Broken-stick components retention rule for
equi-correlated normal population
Yohji Akama∗
December 7, 2023
Abstract
We consider a p-dimensional normal population such that the different variables have a common correlation coefficient ρ ∈ [0, 1). We call such a population an equi-correlated normal population (ENP). We draw a sample ofsize n from an ENP, and form a sample covariance matrix S and a sample
correlation matrix C. For a proportional limiting regime, we deduce thatthe second largest eigenvalue of S converges in distribution to the type-1
Tracy-Widom distribution. From this, the number of principal components(PCs) the Frontier’s broken-stick rule takes from S converges almost surely
to 0 for ρ = 0, and 1 for 0 < ρ < 1. For the limiting distributions, we
compare the second largest eigenvalue of C of ENP, to the eigenvalues ofthe sample correlation matrix of a bounded spiked eigenvalues model.

1 Introduction
In multivariate statistical analysis, we often employ principal component analysis(PCA) ([20],[24]) and factor analysis (FA) ([22],[33]) to find principal components (PCs)and factors to illustrate a great part of the variance or the correlation in the pdimensional data by using the sample covariance matrix S or the sample correlation
matrix C of that data. The PCs and the factors are “significant” if the corresponding
eigenvalues of S or C of the data are large.
Estimating the number of significant PCs and factors in PCA and FA are importantin finance ([11, 12],[30],[13]), biology ([35]), engineering ([39]), psychometrics [25], andso on. Components retention rules are methods to determine the number q of the mostimportant PCs. Components retention rules not only balance the accuracy (or fit) of
the model with ease of analysis and the potential loss of information, but also identify
the cause of phenomena. Many components retention rules have been proposed andcompared in [20],[24],[34],[8],[15], to cite a few.
In [2], the author and Husnaqilati handled Kaiser’s intercorrelation [26], by assuming an equi-correlated normal population (ENP), that is, the multinormal populationsuch that the variables have a common correlation coefficient ρ ≥ 0. Let us call ρ a population equi-correlation coefficient (PECC). [2] assumed that both the data dimension
p and the sample size n tend to infinity with p/n → c > 0, and identified the limiting
spectral distribution of the sample correlation matrix C. Then, they studied the limiting behavior of Kaiser’s rule in the limit c → 0, and explained Kaiser’s observation [27]as the discontinuity of the limiting value at ρ = 0. The phase transitions of various components retention rules concerning the PECC ρ could be interesting from the∗

Mathematical Institute, Graduate School of Science, Tohoku University, Sendai, 980-8578, Japan.
yoji.akama.e8@tohoku.ac.jp
2023 年度科学研究費シンポジウム International Symposium on Recent Advances in Theories andMethodologies for Large Complex Data (2023年12月7–9日)
Keywords: Equi-correlated normal population, sample covariance matrix, the 2nd largest eigenvalue.

viewpoint of random matrix theory in relation to the recent study ([31],[7]) on a sample covariance matrix S having the unbounded largest eigenvalue (See [4] for boundedlargest eigenvalue case). Above all, an ENP is related to intraclass correlation [23].
The broken-stick rule (Frontier [14], Jackson [19]) is a components retention rulemotivated by the study of MacArthur [29] on animal community structure. Let BS(C)(BS(S), resp.) be the number of PCs of a sample correlation matrix C ∈ Rp×p (sample
covariance matrix S ∈ Rp×p , resp.) that the broken-stick rule retains. The i-th (1 ≤i ≤ p) largest eigenvalue of a symmetric matrix M ∈ Rp×p is written by λi (M).
)
(
p
X
(1)
BS(C) := inf i ∈ [1, p] λi (C) ≤k −1 − 1.
k=i

