,L_N(s)⟩-⟨f^{\prime }b,{\mu }_s⟩ds-{\int }_0^t{\mathcal{L}}_s^N(f^{\prime }b)ds\right|$$

$$\le$$

$$N{\int }_0^t⟨f^{\prime \prime }{g}_N^2{h}_N^2,L_N(s)⟩ds=\frac{N}{2}{\int }_0^t⟨f^{\prime \prime }(x)G_N(x,x),L_N(s)⟩ds$$

$$=$$

$$\left|N{\int }_0^t⟨f^{\prime \prime }{g}_N^2{h}_N^2,L_N(s)⟩ds-\frac{1}{2}{\int }_0^t⟨f^{\prime \prime }(x)G(x,x),{\mu }_s⟩ds\right|$$

$$\le$$

Full text

60H15, 60F05,

Dyson’s Brownian motion,

Wishart process,

Generalized Wishart process,

Squared Bessel particle system,

Central limit theorem,

Ornstein-Uhlenbeck matrix process,

1 Introduction

Recently general stochastic differential equations (SDEs) on the group of symmetric matrices have attracted much interest. A prominent example is the following generalized Wishart process introduced in Graczyk and Małecki, (2013),

[TABLE]

Here, $B_{t}$ is a Brownian matrix of dimension $N\times N$, and the continuous functions $g_{N},h_{N},b_{N}:\mathbb{R}\rightarrow\mathbb{R}$ act on the spectrum of $X_{t}^{N}$ (a function $f$ acts on the spectrum of a symmetric matrix $X=\sum_{j=1}^{N}\alpha_{j}u_{j}u_{j}^{\intercal}$ with eigenvalues $(\alpha_{j})$ and eigenvectors $(u_{j})$ if $f(X)=\sum_{j=1}^{N}f(\alpha_{j})u_{j}u_{j}^{\intercal}$). The generalized Wishart process (1.1) includes as simple examples the following well-known matrix-valued stochastic processes: the celebrated symmetric Brownian motion (Dyson,, 1962), the Wishart process (Bru,, 1991), and the symmetric matrix process whose entries are independent Ornstein-Uhlenbeck processes (Chan,, 1992).

Suppose that $\lambda_{1}^{N}(t)\leq\lambda_{2}^{N}(t)\leq\ldots\leq\lambda_{N}^{N}(t)$ are the eigenvalues of $X_{t}^{N}$. According to Theorem 3 in Graczyk and Małecki, (2013), if $\lambda_{1}^{N}(0)<\lambda_{2}^{N}(0)<\cdots<\lambda_{N}^{N}(0)$, then before the first collision time

[TABLE]

the eigenvalues satisfy the following system of SDEs: for $1\leq i\leq N$,

[TABLE]

where $\{W_{i},1\leq i\leq N\}$ are independent Brownian motions and

[TABLE]

In Graczyk and 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 $\tau_{N}$ is infinite almost surely.

Let $L_{N}(t)$ be the empirical measure of the eigenvalues $\{\lambda_{i}^{N}(t),1\leq i\leq N\}$, i.e.,

[TABLE]

In connection with the theory of random matrices, it is of interest to investigate possible limits of these empirical measures $\{L_{N}(t)\}$ when $N$ grows to infinity (high-dimensional limits). The literature on such high-dimensional limits is sparse. 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 process in Chan, (1992), where the symmetric matrix process has independent Ornstein-Uhlenbeck processes as its entries. The results were later generalized in Rogers and Shi, (1993) to the following interacting particle system

[TABLE]

Cépa and Lépingle, (1997) further generalized these SDEs to

[TABLE]

with some coefficient functions $b$, $\sigma$ and constant $\gamma$. Another important case is the Marčenko-Pastur law for the eigenvalue empirical measure process derived in Cabanal-Duvillard and Guionnet, (2001).

The eigenvalue SDEs (1.2) generalize the eigenvalue SDEs in Chan, (1992) and Cabanal-Duvillard and Guionnet, (2001), as well as the particle system in Rogers and Shi, (1993). High-dimensional limits for these eigenvalue SDEs appeared very recently in Song et al., (2019) and Małecki and Pérez, (2019). Particularly in the former article, it was proved that under proper conditions, $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ is relatively compact in $(C[0,T],M_{1}({\mathbb{R}}))$ almost surely. Here $M_{1}({\mathbb{R}})$ is the set of probability measures on ${\mathbb{R}}$ endowed with the topology induced by the weak convergence of measures. Furthermore, any limit measure $\{\mu_{t},t\in[0,T]\}$ from a converging subsequence satisfies

[TABLE]

with

[TABLE]

uniformly. Note that Song et al., (2019) provided examples where such limit $\{\mu_{t},t\in[0,T]\}$ is unique. However, conditions for the uniqueness are still unknown for the general system (1).

In this paper, we study the fluctuations of $\{L_{N}(t),t\in[0,T]\}$ around the limit $\{\mu_{t},t\in[0,T]\}$. Up to considering a subsequence, the theory is here developed, without loss of generality, by assuming the convergence of the whole sequence $\{L_{N}(t),t\in[0,T]\}$ to $\{\mu_{t},t\in[0,T]\}$. Consider the random fluctuations

[TABLE]

for $f\in\mathbb{F}$, where $\mathbb{F}$ is an appropriate space of test functions given by (2.1) or (2.22) in Section 2. The main purpose of the paper is to find a Gaussian limit for the centered process

[TABLE]

as $N$ goes to infinity. To our best knowledge, the literature on this topic is quite limited, and we only refer to Cabanal-Duvillard, (2001); Anderson et al., (2010) which concern the cases of Dyson’s Brownian motion and Wishart process.

Now, we briefly explain the structure of this paper as follows.

The main results in this paper are presented in Section 2. The central limit theorem (CLT) for the empirical measure of the eigenvalues (1.2) is obtained in Section 2.1. The same techniques allow to establish the CLT in Scetion 2.2 for the empirical measure of a class of particle system (2.21) which was introduced in Graczyk and Małecki, (2014) as an generalization of (1.2). Note that in particular (2.21) includes the particle system studied in Cépa and Lépingle, (1997) as a special example.

In Section 3, we apply the results in Section 2 to obtain the CLTs for the eigenvalues of Wishart process in Section 3.2, for the Dyson’s Brownian motion in Section 3.3, and for the eigenvalues of symmetric Ornstein-Uhlenbeck matrix process in Section 3.4, respectively. Note that for these three cases, under proper initial conditions, we can obtain the boundedness for the eigenvalues/particles, which enables us to obtain more precise CLTs for a wider class of test functions. In order to obtain such bounds starting from more general initial conditions, inspired by Śniady, (2002) and Anderson et al., (2010), in Section 3.1 we develop a comparison principle for SDE (1.2) and particle system (2.21). This comparison principle also allows to extend the CLTs developed in Section 3 to a wider class of particles systems (Corollaries 3.2, 3.3 and 3.4).

Furthermore, due to the special structures of the Wishart process, the Dyson’s Brownian motion, and the Ornstein-Uhlenbeck matrix process, we are able to directly characterize the fluctuations $\{\mathcal{L}_{t}(x^{n}),t\in[0,T]\}_{n\in\mathbb{N}}$, where $\mathcal{L}_{t}(x^{n})$ is the limit of $\mathcal{L}_{t}^{N}(x^{n})$, by recursive formulas (See Theorems 3.2, 3.3, 3.4 and the remarks thereafter). For the Dyson’s Brownian motion, the CLT was obtained in Cabanal-Duvillard, (2001) with null initial condition, and the restriction on the initial condition was later relaxed in Anderson et al., (2010). This CLT is recovered in Section 3.3 with slightly more general initial condition. For the eigenvalue processes of Wishart process, the CLT was obtained in Cabanal-Duvillard, (2001) again with null initial condition, and it is now extended in Section 3.2 allowing more general initial conditions. Lastly, the CLT obtained in Section 3.4 for the eigenvalue process of Ornstein-Uhlenbeck matrix process seems new.

Finally, in Section 4 some useful lemmas are provided.

2 Central limit theorems

In this section, we prove our main results of the CLTs for eigenvalues of general Wishart processes in Section 2.1 and for particle systems in Section 2.2, repsectively.

