arXiv:1706.06638v1 [] 20 Jun 2017

On convergence of the sample correlation matrices
in high-dimensional data
Sévérien Nkurunziza∗

and

Yueleng Wang†

Abstract
In this paper, we consider an estimation problem concerning the matrix of correlation coefficients in context of high dimensional data settings. In particular, we revisit
some results in Li and Rolsalsky [Li, D. and Rolsalsky, A. (2006). Some strong limit
theorems for the largest entries of sample correlation matrices, The Annals of Applied
Probability, 16, 1, 423–447]. Four of the main theorems of Li and Rolsalsky (2006)
are established in their full generalities and we simplify substantially some proofs of
the quoted paper. Further, we generalize a theorem which is useful in deriving the
existence of the pth moment as well as in studying the convergence rates in law of
large numbers.

Keywords: Convergence almost surely; Correlation coefficient; Strong Law of Large numbers; convergence rates.
∗

University of Windsor, 401 Sunset Avenue, Windsor, Ontario, N9B 3P4. Email: severien@uwindsor.ca

†

University of Windsor, 401 Sunset Avenue, Windsor, Ontario, N9B 3P4. Email: wang1gv@uwindsor.ca

1 Introduction
As in Li and Rolsalsky (2006), we consider an estimation problem concerning the matrix of correlation coefficients in context of high-dimensional data. In particular, as in
the quoted paper, we are interested in asymptotic properties of the largest entries of the
matrix of the correlation coefficients when the sample size may be smaller than the parameters. Thus, we use the same notations as in Li and Rolsalsky (2006). Namely, let
X = (X1 , X2, ..., X p) be a p-variate random vector, and Mn,pn = (Xk,i )1≤k≤n,1≤i≤pn , rows
of Mn,pn are independent copies of X. Let M = {Xk,i ; i ≥ 1, k ≥ 1} be an array of i.i.d.
(n)

random variables. Further, let X̄i

n

=

max

means ith column of Mn,pn , and let

k=1

n

Wn =

(n)

∑ Xk,i/n, Xi

∑ Xk,iXk, j .

1≤i< j≤pn k=1

Li and Rosalsky (2006) studied the limit behavior of Wn and

that of Ln which is defined as
n

Ln =

(n)

max |ρ̂i, j | with

1≤i< j≤pn

(n)

ρ̂i, j =

(n)

∑ (Xk,i − X̄i

k=1
n

∑
k=1

(n) 1/2
(Xk,i − X̄i )2

(n)

)(Xk, j − X̄ j )
n

∑
k=1

.

(n) 1/2
(Xk, j − X̄ j )2

In Jiang (2004), the author derived the asymptotic properties of the statistic Ln and he
proposed a test for testing if the components of the p-column vector X are uncorrelated.
In this paper, we establish the asymptotic results which refine the analysis in Li and
Rosalsky (2006). More specifically, with respect to the similar work in literature, we extend
the existing findings in three ways. First, Theorems 2.1 and 2.3 of Li and Rosalsky (2006)
are established in their full generalities. Thanks to the established results, we also revisit
the statement given in Remark 2.1 in Li and Rosalsky (2006). Further, from the established
results, we simplify remarkably the proof of the result stated in Remark 2.3 of Li and
2

Rosalsky (2006). More precisely, we prove that the condition which is stated as sufficient
is also necessary. Second, we refine Theorems 3.2 and Theorem 3.3 of Li and Rosalsky
(2006) and, as compared to the proofs given in Li and Rosalsky (2006), we provide the
proofs which are significantly shorter than that in the quoted paper. Third, we generalize
Theorem 3.2.1 in Chung (1974) which is useful in establishing the existence of the pth moment as well as in studying the convergence rates in Law of Large numbers. Specifically,
the established result is useful in deriving the main results in Erdös (1949), Baum and Katz
(1965) and Katz (1963) among others.
The remainder of this paper is organized as follows. Section 2 presents some preliminary results. In Section 3, we present the main results of this paper. Finally, for the
convenience of the reader, we present some proofs in the Appendix A and, we recall in the
Appendix B some existing results used in this paper.

