CLT for random quadratic forms based on sample means and sample covariance matrices

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arXiv:2210.11215v1 [math.ST] 20 Oct 2022

CLT for random quadratic forms based on

sample means and sample covariance

matrices

Wenzhi Yang1, Yiming Liu 2, Guangming Pan3and Wang Zhou 4

1School of Big Data and Statistics, Anhui University, e-mail: *wzyang@ahu.edu.cn

2School of Economics, Jinan University, e-mail: **liuy0135@e.ntu.edu.sg

3School of Physical and Mathematical Sciences, Nanyang Technological University, e-mail:

†GMPAN@ntu.edu.sg

4Department of Statistics and Data Science, National University of Singapore, e-mail:

‡wangzhou@nus.edu.sg

Abstract: In this paper, we use dimensional reduction technique to study

the central limit theory (CLT) random quadratic forms based on sample

means and sample covariance matrices. Speciﬁcally, we use a matrix de-

noted by Up×q, to map q-dimensional sample vectors to a pdimensional

subspace, where q≥por q≫p. Under the condition of p/n →0as

(p, n)→ ∞, we obtain the CLT of random quadratic forms for the sample

means and sample covariance matrices.

MSC2020 subject classiﬁcations:Primary 62E20.

Keywords and phrases: Random quadratic forms, Sample means, Sam-

ple covariance matrices, Central limit theory.

1. Introduction

Consider the multivariate model

yj=µ+Γxj,1≤j≤n, (1.1)

where µis a mean vector in Rq,Γis a qby mmatrix, q≤m,Σq=ΓΓ⊤

is a positive deﬁnite covariance matrix (denoted by Σq≻0) and x1,...,xn

are independent and identically distributed (i.i.d.)m-dimensional real random

vectors with the mean vector Ex1=0mand Cov(x1) = Im. So y1,...,ynare q-

dimensional real random vectors and sample mean statistic ¯

y=1

nPn

i=1 yiand

sample covariance matrix Sn=1

nPn

i=1(yi−¯

y)(yi−¯

y)⊤are very important

in the mean vector test and covariance matrix test (see Anderson [1]). Here, ⊤

represents the transpose of a vector. However, as the dimension increases, there

are problems in the mean vector test and covariance matrix test. For example,

when q > n −1, the inverse of Sndoes not exist so the Hotelling’s T2test

obtained by Hotelling[13]

T2=n(¯

y−µ)⊤S−1

n(¯

y−µ),(1.2)

/ CLT for random quadratic forms 2

fails to test the high dimensional mean. There are many papers to study the

high dimensional means and covariance matrix. For example, Bai et al.[2], Bai

and Saranadasa[3], Bai and Silverstein[5,6], Chen et al. [11], Chen and Qin[12],

Pan and Zhou[14], Srivastava [15], Srivastava and Du[16], Srivastava and Li[17],

etc.

Inspired by dimension reduction techniques such as principle component anal-

ysis, we aim to project the observations into a low dimensional subspace through

a reduction matrix Up×q(p≪q). The projected observations can be written in

a vector form as

zj=Uyj=Uµ+UΓxj,1≤j≤n, (1.3)

where U=Up×qand Γ=Γq×mare nonrandom matrices and x1,...,xnare

i.i.d. m-dimensional real random vectors with the mean vector 0mand covari-

ance matrix Im. In view of (1.3), the centered sample covariance matrix is

deﬁned by

Sn=1

j=1

(zj−¯

z)(zj−¯

z)⊤=1

j=1

UΓ(xj−¯

x)(xj−¯

x)⊤Γ⊤U⊤(1.4)

where ¯

z=n−1Pn

j=1 zj=Uµ+UΓ¯

xand ¯

x=n−1Pn

j=1 xj.

This is the ﬁrst paper in a series of two papers. In this paper, we will study

the central limit theory (CLT) of random quadratic forms involving sample

means and sample covariance matrices when p/n →0as (p, n)→ ∞ but q

can be arbitrarily large. For the details, please see Theorem 1. In our second

paper, we use Theorem 1to derive the CLT Hotelling’s T2test when p/n →0.

To investigate this limit theory, we recall some basic deﬁnitions from random

matrix theory.