2.1 Central limit theorem for eigenvalues of general Wishart processes

In this subsection, we study the CLT for the empirical measure (1.4) of the eigenvalues (1.2) of generalized Wishart process (1.1).

Recall that the functions $b(x)$ and $G(x,x)$ are defined in (1.6), and $Q_{t}^{N}(f)$ is defined in (1). We use the following space of test functions

[TABLE]

Theorem 2.1**.**

Assume that the limit functions $b(x)$ and $G(x,y)$ are continuous and satisfy

[TABLE]

Also assume that (1.2) has a non-exploding and non-colliding strong solution, such that the sequence of the empirical measures $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ given by (1.4) converges weakly to $\{\mu_{t},t\in[0,T]\}$.

Then, for any $k\in\mathbb{N}$ and any $f_{1},\ldots,f_{k}\in\mathbb{F}$, as $N$ goes to infinity, $(Q_{t}^{N}(f_{1}),\ldots,Q_{t}^{N}(f_{k}))_{t\in[0,T]}$ converges in distribution to a Gaussian process $(G_{t}(f_{1}),\ldots,G_{t}(f_{k}))_{t\in[0,T]}$ with mean zero and covariance

[TABLE]

Proof.

By Itô’s formula (see Song et al., (2019) for more details), for $f\in C^{2}[0,T]$,

[TABLE]

where we use the convention $\frac{f^{\prime}(x)-f^{\prime}(y)}{x-y}=f^{\prime\prime}(x)$ on $\{(x,y)\in\mathbb{R}^{2}:x=y\}$, and $M_{f}^{N}(t)$ is a local martingale,

[TABLE]

with quadratic variation

[TABLE]

On the other hand, for $f\in\mathbb{F}$, under the condition (2.2), one may apply the approach used in the proof of Theorem 2.2 in Song et al., (2019) to get

[TABLE]

(Indeed, the proof of Theorem 2.2 in Song et al., (2019) deals with the special case $f(x)=(z-x)^{-1}$ with $z\in\mathbb{C}\backslash{\mathbb{R}}$.)

Thus, (2.4) and (2.7) yield

[TABLE]

The third term on the right-hand side of (2.8) can be written as

[TABLE]

Thus, we have

[TABLE]

For the fourth term on the right-hand side of (2.8),

[TABLE]

Hence, we have

[TABLE]

as $N\to\infty$, where the last step follows from the weak convergence of $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ and the continuity and boundedness of $G(x,x)f^{\prime\prime}(x)$ for $f\in\mathbb{F}$.

The fifth term on the right-hand side of (2.8) can be written as

[TABLE]

where the last equality follows from the symmetry of $\frac{f^{\prime}(x)-f^{\prime}(y)}{x-y}G(x,y)$. For the first term on the right-hand side of (2.1), we have

[TABLE]

Therefore, by (1), (2.8) and the above estimations (2.1), (2.1), (2.1), and (2.1), we have that the term

[TABLE]

converges to [math] almost surely as $N\rightarrow\infty$, uniformly in $t\in[0,T]$. Note that in (2.1), (2.1) and (2.1), the integrand function is bounded, and hence the convergence is also in $L^{p}$ for all $p\geq 1$. Thus, $Q_{t}^{N}(f)-NM_{f}^{N}(t)$ with $f\in\mathbb{F}$ converges to [math] in $L^{p}$ for all $p\geq 1$ uniformly in $t\in[0,T]$.

Therefore, to prove the desired result, it suffices to show that, for any $k\in\mathbb{N}$ and $f_{1},f_{2},\dots,f_{k}\in\mathbb{F}$, the vector-valued stochastic process $(NM_{f_{1}}^{N}(t),NM_{f_{2}}^{N}(t),\dots,NM_{f_{k}}^{N}(t))_{t\in[0,T]}$ converges in distribution to a centered Gaussian process $(G_{t}(f_{1}),G_{t}(f_{2}),\dots,G_{t}(f_{k}))_{t\in[0,T]}$ with covariance given by (2.3). To this end, by Lemma 4.1 it suffices to prove that $\{NM_{f}^{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ are martingales for $f\in\mathbb{F}$ such that the following limit holds in $L^{1}(\Omega)$,

[TABLE]

By the uniform convergence of $NG_{N}(x,y)$ towards $G(x,y)$, the boundedness of $f^{\prime}(x)^{2}G(x,x)$ and (2.6), one can show that $\{NM_{f}^{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ are martingales. It follows from (2.5) that, for $f_{1},f_{2}\in\mathbb{F}$,

[TABLE]

The term $\int_{0}^{t}\langle f_{1}^{\prime}(x)f_{2}^{\prime}(x)(NG_{N}(x,x)-G(x,x)),L_{N}(s)\rangle ds$ converges to 0 a.s. and in $L^{p}$ for all $p\geq 1$ due to the boundedness of $f_{1}^{\prime}(x)$ and $f_{2}^{\prime}(x)$ and the uniform convergence of $NG_{N}(x,y)$ towards $G(x,y)$. Furthermore, the following convergence

[TABLE]

holds a.s. and in $L^{p}$ for all $p\geq 1$, because of the weak convergence of $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ to $\{\mu_{t},t\in[0,T]\}$ and the boundedness of $f_{1}^{\prime}(x)f_{2}^{\prime}(x)G(x,x)$. Therefore, $\langle NM_{f_{1}}^{N},NM_{f_{2}}^{N}\rangle_{t}$ converges to $2\int_{0}^{t}\langle f_{1}^{\prime}(x)f_{2}^{\prime}(x)G(x,x),\mu_{s}\rangle ds$ a.s. and in $L^{p}$ for all $p\geq 1$.

The proof is concluded. ∎

If the eigenvalues in (1.2) are bounded, the test function space $\mathbb{F}$ can be enlarged by removing the boundedness condition in the above theorem.

Corollary 2.1**.**

Assume the same conditions as in Theorem 2.1. Moreover, for $T<\infty$, assume that

[TABLE]

a.s. for some constant $C(T)$ depending on $T$. Then Theorem 2.1 still holds if the set $\mathbb{F}$ of test function is replaced by $C^{2}(\mathbb{R})$.

Proof.

It follows from (2.14) that all but finitely many terms in $\{\sup_{t\in[0,T]}\max_{1\leq i\leq N}|\lambda_{i}^{N}(t)|\}_{N\in\mathbb{N}}$ are bounded by $A(T)=C(T)+1$ a.s.. Thus there is a measurable set $A\subset\Omega$ with $\mathbb{P}(A)=1$ and a random variable $N_{0}\in\mathbb{N}$, such that for $\omega\in A$, the empirical measure $\{L_{N}(t)(\omega),t\in[0,T]\}$ is supported in $[-A(T),A(T)]$ for all $N\geq N_{0}(\omega)$. Hence the limit $\{\mu_{t},t\in[0,T]\}$ also has the same support. By (Rudin,, 1991, 1.46), there exists a cut-off function $\eta(x)\in C^{\infty}(\mathbb{R})$ equal to $1$ on $[-A(T),A(T)]$, of which the support is $[-2A(T),2A(T)]$. If we replace $f$ by $f\eta$, noting that $f\eta\in\mathbb{F}$ for $f\in C^{2}(\mathbb{R})$ and that $f\eta=f$ on $[-A(T),A(T)]$, we can show that the term $Q_{t}^{N}(f)-NM_{f}^{N}(t)$ in (2.13) converges to 0 a.s. using the same argument as in the proof of Theorem 2.1. Then following the rest part of the proof, it is easy to get the result of Theorem 2.1. ∎

Remark 2.1**.**

Under the conditions in Theorem 2.1, (2.14) yields the almost sure convergence of $Q_{t}^{N}(f)-NM_{f}^{N}(t)$ towards [math] for $f\in C^{2}({\mathbb{R}})$. The next Corollary provides a sufficient condition for the $L^{p}$ convergence for $p\geq 1$.

Corollary 2.2**.**

Assume the same conditions as in Theorem 2.1. For $T<\infty$, for all $p\geq 1$ and all $N\geq cp$ for some positive constant $c$, assume that

[TABLE]

where $C(T)$ is a positive constant depending only on $T$. Furthermore, assume that $G(x,x)$ and its derivative have at most polynomial growth. Then for $f\in C^{3}(\mathbb{R})$ of which the derivatives have at most polynomial growth, $Q_{t}^{N}(f)-NM_{f}^{N}(t)$ converges to [math] in $L^{p}$ uniformly in $t\in[0,T]$ for all $p\geq 1$.