2 Some Preliminary Results
In this section, we derive some results which are useful in establishing the main results
of this paper. In particular, the results of this section generalize Theorem 3.2.1 given in
Chung (1974). For the convenience of the reader, the quoted theorem stipulates that for a
∞

∞

random variable X , we have
∞

E(|X |) < ∞ if and only if

∑ P(|X | > n) 6 E(|X |) 6 1 + ∑ P(|X | > n). Thus,
n=1

n=1

∑ P(|X | > n) < ∞. To introduce some notations, let an = O(bn)

n=1

stand for the sequence an /bn , n=1,2,. . . is bounded.
∞
Theorem 2.1. Let {αn }∞
n=0 be nonnegative sequence of real numbers, let {βn }n=0 be non-

3

negative and nondecreasing sequence of real numbers such that β0 = 0, lim βn = ∞,
n→∞

βn+1 − βn = O(βn) and c−1 αn ≤ βn − βn−1 ≤ cαn

n = 1, 2, . . ., for some

c > 1. Then, for any random variable X ,
∞

∞

c−1 ∑ αn P(|X | > βn ) 6 E(|X |) 6 β1 + (B + 1)c ∑ αn P(|X | > βn ), for some B > 0.
n=1

n=1

∞

Thus, E(|X |) < ∞ if and only if

∑ αnP(|X | > βn) < ∞.

n=1

The proof of this theorem is given in the Appendix A. To illustrate how the established
result generalizes Theorem 3.2.1 in Chung (1974), we note first that, from the quoted the∞

orem, we also have,

∞

∑ P(|X | > n) 6 E(|X |) 6 1 + 2 ∑ P(|X | > n). For the particular case

n=1

n=1

where αn = 1, βn = n, Theorem 2.1 yields Theorem 3.2.1 in Chung (1974) with c = B = 1.
By using this theorem, we establish the following result which is useful in deriving one of
the main result of this paper.
Corollary 2.2. Let α > 0, let β > 0 and let X be a random variable. Then,


 α +1 
∞
α
β
P
|X
|
>
n
<
∞
if
and
only
if
E
|X | β < ∞.
n
∑

n=1

The proof of this corollary is given in the Appendix A. Further, from Theorem 2.1 and
Corollary 2.2, we derive the following corollary which is useful in establishing the second
main result of this paper.
∞
Corollary 2.3. Let α , β > 0, let {αn }∞
n=1 and {βn }n=1 be nonnegative sequences of real

numbers and suppose that αn /nα and βn /nβ are bounded away from 0 and ∞. Then, for a


 α +1 
∞
random variable X , ∑ αn P |X | > βn < ∞ if and only if E |X | β < ∞.
n=1

The proof follows directly from Corollary 2.2. Below, we derive a lemma which plays a
central role in establishing the main results. Thank to the established lemma, we also sim4

plify significantly the proof of the statement in Remark 2.3 of Li and Rosalsky (2006). To
introduce some notations, we consider that X1,i , i = 1, 2, ..., are independent and identically
distributed random variables, and Xk = (Xk,1 , Xk,2, ..., ) is an independent copy of a random
vector X1 = (X1,1 , X1,2 , ..., ).
Lemma 2.4. Let m be a fixed positive integer and let {un }∞
n=1 be a nonnegative sequence
of real numbers. Then,
∞

m

n=1

h=1

∑ nmP ∏ |X1,h| ≥ un

!

∞

< ∞ if and only if

m

∑P

∏ |X1,ih | ≥ un

max

1≤i1 <i2 <···<im ≤n h=1

n=1

!

< ∞.

The proof of this lemma is given in the Appendix A. Note that the if part is obvious
as it follows directly from the sub-additivity of a probability measure. We also prove the
following lemma which is useful in deriving the main results of this paper.
Lemma 2.5. Let m be a fixed positive integer. Let {un }∞
n=1 be a nonnegative and nondecreasing sequence of real numbers such that lim un = ∞. Further, suppose that there exists
n→∞
.
a nonnegative, continuous and increasing function f such that f (un ) nβ is bounded away
from 0 and from infinity, for some β > 0. Then,
!
∞

∑P

n=1

max

∏ |X1,ih | ≥ un

1≤i1 <i2 <···<im ≤n h=1

∞

Proof. From Lemma 2.4, ∑ P
!n=1
∞

∑n

m

!# m+1 
β
 < ∞.
∏ |X1,h|

"

m

m

< ∞ if and only if E  f
m

∏ |X1,ih | ≥ un

max

1≤i1 <i2 <···<im ≤n h=1

!

h=1

< ∞ if and only if

m

P

n=1

∏ |X1,h| ≥ un

h=1

< ∞. Since the function f is increasing and continuous, this last
"
!
#
∞

statement is equivalent to

∑ nm P

n=1

m

f

∏ |X1,h|

h=1



lary 2.3, this last statement is equivalent to E  f

the proof.