Pp
−1

−1
By [18, Theorem 2.2], for i (1 ≤ i ≤ p), p
is the expectation of the length
k=i kof the i-th longest subinterval of I = [0, 1] divided by p−1 points independentlyPobeyingp−1
the uniform distribution on I. Thus BS(C) <P∞, because otherwise
Pp Pλp i (C)−1> k=i kp(1 ≤ i ≤ p) which implies p = trace C == p. Fori=1 λi (C) >
i=1
k=i k
i.i.d.
X1 , . . . , Xn ∼ Np (0, Σp (ρ)) and ρ ∈ [0, 1), letting inf ∅ = ∞, we define BS(S) to be(1) with C replaced by S.
a.s.
P
Let → and → be the almost sure convergence and the convergence in probability,
respectively. For an ENP and the limiting regime n, p → ∞, p/n → c > 0, we will
a.s.
a.s.
prove BS(C) → 0 for ρ = 0, 0 < lim inf BS(C) (a.s.) for 0 < ρ < 1, BS(S) → 0 forPρ = 0, and BS(S) → 1 for 0 < ρ < 1.
This paper is organized as follows: The next section proves above results by studying
the limiting distribution of λ2 (S) of an ENP. Section 3 analyzes the behaviors of thebroken-stick rule and various components retention rules, for real datasets of S&P
500 return time-series, Fama-French 100 portfolios time-series, and binary multiple
sequence alignment [35]. Section 4 does simulation study for the limiting distributionof λ2 (C) of an ENP, in comparison to the theoretical study [32] of a bounded spiked
eigenvalues model.

2 Limiting behavior of broken-stick rule for sample covariance matrix of equi-correlated normal populations
Below µ ∈ Rp , σ 2 > 0, ρ ∈ [0, 1) and c > 0 are deterministic, and Σp (ρ) = (1 −ρ)Ip + [ρ] ∈ Rp×p . We derive p,n→∞lim λ1 (S)/p = ρ (a.s.) for the population Np (0, Σp (ρ))
p/n→cfrom [41], [31, Theorem 2.1], [7, Theorem 2.1] and so on.
i.i.d.
Proposition 2.1 ([1]). Let X1 , . . . , Xn ∼ Np (µ, DΣp (ρ)D) with D ∈ GLp (R) being
deterministic and diagonal. Then p,n→∞lim λ1 (C)/p = ρ (a.s.).
p/n→cGuttman-Kaiser rule ([17],[25],[42]) is a components retention rule that retains the
eigenvalues λi (M) where M ∈ Rp×p is a sample covariance matrix S or a sample
correlation matrix C. Let GK(M) := # { i ∈ [1, p] | λi (M) ≥ trace M/p } .
i.i.d.
Proposition 2.2 ([2]). Let X1 , . . . , Xn ∼ Np (µ, DΣp (ρ)D) with D ∈ GLp (R)being deterministic and diagonal. If (M = S, µ = 0, D = Ip ) or M = C, thenlimp,n→∞ GK(M)/p almost surely converges as c → 0 to 1/2 (ρ = 0); 0 (ρ > 0).
p/n→c

In contrast, limp,n→∞ BS(C) increases from 0 as ρ increases from 0.
p/n→ci.i.d.
Theorem 2.3. Suppose X1 , . . . , Xn ∼ Np (0, σ 2 Σp (ρ)) and n, p → ∞, p/n → c.
a.s.
PThen BS(S) → 0 (ρ = 0) and BS(S) → 1 (ρ > 0).
The proof of the case ρ = 0 is by
Proposition 2.4 ([5, Theorem 3.1]). If { xij | i,j ≥ 1 } are i.i.d., E x11 = 0, E x211 = 1,√
and E |x11 |4 < ∞, then limp,n→∞ λ1 (S) = (1 + c)2 (a.s.).
p/n→c

We demonstrate Theorem 2.3 for the case ρ > 0 by establishing that the limiting
distribution of the second largest eigenvalue of S is the type-1 Tracy-Widom distribution scaled by 1 − ρ. For this, we use [7, Theorem 2.5] of which setting and theassumptions are reviewed below. Let the data matrix be
X = ΓΞ where Ξ = [ξij ]1≤i≤p+l, 1≤j≤n = [ξ1 · · · ξn ] ∈ R(p+l)×n(2)

satisfying the following two conditions:
Assumption 1 ([7, Assumption 1]). { ξj = [ξ1j · · · ξp+l,j ]⊤ | 1 ≤ j ≤ n } are i.i.d. random vectors. { ξij | 1 ≤ i ≤ p + l, 1 ≤ j ≤ n } are independent random variables suchthat E(ξij ) = 0, E |ξij |2 = 1, and supi,j E |ξij |4 < C for some constant C.
Assumption 2 ([7, Assumption 7]). supi,j E |ξij |k < ck for some constant ck (k ∈ N).
Let Γ ∈ Rp×(p+l) be a deterministic matrix with l/p → 0, Σ = ΓΓ⊤ ∈ Rp×p ,ℓi = λi (Σ) (1 ≤ i ≤ p), and Λ = diag(ℓ1 , . . . , ℓp ) ∈ Rp×p . Let Γ = VΛ1/2 U be a
singular value decomposition with V ∈ Rp×p , U ∈ Rp×(p+l) , and VV⊤ = UU⊤ = Ip .
For two sequences of positive numbers an and bn , an ≳ bn means an ≥ cbn for someconstant c > 0. Let an ∼ bn be an ≳ bn and bn ≳ an . Let an ≪ bn be an /bn → 0.
Let a deterministic integer K > 0 satisfy the following two assumptions:
Assumption 3 ([7, Assumption 2]). p ≳ n, di := p/(nℓi ) → 0 (n, p → ∞) fori = 1, 2, . . . , K, max ℓi is bounded by a constant C, Kn−1/6 → 0, and K 2 dK → 0.
K+1≤i≤p