As a consequence, Theorem 2.1 holds for such test functions $f$.

Proof.

By the analysis in the proof of Theorem 2.1, it suffices to show

[TABLE]

for $p\geq 1$ and $g\in C^{1}(\mathbb{R})$ with $|g^{\prime}(x)|\leq a|x|^{n-1}+b$ for some $a,b\in\mathbb{R},\ n\in\mathbb{N}_{+}$. More precisely, one can check that under the conditions (2.15) and (2.16), the convergences to 0 in (2.1), (2.1) and (2.1) are uniform in $L^{p}$, and hence $Q_{t}^{N}(f)-NM_{f}^{N}(t)$ in (2.13) converges to 0 in $L^{p}$ uniformly.

By Markov inequality and (2.15),

[TABLE]

Choosing $p=\ln^{2}N$, we have

[TABLE]

By Borel-Cantelli lemma, we get that almost surely,

[TABLE]

By the proof of Corollary 2.1, the limit measure $\{\mu_{t},t\in[0,T]\}$ is supported in $[-C(T)-1,C(T)+1]$.

For $g\in C^{1}(\mathbb{R})$ with $|g^{\prime}(x)|\leq a|x|^{n-1}+b$ for some $a,b\in\mathbb{R},\ n\in\mathbb{N}_{+}$, define

[TABLE]

for $\delta>0.$ Then $g_{\delta}(x)$ is a bounded continuous function, and hence

[TABLE]

almost surely. By dominated convergence theorem,

[TABLE]

Note that $g^{\prime}(x)$ grows no faster than polynomials of degree $n-1$, by the mean value theorem, it is not difficult to show $|g(x)-g_{\delta}(x)|\leq C\delta(|x|^{n+2}+|x|^{3})$, which implies that $g_{\delta}$ converges to $g$ uniformly in any compact interval as $\delta\rightarrow 0^{+}$. Thus,

[TABLE]

Finally, by the Jensen’s inequality and (2.15), we obtain that, as $\delta\to 0^{+}$,

[TABLE]

uniformly in $N\in\mathbb{N}_{+}$.

By (2.17), (2.18), (2.1) and the triangle inequality, we can obtain (2.16), and the proof is concluded. ∎

Proposition 2.1**.**

Consider the centered Gaussian family $\{G_{t}(f),f\in\mathbb{F}\}$ in Theorem 2.1 with covariance

[TABLE]

We have the following linear property, for $f_{1},f_{2}\in\mathbb{F}$ and $\alpha_{1},\alpha_{2}\in{\mathbb{R}}$,

[TABLE]

almost surely.

Proof.

For $f_{1},f_{2}\in\mathbb{F}$ and $\alpha_{1},\alpha_{2}\in{\mathbb{R}}$, it is easy to check that $\alpha_{1}f_{1}+\alpha_{2}f_{2}\in\mathbb{F}$. By the proof of Theorem 2.1, the random vector $(NM_{f_{1}}^{N}(t),NM_{f_{2}}^{N}(t),NM_{\alpha_{1}f_{1}+\alpha_{2}f_{2}}^{N}(t))_{t\in[0,T]}$ converges in distribution to $(G_{t}(f_{1}),G_{t}(f_{2}),G_{t}(\alpha_{1}f_{1}+\alpha_{2}f_{2}))_{t\in[0,T]}$. Hence, the linear combination $(\alpha_{1}NM_{f_{1}}^{N}(t)+\alpha_{2}NM_{f_{2}}^{N}(t)-NM_{\alpha_{1}f_{1}+\alpha_{2}f_{2}}^{N}(t))_{t\in[0,T]}$ converges in distribution to $(\alpha_{1}G_{t}(f_{1})+\alpha_{2}G_{t}(f_{2})-G_{t}(\alpha_{1}f_{1}+\alpha_{2}f_{2}))_{t\in[0,T]}$.

By (2.5), we can see that the martingale $M_{f}^{N}(t)$ is linear with respect to the function $f$, so $\alpha_{1}NM_{f_{1}}^{N}(t)+\alpha_{2}NM_{f_{2}}^{N}(t)=NM_{\alpha_{1}f_{1}+\alpha_{2}f_{2}}^{N}(t)$ for all $t\in[0,T]$ and all $N\in\mathbb{N}$, which implies that the process $(\alpha_{1}NM_{f_{1}}^{N}(t)+\alpha_{2}NM_{f_{2}}^{N}(t)-NM_{\alpha_{1}f_{1}+\alpha_{2}f_{2}}^{N}(t))_{t\in[0,T]}$ is actually a zero process. Thus, as the limit of the convergence in distribution, $(\alpha_{1}G_{t}(f_{1})+\alpha_{2}G_{t}(f_{2})-G_{t}(\alpha_{1}f_{1}+\alpha_{2}f_{2}))_{t\in[0,T]}$ is also a zero process, which implies (2.20). ∎

2.2 Central limit theorem for particle systems

In this subsection, we provide the central limit theorem for the empirical measure of the following particle system: for $1\leq i\leq N$,

[TABLE]

with $H_{N}(x,y)$ being a symmetric function. This particle system was introduced in Graczyk and Małecki, (2014) as a generalization of (1.2). 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., (2019) that the family of empirical measure $\{L_{N}(t),t\in[0,T]\}$ is tight almost surely, and any limit $\{\mu_{t},t\in[0,T]\}$ satisfies

[TABLE]

where, $b(x)$, $\sigma(x)$ and $H(x,y)$ are the uniform limits of $b_{N}(x)$, $\sigma^{N}(x)$ and $NH_{N}(x,y)$, respectively.

Now we adopt the following set of test functions

[TABLE]

where $\tilde{\sigma}(x)$ is the uniform limit of $\sqrt{N}\sigma^{N}(x)$. Considering the centered fluctuation process, for $f\in\widetilde{\mathbb{F}}$,

[TABLE]

as an extension of Theorem 2.1, we have the following result.

Theorem 2.2**.**

Suppose that the limit functions $\tilde{\sigma}(x)$, $b(x)$ and $H(x,y)$ are continuous and the following conditions hold,

[TABLE]

Also assume that (2.21) has a non-exploding and non-colliding strong solution, such that the sequence of the empirical measures $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ converges weakly to $\{\mu_{t},t\in[0,T]\}$.

Then, for any $k\in\mathbb{N}$ and any $f_{1},\ldots,f_{k}\in\widetilde{\mathbb{F}}$, $(\widetilde{Q}_{t}^{N}(f_{1}),\ldots,\widetilde{Q}_{t}^{N}(f_{k}))_{t\in[0,T]}$ converges in distribution to a centered Gaussian process $(\widetilde{G}_{t}(f_{1}),\ldots,\widetilde{G}_{t}(f_{k}))_{t\in[0,T]}$ with covariance

[TABLE]

Results analogous to Corollary 2.1, Corollary 2.2 and Proposition 2.1 are as follows.

Corollary 2.3**.**

Assume the same conditions as in Theorem 2.2. Moreover, for $T<\infty$, assume that

[TABLE]

almost surely for some constant $C(T)$ depending on $T$. Then Theorem 2.2 still holds if the set $\,\widetilde{\mathbb{F}}$ of test function is replaced by $C^{2}(\mathbb{R})$.

Corollary 2.4**.**

Assume the same conditions as in Theorem 2.2. For $T<\infty$ and all $p\geq 1$, assume that

[TABLE]

for some positive constant $C(T)$ which depends only on $T$. Furthermore, assume that $(\frac{1}{2}\tilde{\sigma}(x)^{2}-H(x,x))f^{\prime\prime}(x)$ and its derivative have at most polynomial growth. Then for $f\in C^{3}(\mathbb{R})$ of which the derivatives have at most polynomial growth, $\widetilde{Q}_{t}^{N}(f)-NM_{f}^{N}(t)$ converges to [math] in $L^{p}$ for all $p\geq 1$ uniformly in $t\in[0,T]$.