5

≥ f (un ) < ∞. Then, by using Corol!! m+1 
m

∏ |X1,h|

h=1

β

 < ∞, this completes

∞

Corollary 2.6. We have

,

m

∑P

n=1

m

E  ∏ |X1,h|2(m+1)
h=1

∏ |X1,ih | ≥

max

1≤i1 <i2 <···<im ≤n h=1
m

ln e + ∏ |X1,h |
h=1

!!m+1 

p

n ln(n)

!

< ∞ if and only if

 < ∞, with m a fixed positive integer.

The proof of this corollary is given in the Appendix A. From this corollary, we establish

the following result which improves the statement in Remark 2.3 of Li and Rosalsky (2006).
For the convenience of the reader, we recall that, in Remark 2.3 in Li and Rosalsky (2006),

∞ 
p
the authors conclude that ∑ P max |X1X2 | ≥ n log n < ∞ implies E|X1 |β < ∞, for
1≤i< j≤n

n=1

0 6 β < 6. This becomes a special case of the following result by taking m = 2.
∞

m

!

p

max
Corollary 2.7. Suppose that ∑ P
∏ |X1,ih | ≥ n ln(n) < ∞, for a fixed
1≤i1 <i2 <···<im ≤n h=1
n=1
i
h
positive integer m. Then, E |X1,1 |β < ∞, for all 0 6 β < 2(m + 1).
The proof of this corollary follows directly from Corollary 2.6 along with classical
properties of expected value of random variables.
Corollary 2.8. Suppose that the conditions of Lemma 2.5 hold with un /nβ bounded away
!
∞

m

from 0 and from infinity, for some β > 0. Then, ∑ P
n=1

m+1 
if and only if E |X1,1 | β < ∞.

max

∏ |X1,ih | ≥ un

1≤i1 <i2 <···<im ≤n h=1

<∞

Proof. This result follows immediately from Lemma 2.5 with f being an identity function,
and the fact that X1,i , i = 1, 2, ... are i.i.d. random variables.
Corollary 2.9. Let {un }∞
n=1 be a nonnegative sequence of real numbers. Then,

∞
∞ 
2
∑ n P (|X1,1X1,2| ≥ un ) < ∞ if and only if ∑ P max |X1,iX1, j | ≥ un < ∞.
n=1

n=1

1≤i< j≤n

Proof. The proof follows directly from Lemma 2.4 by taking m = 2.
6

3 Main Results
In this section, we present the main results of this paper. In particular, Theorems 2.1 and 2.3
of Li and Rosalsky (2006) are established in their full generalities. We also refine Theorem 3.2 and 3.3 in Li and Rosalsky (2006) and we provide the proofs which are significantly
shorter than that given in the quoted paper. In the sequel, as in Li and Rosalsky (2006), let

(Uk,i ,Vk,i ); i > 1, k > 1 be iid two-dimensional random vectors, and let {pn , n > 1} be
n

a sequence of positive integers. Further,let Tn =

max

∑ Uk,iVk, j , n = 1, 2, . . ., let

16i6= j6pn k=1

{Yn , n = 1, 2, . . .} be a sequence of iid random variables where Y1 is distributed as U1,1V1,2
n

and let Sn =

∑ Yk , n = 1, 2, . . .
k=1

Theorem 3.1. Suppose that n/pn is bounded away from 0 and ∞, and let 1/2 < α ≤ 1.
Then, the following conditions are equivalent.
Tn
n→∞ nα
∞ 

(1.) lim
(2).

∑P

n=1

= 0 a.s.
max |U1,iV1, j | ≥ n

1≤i6= j≤n

α



<∞

and E(U1,1 )EV1,1 ) = 0.