Assumption 4 ([7, Assumption 3]). There is a constant s not depending on n suchthat ℓi−1 /ℓi ≥ s > 1, i = 2, . . . , K.
Let ΛS = diag(ℓ1 , . . . , ℓK ), ΛP = diag(ℓK+1 , . . . , ℓp ), and(p+l)×(p+l)Σ1 = U⊤
2 ΛP U2 ∈ R 
U1where U =∈ Rp×(p+l) with U1 ∈ RK×(p+l) and U2 ∈ R(p−K)×(p+l) . The Stieltjes
U2R(z ∈ C\supp(µ)).
transform of a finite measure µ on R is, by definition, sµ (z) := dµ(x)x−z+
Let C be the complex upper half plane. Finally, we consider the following assumption:
Assumption 5 ([7, Assumption 8]). There are probability measures (µn )n∈N on R such
−1that sµn (z) = − z − n−1 trace (Ip + sµn (z)Σ1 )−1 Σ1(z ∈ C+ , n ∈ N) and
lim sup ℓK+1 · − limsµn (z) < 1 where tn := inf { t ∈ R | µn ((−∞, t]) = 1 } .
+nz∈C →tn

Let E be an event depending on n. Following [38], we say E holds with highprobability if P(E) ≥ 1 − O(n−c ) for some constant c > 0 (independent of n).
Proposition 2.5 ([7, Theorem 2.5]). Suppose Assumptions 1-5, l ≪ n1/6 , and p ∼ n.
Let ϵ > 0 be sufficiently small. Assume 1 ≤ i − K ≤ log n. Then, with high probability,1 ⊤Ξ Σ1 Ξ ≤ n−2/3−ϵ .
λi (S) − λi−Kn
In particular, λK+1 has limiting Type-1 Tracy-Widom distribution.
We will verify all the assumptions of Proposition 2.5 are satisfied by Theorem 2.3
with l = 0, K = 1, i = 2, ΛS = (p − 1)ρ + 1, and Σ1 = ΛP = (1 − ρ)Ip . Then,the limiting distribution of the second largest eigenvalue λ2 (S) of S will be that of
λ1 ((1 − ρ)n−1 ΞΞ⊤ ) for Ξ introduced in (2), so we will apply the following results ofD
Soshnikov to Ξ, where the convergence in distribution is denoted by −→ andµn,p := (n1/2 + p1/2 )2 ,σn,p := (n1/2 + p1/2 )(n−1/2 + p−1/2 )1/3 .

Proposition 2.6 ([37, Theorem 1, Corollary 1]). Let X = [xij ] ∈ Rp×n have independent random entries such that E xij = 0, E x2ij = 1, E xkij < ck for some constantck (k ∈ N), E(xij )2m ≤ (const m)m , and xij follow symmetric laws of distribution⊤D(1 ≤ i ≤ p, 1 ≤ j ≤ n). Suppose n, p → ∞, p/n → c. Then λ1 (XXσn,p)−µn,p −→ T W1 .