Proposition 2.2**.**

Consider the centered Gaussian family $\{\widetilde{G}_{t}(f),f\in\widetilde{\mathbb{F}}\}$ with covariance

[TABLE]

We have the following linear property, for $f_{1},f_{2}\in{\widetilde{\mathbb{F}}}$ and $\alpha_{1},\alpha_{2}\in{\mathbb{R}}$,

[TABLE]

almost surely.

The proofs of Theorem 2.2, Corollary 2.3, Corollary 2.4 and Proposition 2.2 are similar to those of Theorem 2.1, Corollary 2.1, Corollary 2.2 and Proposition 2.1, respectively, and thus omitted.

3 Applications

In this section, we apply our main results obtained in Section 2 to the eigenvalues of Wishart process (Section 3.2), the Dyson’s Brownian motion (Section 3.3) and the eigenvalues of symmetric Ornstein-Uhlenbeck matrix process (Section 3.4). In particular, for these three cases, we will show the boundedness of the moments of the empirical measures assuming proper initial conditions. This enables us to apply Corollaries 2.1, 2.2, 2.3 and 2.4 to study the flunctuations $\mathcal{L}_{t}(f)$ for polynomial functions $f\in{\mathbb{R}}[x]$, and recursive formulas are obtained for the basis $\{\mathcal{L}_{t}(x^{n}),t\in[0,T]\}_{n\in\mathbb{N}}$ of $\{\mathcal{L}_{t}(f),t\in[0,T]\}_{f\in{\mathbb{R}}[x]}$. Note that these results are more precise than the general results in Section 2, where we study the centered process $\{Q_{t}^{N}(f)\}$ for more restricted test function $f$.

3.1 Comparison principle

In this subsection, we provide a comparison principle for SDE (1.2) and particle system (2.21), which allows us to obtain the boundedness of the eignenvalues/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: for $1\leq i\leq N,\ t\geq 0$,

[TABLE]

and

[TABLE]

with non-colliding initial values $x(0)=(x_{1}(0),\ldots,x_{N}(0))$ and $y(0)=(y_{1}(0),\ldots,y_{N}(0))$, respectively. Here, the functions $\sigma_{i}(x)$, $b_{i}(x)$ and $\tilde{b}_{i}(x)$ for $1\leq i\leq N$ are continuous, and $H_{ij}(x,y)$ with $i\neq j$ is a continuous, non-negative and symmetric function satisfying the condition (Graczyk and Małecki,, 2014, (A1)):

[TABLE]

Note that conditions for the existence and uniqueness of a non-colliding and non-exploding 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.

Theorem 3.1**.**

Suppose $x(t)=(x_{1}(t),\ldots,x_{N}(t))$ and $y(t)=(y_{1}(t),\ldots,y_{N}(t))$ are the non-exploding and non-colliding unique strong solutions to (3.1) and (3.2), respectively. Assume that there exists a strictly increasing function $\rho:[0,\infty)\rightarrow[0,\infty)$ with $\rho(0)=0$ and

[TABLE]

such that

[TABLE]

If we further assume that $b_{i}(u)\leq\tilde{b}_{i}(u)$ for all $u\in\mathbb{R}$, and $x_{i}(0)\leq y_{i}(0)$ a.s., $1\leq i\leq N$, then

[TABLE]

Proof.

The continuity of the functions $H_{ij}$ and the condition (3.3) implies that for all $1\leq i\neq j\leq N$,

[TABLE]

and

[TABLE]

Hence, the drift functions

[TABLE]

satisfy the quasi-monotonously increasing condition in Lemma 4.2.

In order to apply Lemma 4.2 to get the desired result, we use an approximation argument to remove the singularities of the drift functions $F$ and $\widetilde{F}$. For $\epsilon>0$, let

[TABLE]

and define the stopping time

[TABLE]

One can find continuous quasi-monotonously increasing functions $F_{\epsilon}$ and $\widetilde{F}_{\epsilon}$, such that they coincide with $F$ and $\widetilde{F}$ in $\Delta_{\epsilon}$, repspectively. Before time $\tau_{\epsilon}$, both $x$-particles and $y$-particles stay in $\Delta_{\epsilon}$ and thus satisfy (3.1) and (3.2) with drift functions $\widetilde{F}_{\epsilon}$ and $\widetilde{F}_{\epsilon}$, respectively.

Applying Lemma 4.2 to the processes $x^{\epsilon}$ and $y^{\epsilon}$, we have

[TABLE]

which implies

[TABLE]

The desired result now follows from the non-colliding property $\lim_{\epsilon\to 0^{+}}\tau_{\epsilon}=\infty$.

∎

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).

Corollary 3.1**.**

Suppose that the following systems of eigenvalue SDEs

[TABLE]

and

[TABLE]

with non-colliding initial values $\lambda(0)=(\lambda_{1}(0),\ldots,\lambda_{N}(0))$ and $\theta(0)=(\theta_{1}(0),\ldots,\theta_{N}(0))$, respectively, have non-exploding and non-colliding unique strong solutions $\lambda(t)=(\lambda_{1}(t),\ldots,\lambda_{N}(t))$ and $\theta(t)=(\theta_{1}(t),\ldots,\theta_{N}(t))$, respectively. Here, $g_{N}(x)$, $h_{N}(x)$, $b_{N}(x)$ and $\tilde{b}_{N}(x)$ are continuous functions, and $G_{N}(x,y)=g_{N}^{2}(x)h_{N}^{2}(y)+g_{N}^{2}(y)h_{N}^{2}(x)$ satisfies

[TABLE]

Assume that there exists a strictly increasing function $\rho:[0,\infty)\rightarrow[0,\infty)$ with $\rho(0)=0$ and

[TABLE]

such that

[TABLE]

Furthermore, we assume that $b_{N}(u)\leq\tilde{b}_{N}(u)$ for all $u\in\mathbb{R}$. If $\lambda_{i}(0)\leq\theta_{i}(0)$ for all $1\leq i\leq N$ almost surely, then

[TABLE]

3.2 Application to eigenvalues of Wishart process

In this subsection, we discuss the limit theorem for the Wishart process. As illustrated in Graczyk and Małecki, (2013) and Song et al., (2019), the scaled Wishart process $X^{N}_{t}=\tilde{B}^{\intercal}(t)\tilde{B}(t)/N$, where $\tilde{B}(t)$ is a $P\times N$ Brownian matrix with $P>N-1$, is the solution to (1.1) with the coefficient functions

[TABLE]

The eigenvalue processes now satisfy

[TABLE]

In this case, we have

[TABLE]

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 $\mathbb{P}^{N}$ be the distribution on $\Delta_{N}=\{x=(x_{1},x_{2},\dots,x_{N})\in\mathbb{R}^{N}:0<x_{1}<\ldots<x_{N}\}$ with density

[TABLE]

where $C_{N,p}>0$ is a normalization constant. Then we have the following estimation on the eigenvalues.

Lemma 3.1**.**

Let $\xi^{N}=(\xi^{N}_{1},\dots,\xi^{N}_{N})$ be a random vector that is independent of $(W_{1},\dots,W_{N})$ and has (3.7) as its joint probability density function. Assume that $(\lambda_{1}^{N}(0),\dots,\lambda_{N}^{N}(0))$ is independent of $(W_{1},\dots,W_{N})$ and that there exists a constant $a>0$, such that $\lambda_{i}^{N}(0)\leq a\xi_{i}^{N}$ for $1\leq i\leq N$ almost surely. Then there exists a stationary stochastic process $u^{N}(t)$ with initial value $u^{N}(0)=\xi^{N}$ satisfying, for $1\leq i\leq N$ and $t\geq 0$,

[TABLE]

Proof.

Consider the following system of SDEs, for $1\leq i\leq N$,

[TABLE]

with initial value $u_{i}^{N}(0)=\xi_{i}^{N}(0)$ distributed according to $\mathbb{P}^{N}$ and $u_{1}^{N}(t)\leq\ldots\leq u_{N}^{N}(t)$.

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 $t\geq 0$. 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 $\mathbb{P}^{N}$, then Lemma 4.3 yields that $\frac{d}{dt}\mathbb{E}[f(u^{N}(t))]$ vanishes for $f\in C_{b}^{2}(\mathbb{R})$. Since $u^{N}(0)$ is distributed according to $\mathbb{P}^{N}$, we can conclude that $(u^{N}(t))_{t\geq 0}$ is a stationary process with marginal distribution $\mathbb{P}^{N}$.