Let A=Ap×pbe any p×psquare matrix with real eigenvalues denoted by

λ1≥ ··· ≥ λp. The empirical spectral distribution (ESD) of Ais deﬁned by

FA(x) = 1

j=1

I(λj≤x), x ∈R,

where I(·)is the indicator function. The Stieltjes transform of FAis given by

mFA(z) = Z1

x−zdF A(x),

where z=u+iv ∈C+≡ {z∈C,I(z)>0}. Let Xn= (x1,...,xn)and

Σm=ΓΓ⊤. The famous Marcěnko Pastur (M-P) law states that the ESD of

Sn=1

nΓXnX⊤

nΓ⊤, i.e., FSn(x), weakly converges to a nonrandom probability

distribution function Fc,H (x)whose Stieltjes transform is determined by the

following equation

m(z) = Z1

λ(1 −c−czm(z)) −zdH(λ)

/ CLT for random quadratic forms 3

for each z∈C+as m/n →c∈(0,∞)and n→ ∞, where His the limiting

spectral distribution of ΓΓ⊤. One can refer to Bai and Silverstein [6] for more

details.

The rest of this paper is organized as follows. Section 2 presents the CLT

for random quadratic forms with dimensionality reduction, and its proof is pre-

sented in Section 3. Some auxiliary proofs are presented in Appendix A.1-A.2.

2. CLT for random quadratic forms

Assumption:

(A.1) Let Xn= (x1,...,xn) = (Xij )be an m×nmatrix whose entries are

i.i.d. real random variables with EX11 = 0, Var(X11) = 1 and EX4

11 <∞.

(A.2) For p≤q≤m, let U=Up×q,Γ=Γq×mand Σpbe nonrandom

matrices satisfying

UΓΓ⊤U⊤=Σp≻0.

(A.3) Let cn=p/n and cn=O(n−η)for some η∈(0,1).

In the following, we will study the limiting distributions of random quadratic

forms involving sample means and sample covariance matrices. Since Σp≻0,

Σ−1/2

pexists. So we take the transform

µ=Σ−1

pUµ,B=Σ−1

pUΓ,(2.1)

zj=Σ−1

pzj=Σ−1

pUyj=˜

µ+Bxj,1≤j≤n, (2.2)

Sn=1

j=1

(˜

zj−¯

z)(˜

zj−¯

z)⊤=1

j=1

B(xj−¯

x)(xj−¯

x)⊤B⊤,(2.3)

where ¯

x=1

nPn

j=1 xjand

z=1

j=1

zj=Σ−1

pUµ+1

j=1

Σ−1

pUΓxj=˜

µ+B¯

x.(2.4)

Let λ1, λ2,...,λpdenote eigenvalues of ˜

Sndeﬁned by (2.3). For any analytic

function f(·), deﬁne

f(˜

Sn) = V⊤diag(f(λ1),...,f(λp))V,

where V⊤diag(λ1,...,λp)Vdenotes the spectral decomposition of ˜

Sn.

Theorem 1. Let assumptions (A.1)-(A.3) be satisﬁed. Assume that g(x)is

a function with a continuous ﬁrst derivative in a neighborhood of 0 such that

g′(0) 6= 0,f(x)is analytic on an open region containing the interval

[1 −δ, 1 + δ],∃δ∈(0,1) (2.5)

/ CLT for random quadratic forms 4

and satisﬁes f(1) 6= 0. Denote ˜

µ=Σ−1/2

pUµand

Xn=n

√pcnh(¯

z−˜

µ)⊤f(˜

Sn)(¯

z−˜

µ)

k¯

z−˜

µk2−f(1)i, Yn=n

√phg((¯

z−˜

µ)⊤(¯

z−˜

µ))−g(cn)i,

where ˜

Snand ¯

zare deﬁned by (2.3)and (2.4), respectively. As min(p, n)→ ∞,

(Xn, Yn)d

−→ (X, Y ),(2.6)

where (X, Y )∼N(0,Γ1)with

Γ1=2f2(1) 2g′(0)f(1)

2g′(0)f(1) 2(g′(0))2.

Remark 1. The conditions in (A.1) are usually used to study the random ma-

trix (see [5,6]). Condition (A.2) requests the reduced dimensional covariance

matrix to be a positive deﬁnite matrix. When q/n →c∈(0,1), Pan and Zhou[14]

obtained the random quadratic forms based on sample means and sample covari-

ance matrices and gave its application to the Hotelling’s T2test. In this paper,

we use Up×qmatrix to map q-dimensional sample vectors to a pdimensional

subspace, where q≥por q≫p. Under the condition of p/n →0as (p, n)→ ∞,

we obtain the CLT of random quadratic forms for the sample means and sample

covariance matrices. In our second paper, we will use Theorem 1to derive the

CLT Hotelling’s T2test when p/n →0.