Moreover, λ1 (XX⊤ ) ≤ µn,p + O p1/2 log p (a.s.).
Our case satisfies all the assumptions of Proposition 2.5. Assumptions 1 and 2
follow from the normality of the population. Assumption 3 is seen as follows: K = 1,
dK = p/(n((p − 1)ρ + 1)) → 0 (n, p → ∞, p/n → c). Besides, ℓi = 1 − ρ (i = 2, . . . , p)
are bounded, Kn−1/6 → 0, and K 2 dK → 0. Assumption 4 is by K = 1. Finally, we
check Assumption 5. By Σ1 = (1 − ρ)Ip , the equation of sµn (z) is quadratic. The
Marčenko-Pastur probability measure νy,t of the dimension-to-sample-size ratio y > 0
and the scale parameter t > 0, has a point mass 1 − y −1 at 0 for y > 1, andpsνy,t (z)is given in [40]. Then µn = (1 − p/n) δ0p+ (p/n)νp/n,1−ρ , so tn = (1 − ρ)(1 + p/n)2 .
Thus ℓK+1 ·(− limz∈C+ →tn sµn (z)) = (1+ p/n)−1 , which assures all the assumptions of
Proposition 2.5 for our case. Hence, by Proposition 2.5, the limiting distribution of thesecond largest eigenvalue λ2 (S) of S is that of the largest eigenvalue λ1 ((1−ρ)n−1 ΞΞ⊤ ).
For ξ ∼ N(0, σ 2 ), E ξ 2k ≤ (2σ 2 k)k and E ξ 2k−1 = 0 (k = 1, 2, . . .), so Proposition 2.6
implies:i.i.d.
Theorem 2.7. Let X1 , . . . , Xn ∼ Np (0, σ 2 Σp(ρ)), ρ ∈ (0, 1), X = [X1 · · · Xn ],and n, p → ∞, p/n → c. Thenλ2 (XX⊤ )− µn,p−1
1−ρD/σn,p−1 −→ T W1 . Moreoverp
λ2 (XX⊤ )≤ µn,p−1 + O
p − 1 log(p − 1)(with high probability).
1−ρ
P
Recall pk=1 k −1 = log p + γ + ϵp where γ is the Euler constant and limp→∞ ϵp = 0.
From the last inequality of Theorem 2.7 and Proposition 2.1, Theorem 2.3 follows.
By the same proof argument,
Theorem 2.8. In the setting of Proposition 2.5, BS(S) ≤ K with high probability.

3 Broken-stick rule for real datasets
We consider the correlation matrices C of the following:
1. the dataset [1] obtained from S&P 500 return stock price time-series by removing
many quotes having missing values during the period. The dataset is available
via https://zenodo.org/record/8253821.
2. the datasets (Fan et al. [13]) obtained by cleaning outliers from the dataset ofthe daily excess returns [21] of Fama-French 100 portfolios [11, 12]. The latterdataset is from Prof. French’s data library http://mba.tuck.dartmouth.edu/
pages/faculty/ken.french/.
3. binary multiple sequence alignment [35] by the courtesy of Prof. Quadeer.
We compute the n, p/n, λ1 (C)/p, p, BS(C), and the numbers GK(C), CPV(C), ACT(C)of PCs the Guttman-Kaiser rule, Cumulative-Percentage-of-Variance rule (Jolliffe [24]),and Adjusted Correlation Thresholding (Fan et al. [13]) retain from C.
)(iX
λk (C)/p ≥ .7 .
CPV(C) = min i ∈ [1, p]
k=1(
ACT(C) = maxi ∈ [1, p]p−i
1ci,n−1 =, mn,i =
n−1
p−i

"

1 − ci,n−1− ci,n−1 mn,iλi (C)
p
X

−1
>1+

√

)
c0,n−1

4
1+
λk (C) − λi (C) λi+1 (C) − λi (C)
k=i+1

(3)

#
(1 ≤ i ≤ p).

The left side of the inequality in (3) is due to Bai-Ding [3], and the ratio against the i-thlargest spiked eigenvalue of the population correlation matrix converges in probability√
to 1 under a mild condition [13]. Besides, the threshold 1+ c0,n−1 in (3) is optimal [13].
3.1 S&P 500 return time-series
In [10], Engle and Kelly employed their Dynamic Equicorrelation to forecast timeseries of economics. As in [1], we considered datasets of S&P 500 stock returns for theperiod 2012-01-04/2021-12-31 (n = 2516=trading days−1), for the 11 sectors (fromeach sector we choose p stocks without missing data) and the totality of the 11 sectors.
We do not clean “outliers” for any days, as “outliers” may contribute to the unboundedness of λ1 (C). The rows of Table 1 are listed in the increasing order of λ1 (C)/p. The
11 sectors have BS(C) = 1 probably because each of them is relatively homogeneous.
3.2 Fama-French 100 portfolios time-series
In asset pricing and portfolio management, Fama and French designed statisticalmodels, 3-factor model [11] and 5-factor model [12], to describe stock returns. For
the datasets of the daily excess returns of 100 companies French chose, Fan et al. [13]computed the estimator ACT and confirmed the three Fama-French factors [11] but
not the momentum factor [9].
(A) Fan et al. [13] used the daily excess returns of 100 industrial portfolios formed on
the basis of size and book-to-market ratio from January 2, 1998, to December 31,
2007. (“Before 2007-2008 financial crisis”)