Now let $v_{i}^{N}(t)=(t+a)u_{i}^{N}(t)$ for $1\leq i\leq N$ and $v^{N}(t)=(v_{1}^{N}(t),\ldots,v_{N}^{N}(t))$. Then the Itô formula shows that $v^{N}(t)$ is a solution to (3.5) with initial value $v^{N}(0)=au^{N}(0)=a\xi^{N}$. 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 (3.4), we can apply the comparison principle in Corollary 3.1 to obtain

[TABLE]

The proof is concluded. ∎

Lemma 3.2**.**

Assume the same conditions as in Lemma 3.1. Then for any $T<\infty$, there exists a positive constant $C(a,T)$ depending only on $(a,T)$, such that for all $p\geq 1$,

[TABLE]

almost surely for $N\geq(2p-1)/\alpha$ for some positive constant $\alpha$.

Proof.

Noting that the probability density of $u^{N}(t)$ considered in Lemma 3.1 is (3.7) for all $t$, we can obtain the following tail probability estimation with $\alpha$ being a positive constant independent of $N$,

[TABLE]

By Lemma 3.1 and (3.9), we have for $t\geq 0$,

[TABLE]

for $k\in[0,\alpha N]$, where $\Gamma(x)$ is the gamma function.

Now we apply (2.4) and (2.6) with $f(x)=x^{n+2}$ for $n\geq-1$ to obtain

[TABLE]

where the martingale term $M_{x^{n+2}}^{N}(t)$ has the quadratic variation

[TABLE]

By the Cauchy-Schwarz inequality, Burkholder-Davis-Gundy inequality, Hölder inequality and the estimation (3.2), for $(2n+3)q\leq\alpha N$, $q\in\mathbb{N}$, and $\Lambda_{q}$ being a positive constant depending only on $q$,

[TABLE]

Defining, for $k\geq 1,$

[TABLE]

it follows from (3.2) that for $n\geq-1$,

[TABLE]

For the third and the fourth terms on the right-hand side of (3.2), we have by (3.2),

[TABLE]

and

[TABLE]

for $n+1\leq\alpha N$. Hence, by (3.2), (3.2), and the above two estimations, for $n\geq-1$ such that $2n+3\leq\alpha N$ and $t\in[0,T]$, we have

[TABLE]

Thus, for all $-1\leq n\leq\frac{\alpha N-3}{2}$, noting that $E_{0}^{N}(n+2)\leq\mathbb{E}[\lambda_{N}^{N}(0)^{n+2}]\leq a^{n+2}$ by (3.2), we have

[TABLE]

for some positive constant $C_{a,T}$ depending on $(a,T)$ only.

The proof is concluded. ∎

Now we are ready to prove the following CLT for the eigenvalues of the scaled Wishart process $X^{N}_{t}=\tilde{B}^{\intercal}(t)\tilde{B}(t)/N$, where $\tilde{B}(t)$ is a $P\times N$ Brownian matrix with $P>N-1$. Noting that under the conditions in Lemma 3.1, Lemma 3.2 implies $\limsup_{N\to\infty}\sup_{N}\lambda_{N}^{N}(0)<\infty$ almost surely. One can check that the conditions (A) - (D) in Song et al., (2019) are satisfied, hence $\{L_{N}(t),t\in[0,T]\}_{n\in\mathbb{N}}$ is tight (see also (Song et al.,, 2019, Remark 3.3)), and we know that it converges to $\{\mu_{t},t\in[0,T]\}$, where $\mu_{t}$ is a scaled Marchenko-Pastur law. Recall that $c=\lim\limits_{N\to\infty}P/N$ and that $\mathcal{L}_{t}^{N}(f)$ is defined by (1.7) in Theorem 2.1.

Theorem 3.2**.**

Assume that $\lim_{N\rightarrow\infty}|P-cN|=0$, and that for any polynomial $f(x)\in\mathbb{R}[x]$, the initial value $\mathcal{L}_{0}^{N}(f)$ converges in probability to a random variable $\mathcal{L}_{0}(f)$. Besides, assume the same condition on $\{\lambda_{i}^{N}(0),i=1,2,\dots,N\}$ as in Lemma 3.1 for all $N\in\mathbb{N}$. Furthermore, assume that for all $n\in\mathbb{N}$,

[TABLE]

for all $q\geq 1$. Then for any $0<T<\infty$, there exists a family of processes $\{\mathcal{L}_{t}(f),t\in[0,T]\}_{f\in\mathbb{R}[x]}$, such that for any $n\in\mathbb{N}$ and any polynomials $P_{1},\ldots,P_{n}\in\mathbb{R}[x]$, the vector-valued process $(\mathcal{L}_{t}^{N}(P_{1}),\ldots,\mathcal{L}_{t}^{N}(P_{n}))_{t\in[0,T]}$ converges to $(\mathcal{L}_{t}(P_{1}),\ldots,\mathcal{L}_{t}(P_{n}))_{t\in[0,T]}$ in distribution, as $N\to\infty$.

The limit process $\{\mathcal{L}_{t}(f),t\in[0,T]\}_{f\in\mathbb{R}[x]}$ is characterized by the following properties.

For $P_{1},P_{2}\in\mathbb{R}[x]$, $\alpha_{1},\alpha_{2}\in\mathbb{R}$, $t\in[0,T]$,

[TABLE] 2. 2.

The basis $\{\mathcal{L}_{t}(x^{n}),t\in[0,T]\}_{n\in\mathbb{N}}$ of $\{\mathcal{L}_{t}(P),t\in[0,T]\}_{P\in\mathbb{R}[x]}$ satisfies

[TABLE]

and for $n\geq 0$,

[TABLE]

where $\{G_{t}(x^{n}),t\in[0,T]\}_{n\in\mathbb{N}}$ is a family of centered Gaussian processes with covariance

[TABLE]

Proof.

First, note that by Lemma 3.2 and Corollary 2.2, $Q_{t}^{N}(x^{n})$ defined by (1) converges in distribution to a centered Gaussian family $\{G_{t}(x^{n}),t\in[0,T]\}_{n\in\mathbb{N}}$ with covariance given by (3.16). Furthermore, by (1.7), (1) and (3.6), for $n\geq-1$, we have

[TABLE]

In Corollary 2.1 and Corollary 2.2, we have shown $Q_{t}^{N}(x^{n+2})-NM_{x^{n+2}}^{N}$ converges to [math] almost surely and in $L^{q}$ for all $q\geq 1$ as $N\rightarrow\infty$, uniformly in $t\in[0,T]$. Thus, by (3.2), (3.17), and the condition (3.14), it is not difficult to show

[TABLE]

for $q\geq 1$ and $n\in\mathbb{N}$ by using an induction argument on $n$.

To estimate the last term on the right-hand side of (3.17), we apply the Cauchy-Schwarz inequality to obtain, for $0\leq k\leq n$,

[TABLE]

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 define

[TABLE]

for $n\geq-1$, then the difference $|\tilde{Q}_{t}^{N}(x^{n+2})-Q_{t}^{N}(x^{n+2})|$ converges to [math] almost surely and in $L^{q}$ for $q>1$. Thus, Corollary 2.1 implies that $(\tilde{Q}_{t}^{N}(x^{k}),\tilde{Q}_{t}^{N}(x^{k-1}),\ldots,\tilde{Q}_{t}^{N}(x))_{t\in[0,T]}$ converges in distribution to $(G_{t}(x^{k}),G_{t}(x^{k-1}),\ldots,G_{t}(x))_{t\in[0,T]}$ with covariance (3.16).