(3). E |U1,1 |3/α |V1,2 |3/α < ∞ and E(U1,1 )EV1,1 ) = 0.
Proof. The equivalence between parts (2) and (3) follows directly from Corollary 2.8 by
taking m = 2 and X1,1 = U1,1V1,2 . As for the equivalence between part (1) and part (2), the
proof of "only if" part is similar to that given in Li and Rosalsky (2006) and thus, we need to
give the proof of the "if" part. To this end, note that by Corollary 2.9 and Corollary 2.3, the


∞
3
α
condition ∑ P
max |U1,iV1, j | ≥ n
< ∞ is equivalent to E|U1,1V1,2 | α < ∞. Then, by
n=1

1≤i6= j≤n

the celebrated theorem of Baum and Katz (1965) (or see Theorem B.1 in the Appendix B),

7

∞



|Sm |
we have, ∑ nP sup α > ε
m>n m
n=1
which means

p2n
n

<

c2 n2
n



<∞

for all ε > 0. Then, since c−1 ≤ pn /n < c, n ≥ 1,

= c2 n, then
∞



|Sn |
p2n
∑ P nα > ε < ∞,
n=1 n

for all ε > 0.

(3.1)

[cn]α
Further, one verifies that lim lim sup α = 1. Futher, note that p2n /n > c−2 n, and then,
n
c↓1 n→∞




P
|S |
|S |
by (3.1), we have c−2 nP nαn > ε → 0 and P nαn > ε → 0, i.e. nSαn −−−→ 0. Hence, by
n→∞

Tn
Theorem 3.1 of Li and Rosalsky (2006), we have lim sup α 6 ε
n→∞ n

a.s. for all ε > 0, and

then, by letting ε ↓ 0, we get the desired result, and this completes the proof.
Note that, by part (3) of Theorem 3.1, we generalizes Theorem 3.2 of Li and Rosalsky (2006). In addition, we present a very short proof of the equivalence between part (1)
and part (2) as compared to the proof given in the quoted paper.
Theorem 3.2. Suppose that n/pn is bounded away from 0 and ∞.
If E(U1,1 )E(V1,1 ) = 0,

E

|U1,1V1,2 |6

2 )E(V 2 ) = 1 and
E(U1,1
1,1

(log(e + |U1,1V1,2 |))3

!

∞

< ∞ or

∑P

n=1




p
max |U1,iV1, j | ≥ n log n < ∞,

1≤i6= j≤n

(3.2)

then
lim sup √
n→∞

Tn
≤ 2 a.s.
n log n

Tn
Conversely, if lim sup √
< ∞ a.s., then (3.2) hold, E(U1,1 )E(V1,1 ) = 0, and
n log n
n→∞

 

E |U1,1 |β E |V1,2 |β < ∞ for all 0 6 β < 6.
Proof. The proof of the second part of the theorem follows from Corollary 2.7 and by
following the same steps as in proof of the only if part of Theorem 3.3 of Li and Rosal8

sky (2006). To prove the first part of the theorem, we note first that the equivalence between
the conditions in (3.2) follows directly from Corollary 2.6 by taking m = 2. Further, if
E

|U1,1V1,2 |6

(log(e + |U1,1V1,2 |))3

!

< ∞,

by Theorem 3 of Lai (1974) (or see Theorem B.2 in the Appendix B), we have
∞



|Sn |
∑ nP √n log n > λ
n=2



<∞

for all λ > 2,

∞



|Sn |
p2n
∑ n P √n log n > λ < ∞ for all λ > 2. Therefore,
n=2
Tn
by Theorem 3.1 of Li and Rosalsky (2006), we get lim sup α 6 λ a.s. for all λ > 2, and
n→∞ n
Sn
P
and then, √
−−−→ 0 and
n→∞
n log n

then, letting λ ↓ 2, we get the desired result, and this completes the proof.
Note that by taking β = 2 in the second part of Theorem 3.2, we have the statement in
Li and Rosalsky (2006). Further, in the first part of Theorem 3.2, the condition (3.2) relaxes
the condition (3.15) of Li and Rosalsky (2006). Moreover, in addition to state in its full
generality the result of Theorem 3.3 of Li and Rosalsky (2006), we simplify substantially
the proof. By using Corollary 2.9, we establish the following result which generalizes
Theorem 2.1 in Li and Rosalsky (2006).
Theorem 3.3. Suppose that n/pn is bounded away from 0 and ∞. Let 1/2 < α ≤ 1. Then,
the following statements are equivalent.
∞

(1).