SectorCommunication Services
Consumer Discretionary
Health Care
Consumer Staples
Information Technology
Industrials
Materials
Real estate
Financials
EnergyUtilities
The totality

n
2516
2516
2516
2516
2516
2516
2516
2516
2516
2516
2516
2516

p/n
.0076
.0207
.0187
.0091
.0246
.0258
.0095
.0119
.025
.0064
.0111
.1705

λ1 (C)/p
.3571
.3843
.3948
.4302
.4648
.4985
.499
.5819
.6086
.6872
.6897
.3819

p
19
52
47
23
62
65
24
30
63
16
28
429

BS
1
1
1
1
1
1
1
1
1
1
1
4

ACT
3
5
5
3
4
5
2
3
4
1
2
10

GK
4
8
7
4
6
6
4
3
5
1
2
46

CPV
7
15
14
7
14
12
6
3
4
2
2
54

Table 1: The stock return datasets of 11 sectors and the totality (2012-01-04/2021-12-31).

(a) (A)(b) (B)(c) (A)(d) (B)

Figure 1: Heat maps of the correlation matrices and the SECC time-series of the daily
excess returns of Fama-French 100 portfolios of (A) 1998-01-02/2007-12-31 and(B) 2010-01-04/2019-04-30. The red lines are λ1 (C)/p. See Subsection 3.2.

(B) The dataset for the daily excess returns of Fama-French 100 portfolios, from January 4, 2010, to April 30, 2019. (“After 2007-2008 financial crisis”)
By Figure 1, the correlation matrix CA of (A) has submatrices of larger positive
correlation coefficients and/or larger sizes than the correlation matrix CB of (B). So
BS(CA ) = 2 and BS(CB ) = 1. These two datasets are clearly compared, by fitting
them to Glosten-Jagannathan-Runkle’s Generalized Autoregressive Conditional Heteroskedasticity model (GJR GARCH) [16], the distribution for the univariate estimation being normal, and the correlation model being the Dynamic Equicorrelation [10].
The resulting time-series of sample equi-correlation coefficient (SECC) of the daily excess returns are in Figure 1. The time-series (A) is lower than the time-series (B), as
the SECC time-series of the daily returns before February 2020 [1] is lower than that ofthe daily returns after then; the worldwide stock markets crashed on February 2020 bygrowing uncertainty due to the Coronavirus disease 2019 pandemic. In Figure 1 (c), thetime-series (A) is lower for 1999-2001 (“dot-com bubble”) than for the other periods.
The time-series (A) consists of two time-series, and thus has BS(C) = 2. the timeseries (B) is less changing, so BS(C) = 1. Table 2 shows that the broken-stick rule issmaller than the ACT and the Guttman-Kaiser rule is almost the same as the ACT.
For SECC time-series, the time-average ρ satisfies ρ ∼ 1.02833 λ1 (C)− 0.09152 withp2
adjusted R being 0.9924, by the linear regression analysis for the dataset combining

Periodnp/n λ1 (C)/p p BS
(A)
2514 .0398
.6582
2
100
(B)
2346 .0426.8062
1

ACT GK
4
5
3
3

CPV

ρ

2
2

.6016
.7511

Table 2: The daily excess returns of Fama-French 100 portfolios. (A) 1998-01-02/2007-1231 and (B) 2010-01-04/2019-04-30. ρ is the time-average of the SECC time-series.

Table 2 and [1, Table 1].
We conjecture limp,n→∞ (ρ − λ1 (C)/p) = 0 (a.s.) for GJR GARCH+DECO model.
p/n→c

3.3 Binary multiple sequence alignment
We consider the binary multiple sequence alignment (MSA) [35] of a p-residue (site)
protein with n sequences where p = 475 and n = 2815. In [35, p. 7628], Quadeer et al.
did “identify groups of coevolving residues within HCV nonstructural protein 3 (NS3)by analyzing diverse sequences of this protein using ideas from random matrix theory
and associated methods.” They also found “Sequence analysis reveals three sectors ofcollectively evolving sites in NS3. ..., there remained α = 9 eigenvalues greater thanλrnd
max , presumably representing intrinsic correlations.”[35, Section Results, p. 7631]
They detected signals by a randomization from the data.
On the statistical model of [35], Morales-Jimenez, one of the authors of Quadeer [36],commented “the majority of variables (protein positions in the genome) are essentiallyindependent, and there are just some small groups of variables which are correlated,giving rise to the different spikes. These group of variables can be modelled with equicorrelation, but the size of these groups is modelled as fixed, i.e., not growing withthe dimension of the protein. That leads to a non-divergent spiked model, like the one
considered in our Stat Sinica paper.” [32].
Nonetheless, for the dataset of MSA [35], the broken-stick rule and the adjusted
correlation thresholding work well. The number 3 of the sectors is detected by then
MSA 2815p/n
λ1 (C)/p0.1687 0.0216