Now we deduce the convergence in distribution of $(\mathcal{L}_{t}^{N}(x^{k}))_{t\in[0,T]}$ for $k\in\mathbb{N}$. First of all, we have $\mathcal{L}_{t}^{N}(1)=0$ and $\mathcal{L}_{t}^{N}(x)=\mathcal{L}_{0}^{N}(x)+\tilde{Q}_{t}^{N}(x)$ converges in distribution since the initial value converges in probability. By induction, if we assume $(\mathcal{L}_{t}^{N}(x^{k}),\ldots,\mathcal{L}_{t}^{N}(x))_{t\in[0,T]}$ convergence in distribution to $(\mathcal{L}_{t}(x^{k}),\ldots,\mathcal{L}_{t}(x))_{t\in[0,T]}$, then the convergence in distribution of $(\tilde{Q}_{t}^{N}(x^{k+1}),\tilde{Q}_{t}^{N}(x^{k}),\ldots,\tilde{Q}_{t}^{N}(x))_{t\in[0,T]}$ implies that $(\tilde{Q}_{t}^{N}(x^{k+1}),\mathcal{L}_{t}^{N}(x^{k}),\ldots,\mathcal{L}_{t}^{N}(x))_{t\in[0,T]}$ converges in distribution, and hence $(\mathcal{L}_{t}^{N}(x^{k+1}),\ldots,\mathcal{L}_{t}^{N}(x))_{t\in[0,T]}$ converges in distribution.

Thus, by (3.2) we have

[TABLE]

where “$\overset{d}{=}$” means equality in distribution. The proof is concluded. ∎

Remark 3.1**.**

By the self-similarity of Brownian motion, when $X_{0}^{N}=0$, we have $X_{t}^{N}\overset{d}{=}tX_{1}^{N}$. Thus, $(\lambda_{1}^{N}(t),\ldots,\lambda_{N}^{N}(t))\overset{d}{=}(t\lambda_{1}^{N}(1),\ldots,t\lambda_{N}^{N}(1))$. Therefore,

[TABLE]

and

[TABLE]

Hence, $\mathcal{L}_{t}^{N}(f(x))\overset{d}{=}\mathcal{L}_{1}^{N}(f(tx))$, and thus, $\mathcal{L}_{t}(f(x))\overset{d}{=}\mathcal{L}_{1}(f(tx))$. With these identities and the linearity of $\mathcal{L}_{t}(\cdot)$, (2) can be simplified as, for $n\geq 0$,

[TABLE]

where the Gaussian family $\{G_{t}(x^{n}),t\in[0,T]\}_{n\in\mathbb{N}}$ has the covariance functions

[TABLE]

Note that the case $t=1$ corresponds to the classical Wishart matrix, and $\mu_{1}$ is the Marchenko–Pastur law. More precisely, recalling that $\mathcal{L}_{1}(1)=0$ and $\mathcal{L}_{1}(x)=G_{1}(x)$, we get by (3.19) $\mathcal{L}_{1}(x^{2})=\langle x,\mu_{1}\rangle+(c+1)G_{1}(x)+G_{1}(x^{2})$, for $m\geq 3$, and more generally $\mathcal{L}_{1}(x^{m})=c_{m,0}+c_{m,1}G_{1}(x)+\ldots+c_{m,m}G_{1}(x^{m})$ for some coefficients $(c_{m,j})_{0\leq j\leq m}$ which are determined recursively by (3.19).

We now study a more general particle systems:

[TABLE]

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 3.2 below. Despite the extension being small, the system (3.20) may not correspond to eigenvalues of a matrix SDE, and may not have an explicit joint density function or stationary distribution, and hence cannot be treated in the same way as for the eigenvalues of Wishart process.

Corollary 3.2**.**

Consider the SDEs (3.20), where $b_{N}(x)$ satisfies, for some constant $c\geq 1$,

[TABLE]

Assume the same initial conditions as in Theorem 3.2. Then the conclusion of Theorem 3.2 still holds.

Proof.

Let $p_{1}=N(c-\|b_{N}(x)-c\|_{L^{\infty}})$ and $p_{2}=N(c+\|b_{N}(x)-c\|_{L^{\infty}})$ be two constants depending on $N$. Then (3.21) implies $p_{2}\geq p_{1}>N-1$ when $N$ is large. Clearly, $p_{1}\leq N\|b_{N}(x)\|_{L^{\infty}({\mathbb{R}})}\leq p_{2}$. Consider the following two systems of SDEs:

[TABLE]

and

[TABLE]

with the initial conditions $x_{i}^{N}(0)=y_{i}^{N}(0)=\lambda_{i}^{N}(0)$. By the comparison principle in Corollary 3.1, we have

[TABLE]

Thus, almost surely,

[TABLE]

where $\{L_{N}^{(x)}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ and $\{L_{N}^{(y)}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ are the empirical measures of the two particle systems $(x_{i}^{N}(t))_{1\leq i\leq N}$ and $(y_{i}^{N}(t))_{1\leq i\leq N}$, respectively.

Noting that $p_{1}/N$ and $p_{2}/N$ converge to $c$ as $N\rightarrow\infty$ by (3.21), we have that Lemma 3.2 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 $b_{N}(x)\to c$ uniformly as $N\to\infty$, and hence (3.17) still holds. Then the rest of the proof follows that of Theorem 3.2. ∎

3.3 Application to Dyson’s Brownian motion

In this subsection, we discuss the CLT for the Dyson’s Brownian motion. It was shown in Anderson et al., (2010); Graczyk and Małecki, (2014); Song et al., (2019), the scaled symmetric Brownian motion $X^{N}_{t}=(\tilde{B}^{\intercal}(t)+\tilde{B}(t))/\sqrt{2N}$, where $\tilde{B}(t)$ is a $N\times N$ Brownian matrix, is the solution of the matrix SDE (1.1) with the coefficient functions

[TABLE]

The system of SDEs of the eigenvalue processes, that is, the Dyson’s Brownian motion, is

[TABLE]

In this case, we have

[TABLE]

Here, we consider the distribution $\mathbb{Q}^{N}$ on $\Delta_{N}^{\prime}=\{x=(x_{1},x_{2},\dots,x_{N})\in\mathbb{R}^{N}:x_{1}<\ldots<x_{N}\}$ with the density function

[TABLE]

where $C_{N}$ is a normalization constant.

Similar to the Wishart process, we can obtain the following central limit theorem.

Theorem 3.3**.**

Let $\xi^{N}=(\xi^{N}_{1},\dots,\xi^{N}_{N})$ be a random vector that is independent of $(W_{1},\dots,W_{N})$ and has (3.27) as its joint probability density function. Assume that $(\lambda_{1}^{N}(0),\dots,\lambda_{N}^{N}(0))$ is independent of $(W_{1},\dots,W_{N})$ and that there exist constants $a,b\geq 0$, such that

[TABLE]

for $1\leq i\leq N$ almost surely. Besides, assume that for any polynomial $f(x)\in\mathbb{R}[x]$, the initial value $\mathcal{L}_{0}^{N}(f)$ converges in probability to a random variable $\mathcal{L}_{0}(f)$. Furthermore, assume that for all $n\in\mathbb{N}$,

[TABLE]

for all $p\geq 1$.

Then for any $0<T<\infty$, there exists a family of processes $\{\mathcal{L}_{t}(f),t\in[0,T]\}_{f\in\mathbb{R}[x]}$, such that for any $n\in\mathbb{N}$ and any polynomial $P_{1},\ldots,P_{n}\in\mathbb{R}[x]$, the vector-valued process $(\mathcal{L}_{t}^{N}(P_{1}),\ldots,\mathcal{L}_{t}^{N}(P_{n}))_{t\in[0,T]}$ converges to $(\mathcal{L}_{t}(P_{1}),\ldots,\mathcal{L}_{t}(P_{n}))_{t\in[0,T]}$ in distribution.

The limit process $\{\mathcal{L}_{t}(f),t\in[0,T]\}_{f\in\mathbb{R}[x]}$ is characterized by the following properties.

For $P_{1},P_{2}\in\mathbb{R}[x]$, $\alpha_{1},\alpha_{2}\in\mathbb{R}$, $t\in[0,T]$,

[TABLE] 2. 2.

The basis $\{\mathcal{L}_{t}(x^{n}),t\in[0,T]\}_{n\in\mathbb{N}}$ of $\{\mathcal{L}_{t}(f),t\in[0,T]\}_{f\in\mathbb{R}[x]}$ satisfies

[TABLE]

and for $n\geq 0$,

[TABLE]

where $\{G_{t}(x^{n}),t\in[0,T]\}_{n\in\mathbb{N}}$ is a centered Gaussian family with the covariance

[TABLE]

Proof.