∑ n2P(|X1,1X1,2| ≥ nα ) < ∞

and

E(X1,1 ) = 0.

n=1
∞

(2).

max |X1,i X1, j | ≥ nα ) < ∞
∑ P(1≤i<
j≤n

and

n=1

Wn
= 0. a.s.
n→∞ nα

(3). lim

9

E(X1,1 ) = 0.



(4). E |X1,1|3/α < ∞

and

E(X1,1 ) = 0.

Proof. The equivalence between (1) and (2) follows directly form Corollary 2.9 by taking
un = nα . Further, the equivalence between the statements in (2) and (3) is established in Li
and Rosalsky (2006). Finally, the equivalence between the statements (2) and (4) follows
from Corollary 2.8 by taking m = 2, this completes the proof.
Further, by using Lemma 2.4 and Corollary 2.6, we establish in its full generality Theorem 2.3 of Li and Rosalsky (2006).
Theorem 3.4. Suppose that n/pn is bounded away from 0 and ∞. Then, the following
statements are equivalent.
∞

(1). E(X1,1 ) = 0,

2 )=
E(X1,1



p
1and ∑ n P |X1,1 X1,2 | ≥ n log n < ∞.
2

n=1
∞

(2). E(X1,1 ) = 0,

2 )=
E(X1,1

1 and

∑P

n=1

(3). lim √
n→∞

Wn
= 2 a.s.
n log n

(4). E(X1,1 ) = 0,

2 )=
E(X1,1


1and E



max |X1,i X1, j | ≥

1≤i< j≤n

p



n log n < ∞.


(X1,1 X1,2)6
< ∞.
log3 (e + |X1,1X1,2 |)

Proof. The Equivalence between (1) and (2) follows directly from Lemma 2.4 by taking
m = 2 and un =

p

n ln(n). Further, the equivalence between (2) and (3) is established in

Li and Rosalsky (2006), and the equivalence between (2) and (4) follows directly from
Corollary 2.6 by taking m = 2, this completes the proof.

10

A Proofs of some preliminary results
Proof of Theorem 2.1. Let Λn = {βn ≤ |X | < βn+1 }, n = 0, 1, . . .. We have
Z

E(|X |) = [
∞

Λn

|X | dP =

∞ Z

∑

n=0 Λn

|X | dP,

n=0

and then,
∞

∞

∞

∞

∑ βnP(Λn ) ≤ E(|X |) ≤ ∑ βn+1P(Λn ) = ∑ (βn+1 − βn)P(Λn) + ∑ βnP(Λn ).

n=0

n=0

(A.1)

n=0

n=0

Further, there exists B > 0 such that (βn+1 − βn ) 6 Bβn , n = 1, 2, . . .. Then,
∞

∞

∞

∞

∑ (βn+1 − βn)P(Λn) + ∑ βnP(Λn) 6 β1P(Λ0) + B ∑ βnP(Λn) + ∑ βnP(Λn )
n=0

n=0

n=1
∞

n=1

= β1 P(Λ0 ) + (B + 1) ∑ βn P(Λn ).
n=1

Then, from (A.1), we have
∞

∞

∑ βnP(Λn) ≤ E(|X |) ≤ β1 + (B + 1) ∑ βnP(Λn).

n=1

(A.2)

n=1

Observe that
N

N

∑ βnP(Λn) = ∑ (βn − βn−1 )P(|X | ≥ βn) − βN P(|X | ≥ βN+1 ),

n=1

N = 1, 2, . . .,

(A.3)

n=1

then,
N

N

∑ βnP(Λn ) 6

n=1

∑ (βn − βn−1 )P(|X | ≥ βn),

N = 1, 2, . . .,

(A.4)

n=1

this gives
∞

∑ βn P(Λn) 6

n=1

∞

∞

∑ (βn − βn−1 )P(|X | ≥ βn) 6 c ∑ αnP(|X | ≥ βn),

n=1

n=1

11

(A.5)

and then, combining (A.2) and (A.5), we have
∞

∞

∞

∑ βnP(Λn ) ≤ E(|X |) ≤ β1 + (B + 1) ∑ βnP(Λn ) 6 β1 + (B + 1)c ∑ αnP(|X | ≥ βn).