p BS
475 3

ACT
10

GK
188

CPV
193

Table 3: The multiple sequence alignment dataset.

broken-stick rule BS(C) = 3. The number α = 9 of eigenvalues greater than λrnd
max isnearly ACT(C) = 10.

4 Comparison of ENP to a bounded spiked eigenvalues model
As for the asymptotic behaviors of the eigenvalues of S and C, the first-order
behavior (i.e., location) are the same under a certain condition (El Karoui [28]), but
the second-order behavior (i.e., fluctuation) can be different [32].
For the null case, a sample covariance matrix S and a sample correlation matrix C
have the same limiting distribution of the largest eigenvalue:
Proposition 4.1 ([6, Theorem 1.6]). Let the population be Np (0, Ip ). Suppose p, n →D∞, p/n → c ∈ (0, 1). Then (nλ1 (C) − µn,p )/σn,p −→ T W1 .
An ENP with the population covariance matrix Σp (ρ) (ρ > 0) is a divergent spiked
eigenvalues model. For such an ENP, λ2 (C) (λ2 (S), resp.) is effectively, so to say, thelargest eigenvalue of a sample correlation matrix (a sample covariance matrix, resp.)of a population having covariance matrix (1 − ρ)Ip−1 . Table 4 and Figure 2 (a∼c) showthat the dependence of the (variance of the) fluctuation of λ2 (C) on the equi-correlation
coefficient ρ > 0 is stronger than the dependence (Proposition 2.5) of that of λ2 (S)on ρ. Let τ (ρ) be the histogram obtained by 8000 replications of (nλ2 (C)/(1 − ρ) −µn,p−1 )/σn,p−1 from the population Np (0, Σp (ρ)) for n = 1200 and p = 150. Then, themeans of τ (0.2), τ (0.3), and τ (0.4) are smaller than E[T W1 ].
Figure 2 (d) is the plot of c = 1/64, 1/50, 1/40, 1/32, 1/25, 1/20, 1/16, 1/10, 1/8, 1/5
vs. the means of the histogram obtained by 8000 replications of (8000λ2 (C)/(1 −0.1) − µ8000,8000c−1 )/σ8000,8000c−1 from the population N8000c (0, Σ8000c (0.1)). The meansof the histograms seem nonlinear and bounded in c. It may be limp,n→∞ E λ2 (C) =p/n→c√(1 − ρ)(1 + c)2 for any ρ ∈ [0, 1).

τ (0.2)
τ (0.3)
τ (0.4)
T W1

Mean−1.599−1.512−1.470−1.206

Variance
2.050
2.857
3.970
1.607

Skewness
0.207
0.169
0.125
0.293

Excess kurtosis
0.061
0.059
0.011
0.165

Table 4(a) τ (0.2)(b) τ (0.25)(c) τ (0.4)(d) ρ = 0.1

Figure 2

However, for a bounded spiked eigenvalue model, λ2 (C) converges almost surely
to a value larger than the corresponding eigenvalue
λ of the population correlation√1
matrix, if λ is simple and larger than 1 + c: ξProposition 4.2 ([32, Theorem 1]). Let x =∈ Rm+p be a random vector withη
(4 + δ)-th moment for some δ > 0. Assume that ξ ∈ Rm has mean zero, and isindependent of η ∈ Rp , which has i.i.d. components ηi with mean zero and unit variance.
Let R be the correlation matrix of x such that λi (R) = 1 (m + 1 ≤ i ≤ m + p). Then,λi (R) > 1 +
1√cλi (R)c is simple =⇒ p,n→∞lim λi (C) = λi (R) +(a.s.).
λi (R) − 1
p/n→c

In an ENP with ρ > 0 and p > 2, R = Σp (ρ) but λi (R) = 1 − ρ (2 ≤ i ≤ p) is not simple.

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