The proof is similar to the proofs of the Wishart case (Lemma 3.1, Lemma 3.2 and Theorem 3.2), which is sketched below.

Consider the following SDE, for $1\leq i\leq N$,

[TABLE]

Then $\frac{d}{dt}\mathbb{E}[f(u^{N}(t))]$ vanishes for any $f\in C_{b}^{2}(\mathbb{R})$ if $u^{N}(t)$ has the distribution $\mathbb{Q}^{N}$ given in (3.27), and hence the process $u^{N}(t)$ with initial value $u^{N}(0)=\xi^{N}$ is stationary (see (Anderson et al.,, 2010, Lemma 4.3.17 )). Let $v_{i}^{N}(t)=\sqrt{t+a}u_{i}^{N}(t)+b$ for $1\leq i\leq N$. Then $v^{N}(t)$ and $\lambda^{N}(t)$ solve the same SDEs (3.25), and by the comparison principle in Corollary 3.1, we have

[TABLE]

A similar argument leads to

[TABLE]

Therefore,

[TABLE]

Using the tail probability estimation based on the density function (3.27) of $u_{i}^{N}(t)$,

[TABLE]

where $\alpha$ is positive constant independent of $N$, we obtain

[TABLE]

for $k\in[0,\alpha N]$. Then a similar argument in the proof of Lemma 3.2 leads to

[TABLE]

for some positive constant $C(a,b,T)$ depending only on $(a,b,T)$ and all $p\geq 0$, $N\geq\alpha p$ for some positive constant $\alpha$.

Then applying Corollary 2.2 and following the approach in the proof of Theorem 3.2, we may get the desired result. ∎

Remark 3.2**.**

The above result was obtained in (Anderson et al.,, 2010, Theorem 4.3.20), under a slightly stronger condition on the initial value. We would like to point out that there should be a constant factor $2/\beta$ 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).

Similar to the Wishart case, the self-similarity of the Brownian motion implies $\mathcal{L}_{t}(f(x))\overset{d}{=}\mathcal{L}_{1}(f(\sqrt{t}x))$ and $\langle f(x),\mu_{t}\rangle=\langle f(\sqrt{t}x),\mu_{1}\rangle$ when the initial value $X_{0}^{N}=0$. Thus, (2) can be simplified as, for $n\geq 0$,

[TABLE]

with covariance functions

[TABLE]

The case $t=1$ corresponds to the classical GOE matrix, and $\mu_{1}$ is the semicircle law. Some beginning terms are $\mathcal{L}_{1}(1)=0,\mathcal{L}_{1}(x)=G_{1}(x)$ and $\mathcal{L}_{1}(x^{2})=1+G_{1}(x^{2})$. By (3.31), for $m\geq 2$, $\mathcal{L}_{1}(x^{m})$ has the distribution of a linear combination of central Gaussian variables $\{G_{1}(x^{j}),1\leq j\leq m\}$.

The following Corollary extends the result of Theorem 3.3.

Corollary 3.3**.**

Consider the following SDEs

[TABLE]

where $b_{N}(x)$ satisfies, for some constant $c\in{\mathbb{R}}$,

[TABLE]

Furthermore, assume the same initial conditions as in Theorem 3.3. Then the conclusion of Theorem 3.3 still holds with (2) replaced by

[TABLE]

for $n\geq-1$.

Proof.

Set $c_{1}=c-1$ and $c_{2}=c+1$. Then by (3.33), there exist $N_{0}\in\mathbb{N}$ such that for $N\geq N_{0}$, $c_{1}\leq\|b_{N}(x)\|_{L^{\infty}({\mathbb{R}})}\leq c_{2}$. Without loss of generality, we assume $c_{1}\leq\|b_{N}(x)\|_{L^{\infty}({\mathbb{R}})}\leq c_{2}$ for all $N\geq 1$.

Consider the following two systems of SDEs:

[TABLE]

and

[TABLE]

with the initial conditions $x_{i}^{N}(0)=y_{i}^{N}(0)=\lambda_{i}^{N}(0)$ for $1\leq i\leq N$. By the comparison principle Theorem 3.1, we have

[TABLE]

Thus, for $p\geq 1$, we have

[TABLE]

almost surely, where $\{L_{N}^{(x)}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ and $\{L_{N}^{(y)}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ are the empirical measures of the two particle systems $(x_{i}^{N}(t)-c_{1}t)_{1\leq i\leq N}$ and $(y_{i}^{N}(t)-c_{2}t)_{1\leq i\leq N}$, respectively.

It is easy to verify that both $(x_{i}^{N}(t)-c_{1}t)_{1\leq i\leq N}$ and $(y_{i}^{N}(t)-c_{2}t)_{1\leq i\leq N}$ solve the Dyson’s SDEs (3.25). By (3.30) in the proof Theorem 3.3, we have

[TABLE]

and consequently, by (3.37)

[TABLE]

for some positive constant $C(a,b,T)$ depending only on $(a,b,T)$ and all $p\geq 1$, $N\geq\alpha p$ for some positive constant $\alpha$.

Note that (3.33) also implies that $b_{N}(x)$ converges to the constant $c$ uniformly as $N\rightarrow\infty$. Then applying Corollary 2.2 and following the approach in the proof of Theorem 3.2, we get the desired result.

∎

3.4 Application to eigenvalues of symmetric OU matrix

In this subsection, we discuss the CLT for the eigenvalues of a symmetric Ornstein-Uhlenbeck matrix process. It was shown in Chan, (1992), the symmetric $N\times N$ matrix $X^{N}(t)$, whose entries $\{X_{ij}^{N}(t),i\leq j\}$ are independent Ornstein-Uhlenbeck processes with invariant distribution $N(0,(1+\delta_{ij})/(2N))$, where $\delta_{ij}$ is the Kronecker delta function, is the solution of the matrix SDE (1.1) with the coefficient functions

[TABLE]

The SDEs of the eigenvalue processes are

[TABLE]

In this case, we have

[TABLE]

Similar to the eigenvalues of Wishart process and Dyson’s Brownian motion, we have the following CLT.

Theorem 3.4**.**

Let $\xi^{N}=(\xi^{N}_{1},\dots,\xi^{N}_{N})$ be a random vector that is independent of $(W_{1},\dots,W_{N})$ and has (3.27) as its joint probability density function. Assume that $(\lambda_{1}^{N}(0),\dots,\lambda_{N}^{N}(0))$ is independent of $(W_{1},\dots,W_{N})$ and that there exist constants $a,b\geq 0$, such that

[TABLE]

for $1\leq i\leq N$ almost surely.

Besides, assume that for any polynomial $f(x)\in\mathbb{R}[x]$, the initial value $\mathcal{L}_{0}^{N}(f)$ converges in probability to a random variable $\mathcal{L}_{0}(f)$. Furthermore, assume that for all $n\in\mathbb{N}$,

[TABLE]

for all $p\geq 1$.

Then for any $0<T<\infty$, there exists a family of processes $\{\mathcal{L}_{t}(f),t\in[0,T]\}_{f\in\mathbb{R}[x]}$, such that for any $n\in\mathbb{N}$ and any polynomial $P_{1},\ldots,P_{n}\in\mathbb{R}[x]$, the vector-valued process $(\mathcal{L}_{t}^{N}(P_{1}),\ldots,\mathcal{L}_{t}^{N}(P_{n}))_{t\in[0,T]}$ converges to $(\mathcal{L}_{t}(P_{1}),\ldots,\mathcal{L}_{t}(P_{n}))_{t\in[0,T]}$ in distribution.

The limit process $\{\mathcal{L}_{t}(f),t\in[0,T]\}_{f\in\mathbb{R}[x]}$ is characterized by the following properties.

For $P_{1},P_{2}\in\mathbb{R}[x]$, $\alpha_{1},\alpha_{2}\in\mathbb{R}$, $t\in[0,T]$,

[TABLE] 2. 2.

The basis $\{\mathcal{L}_{t}(x^{n}),t\in[0,T]\}_{n\in\mathbb{N}}$ of $\{\mathcal{L}_{t}(f),t\in[0,T]\}_{f\in\mathbb{R}[x]}$ satisfies

[TABLE]

and for $n\geq 0$,

[TABLE]

where

[TABLE]

and $\{G_{t}(x^{n}),t\in[0,T]\}_{n\in\mathbb{N}}$ is a centered Gaussian family with the covariance

[TABLE]

Proof.