3. Outline of the proofs

The proof of Theorem 1relies on Lemma 1below that deals with the asymp-

totic joint distribution of

Xn(z) = n

√pcnh¯

x⊤B⊤(˜

Sn−zIp)−1B¯

kB¯

xk2−m(z)i,

Yn=n

√phg(¯

x⊤B⊤B¯

x)−g(cn)i,

where B=Σ−1

2UΓ,cn=p/n,m(z) = R1

x−zdH(x)and H(x) = I(1 ≤x)for

x∈R. The stochastic process Xn(z)is deﬁned on a contour Cas follows: Let

v0>0be arbitrary and set Cu={u+iv0, u ∈[ul, ur]}, where ul= 1 −δand

ur= 1 + δfor some δ∈(0,1). Then, we deﬁne

C+={ul+iv :v∈[0, v0]} ∪ Cu∪ {ur+iv :v∈[0, v0]}

and denote C−to be the symmetric part of C+about the real axis. Then, let

C=C+∪ C−. Let

Sn=1

j=1

Bxjx⊤

jB⊤.(3.1)

/ CLT for random quadratic forms 5

By random matrix theory, ˜

Sndeﬁned by (2.3) can be replaced by ˜

Sndeﬁned by

(3.1). It is diﬃcult to control the spectral norm of (˜

Sn−zIp)−1or (˜

Sn−zIp)−1

on the whole contour C(for example v= 0), we thus deﬁne a truncated version

Xn(z)of Xn(z)(see in Bai and Silverstein [5]). For some ϑ∈(0,1), we choose

a positive number sequence {ρn}satisfying

ρn↓0and ρn≥n−ϑ.(3.2)

Let Cl={ul+iv :v∈[n−1ρn, v0]}and Cr={ur+iv :v∈[n−1ρn, v0]}. Write

C+

n=Cl∪Cu∪ Cr. Consequently, for z=u+iv ∈ C, a truncated process ˆ

Xn(z)

is deﬁned as follows

Xn(z) = 









Xn(z),if z∈ C+

n∪ C−

nv +ρn

2ρn

Xn(zr1) + ρn−nv

2ρn

Xn(zr2),if u=ur, v ∈[−ρn

n,ρn

n];

nv +ρn

2ρn

Xn(zl1) + ρn−nv

2ρn

Xn(zl2),if u=ul>0, v ∈[−ρn

n,ρn

n].

(3.3)

Here, zr1=ur+iρn

n,zr2=ur−iρn

n,zl1=ul+iρn

n,zl2=ul−iρn

nand C−

nis

the symmetric part of C+

nabout the real axis.

We now give the asymptotic joint distribution of (ˆ

Xn(z), Yn)in Lemma 1.

Lemma 1. Under the conditions of Theorem 1, for z∈ C, we have

(ˆ

Xn(z), Yn)d

−→ (X(z), Y ),(3.4)

where (X(z), Y )∼N(0,Γ2),

Γ2="2

(1−z)2

2g′(0)

1−z

2g′(0)

1−z2(g′(0))2#.

To transfer Lemma 1to Theorem 1, we introduce a new ESD function

F˜

2(x) =

j=1

jI(λj≤x), x ∈R,

where t= (t1, . . . , tp)⊤=VB¯

x/kB¯

xk,B=Σ−1

pUΓ and Vis the eigenvector

matrix of ˜

Sndeﬁned by (2.3) (see Bai et al.[2]). Following Theorem 1 in Bai

et al.[2] and Remark 3 in Pan and Zhou [14], one can similarly obtain that as

p/n →0,

F˜

2(x)→H(x),a.s.,

where H(x) = I(1 ≤x)for x∈R. Then, by analyticity of f(x),¯

x⊤B⊤f(˜

Sn)B¯

kB¯

xk2in

Theorem 1is transferred to ¯

x⊤B⊤(˜

Sn−zIp)−1B¯

kB¯

xk2and Stieljes transform of F˜

2(x).

Let A−1

n(z) = (˜

Sn+zIp)−1. Note that

x⊤B⊤A−1

n(z)B¯

1−¯

x⊤B⊤A−1

n(z)B¯

x=¯

x⊤B⊤(˜

Sn−zIp)−1B¯

x,(3.5)

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摘要：

arXiv:2210.11215v1[math.ST]20Oct2022CLTforrandomquadraticformsbasedonsamplemeansandsamplecovariancematricesWenzhiYang1,YimingLiu2,GuangmingPan3andWangZhou41SchoolofBigDataandStatistics,AnhuiUniversity,e-mail:*wzyang@ahu.edu.cn2SchoolofEconomics,JinanUniversity,e-mail:**liuy0135@e.ntu.edu.sg3Schoolof...

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