Consider the symmetric OU matrix $X^{N}_{t}$, of which the entries $\{X_{ij}^{N}(t)\}$ satisfy

[TABLE]

where $\{B_{ij}(t),i\leq j\}$ is a family of independent Brownian motions. Denoting by

[TABLE]

the solution to (3.42) is given by

[TABLE]

The stochastic integral is a martingale with quadratic variation

[TABLE]

By Knight’s Theorem, there exists a family of independent standard one-dimensional Brownian motions $\{\tilde{B}_{ij}(t),i\leq j\}$, such that

[TABLE]

Thus, we have

[TABLE]

Let $Y^{N}_{t}$ be a matrix-valued stochastic process whose entries $\{Y_{ij}^{N}(t),i\leq j\}$ are given by

[TABLE]

with $Y_{ij}^{N}(0)=\sqrt{2}X_{ij}^{N}(0),1\leq i\leq j\leq N$. Then $Y^{N}$ is the scaled symmetric Brownian motion introduced in section 3.3. By (3.43) and (3.44),

[TABLE]

and hence

[TABLE]

where $\{\lambda_{i}^{N}(t)\}$ and $\{\tilde{\lambda}_{i}^{N}(t)\}$ are the eigenvalues of $X^{N}(t)$ and $Y^{N}(t)$, respectively.

Thus, almost surely, we have

[TABLE]

where $L_{N}(t)$ and $\tilde{L}_{N}(t)$ are the empirical measures of $\{\lambda_{i}^{N}(t)\}$ and $\{\tilde{\lambda}_{i}^{N}(t)\}$, respectively. Note that $\tilde{\lambda}_{i}^{N}(0)=\sqrt{2}\lambda_{i}^{N}(0)$ satisfies condition (3.28) in Theorem 3.3 with the constants $a$ and $b$ replaced by $2a$ and $\sqrt{2}b$. By the estimation (3.30), for all $p\geq 1$ and $N\geq\alpha p$ for some positive constant $\alpha$, we have

[TABLE]

where $C^{\prime}(a,b,T)$ is positive constant depending only on $(a,b,T)$.

Thus, by Lemma 3.2 and Corollary 2.2, $Q_{t}^{N}(x^{n})$ defined by (1) converges in distribution to a centered Gaussian family $\{G_{t}(x^{n}),t\in[0,T]\}_{n\in\mathbb{N}}$ with covariance given by (3.41). Similar to (3.17), for $n\geq-1$, we have

[TABLE]

Letting $N\to\infty$, we have

[TABLE]

where $R_{t}(n)$ is given in (3.40). Without loss of generality, we may replace “$\overset{d}{=}$” by “$=$” in the above equation. Thus we have

[TABLE]

whose solution is given by (2).

The proof is concluded. ∎

Now we extend the result of Theorem 3.4 to a generalized system of (3.38).

Corollary 3.4**.**

Consider the following SDEs

[TABLE]

where $b_{N}(x)$ satisfies, for some constant $c\in\mathbb{R}$,

[TABLE]

Furthermore, assume the same initial conditions as in Theorem 3.4. Then the conclusion of Theorem 3.4 still holds with $R_{t}(n)$ in (3.40) replaced by

[TABLE]

Proof.

The proof is similar to the proof of Corollary 3.3, which is sketched below.

By (3.47), without loss of generality, we assume

[TABLE]

for all $N\geq 1$. Then we have

[TABLE]

where the processes $(x_{i}^{N}(t))_{1\leq i\leq N}$ and $(y_{i}^{N}(t))_{1\leq i\leq N}$ are the solutions of the following systems of SDEs respectively:

[TABLE]

and

[TABLE]

with the initial conditions $x_{i}^{N}(0)=y_{i}^{N}(0)=\lambda_{i}^{N}(0)$ for $1\leq i\leq N$. Noting that $(x_{i}^{N}(t)-2c+2)_{1\leq i\leq N}$ and $(y_{i}^{N}(t)-2c-2)_{1\leq i\leq N}$ 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 2.2 and following the approach in the proof of Theorem 3.2, we get the desired result. ∎

4 Useful 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 2.1.

Lemma 4.1** (Rebolledo’s Theorem).**

Let $n\in\mathbb{N}$, and let $\{M_{N}\}_{N\in\mathbb{N}}$ be a sequence of continuous centered martingales with values in $\mathbb{R}^{n}$. If the quadratic variation $\langle M_{N}\rangle_{t}$ converges in $L^{1}(\Omega)$ to a continuous deterministic function $\phi(t)$ for all $t>0$, then for any $T>0$, as a continuous process from $[0,T]$ to $\mathbb{R}^{n}$, $(M_{N}(t),t\in[0,T])$ converges in law to a Gaussian process $G$ with mean 0 and covariance

[TABLE]

Section 3.1 was based on the following comparison principle for multi-dimensional SDEs which is a direct consequence of (Geiß and Manthey,, 1994, Theorem 1.1 and Theorem 1.2).

Lemma 4.2**.**

On a certain complete probability space equipped with a filtration that satisfies the usual conditions ((Karatzas and Shreve,, 1991, Definition 2.25)), consider the following SDEs

[TABLE]

where $\{W(t),t\geq 0\}$ is a $d$-dimensional Brownian motion. Assume the solutions to SDEs (4.1) are pathwisely unique and non-exploding. If the following conditions are satisfied,

the drift functions $b^{(1)}(t,x)$ and $b^{(2)}(t,x)$ are continuous mappings from $[0,\infty)\times\mathbb{R}^{n}$ to $\mathbb{R}^{n}$. Besides, they are quasi-monotonously increasing in the sense that for $1\leq i\leq n$ and $j=1,2$, $b_{i}^{(j)}(t,x)\leq b_{i}^{(j)}(t,y)$, whenever $x_{i}=y_{i}$ and $x_{l}\leq y_{l}$ for $l\in\{1,\ldots,n\}\setminus\{i\}$; 2. 2.

the dispersion matrix $\sigma(t,x)$ is a continuous mapping from $[0,\infty)\times\mathbb{R}^{n}$ to $\mathbb{R}^{n\times d}$ that satisfies the following condition

[TABLE]

for all $t\geq 0$ and $x=(x_{1},\ldots,x_{n})^{\intercal},y=(y_{1},\ldots,y_{n})^{\intercal}\in\mathbb{R}^{n}$, where $\rho:[0,\infty)\rightarrow[0,\infty)$ is a strictly increasing function with $\rho(0)=0$ and

[TABLE] 3. 3.

$b_{i}^{(1)}(t,x)\leq b_{i}^{(2)}(t,x)$* for all $1\leq i\leq n$, $t\geq 0$, $x\in\mathbb{R}^{n}$;* 4. 4.

for $1\leq i\leq n$, $Y_{i}(0)\leq Z_{i}(0)$ almost surely,

then we have

[TABLE]

The following lemma was employed in the proof of Proposition 3.1.

Lemma 4.3**.**

Let $u^{N}(t)$ be the strong solution to (3.8). If $u^{N}(t)$ is distributed according to $\mathbb{P}^{N}$ in (3.7), then for $f\in C_{b}^{2}(\mathbb{R}^{N})$,

[TABLE]

Proof.

For $f\in C_{b}^{2}(\mathbb{R}^{N})$, applying Itô’s formula to (3.8), we have

[TABLE]

Here, $\partial_{i}$ is the partial derivative with respect to the $i$-th component $x_{i}$. Therefore, for $t\geq 0$,

[TABLE]

Thus it suffices to show, with the density function $p(x)$ in (3.7),

[TABLE]

where $\Delta_{N}=\{x\in\mathbb{R}^{N}:0<x_{1}<\ldots<x_{N}\}$ is the support of $\mathbb{P}^{N}$. Noting that $p(x)$ vanishes on $\partial\Delta_{N}$, we have by the integration by parts formula,

[TABLE]

Hence, to show (4), it is sufficient to verify

[TABLE]

By the chain rule,

[TABLE]

Hence,

[TABLE]

which gives the desired result. ∎