n=1

n=1

n=1

(A.6)
∞

First, suppose that E(|X |) = ∞, by (A.6), we have
∞

∞

n=1

n=1

∑ αn P(|X | ≥ βn) = ∞ and then,

n=1

c−1 ∑ αn P(|X | ≥ βn ) = E(|X |) = (B+1)c ∑ αn P(|X | ≥ βn ) = ∞, this proves the statement.
Second, suppose that E(|X |) < ∞. By Lebesgue dominated convergence Theorem, we
have lim E(|X |I{|X|≥βN+1} ) = 0 and then,
N→∞

βN P(|X | ≥ βN+1 ) ≤ βN+1 P(|X | ≥ βN+1 ) ≤ E(|X |I{|X|≥βN+1} ) −−−→ 0.
N→∞

Thus, from (A.3), we have
∞

∞

∑ (βn − βn−1 )P(|X | ≥ βn) =

n=1

∑ βnP(Λn),

(A.7)

n=1

then, since c−1 αn 6 βn − βn−1 6 cαn , by combining (A.6) and (A.7), we have
c−1

∞

∞

∞

∑ αnP(|X | ≥ βn) 6 ∑ βn P(Λn) 6 E(|X |) 6 β1 + (B + 1)c ∑ αnP(|X | ≥ βn,

n=1

n=1

n=1

this completes the proof.
∞

Proof of Corollary 2.2. The condition that
∞



α
β
< ∞ is equivalent with
P
|X
|
>
n
n
∑

n=1

 α +1

α
β > nα +1 < ∞. Now let α := nα and β := nα +1 . We have
n
P
|X
|
n
n
∑
n=1


α +1 − 1 ≤ (2α +1 − 1) · nα +1 = O(nα +1 ),
βn+1 − βn = (n + 1)α +1 − nα +1 = nα +1 ( n+1
n )
and

h
i
n−1 α
βn − βn−1 = nα +1 − (n − 1)α +1 = nα n − (
) · (n − 1) ≥ nα = αn
n
12

α
Also, since n − ( n−1
n ) (n − 1) → 1, by the fact every convergent sequence is bounded, there
α
exists a constant C such that for all n ≥ 1, n − ( n−1
n ) (n − 1) ≤ C. Let c = max{2,C}. We

verified that c−1 αn ≤ βn − βn−1 ≤ cαn

∀n, for some c ≥ 1. Therefore, the rest of the

proof follows directly from Theorem 2.1.
Proof of Corollary 2.3. The proof follows form Corollary 2.2. Since αn /nα and βn /nβ are
bounded away from 0 and ∞. Then exists a > 1 s.t. a−1 < nααn < a and a−1 < nββn < a. Thus




∞
∞
−1 α
β
α
β
P
|X
|
>
an
P
|X
|
>
<
∞
implies
a
n
< ∞, and this is equivalent to
n
∑
∑ n
n=1
∞

n=1

   α +1 
 α +1 
β
|X|
<
∞,
i.e.
E
|X | β < ∞.
> nβ < ∞. Now by Corollary 2.2, E
a
a
n=1

α +1
α +1 
β
Conversely, if E(|X | ) < ∞, we have E (a · |X |) β < ∞. Then, by Corollary 2.2,



∞
∞
nβ 
α
∑ n P |X | > a < ∞, which implies ∑ αn P |X | > βn , this complete the proof.
n=1
n=1

∑ nα P

 |X |



Proof of Lemma 2.4. The sufficient condition follows directly from the sub-additivity. To


m
|X
|
≥
u
prove the necessary condition, let A = {n : P
max
∏ 1,ih n = 0}. We
1≤i1 <i2 <···<im ≤n h=1

have
∞

∑P

n=1



m


|X
|
≥
u
=
n
1,i
∏ h

max

1≤i1 <i2 <···<im ≤n h=1

and note that if n ∈ A, P
Then,
∞

∑ nm P

n=1





m

∏ |X1,h| ≥ un

h=1



∑
n∈N\A



m


|X
|
≥
u
∏ 1,h n =

∏ |X1,ih | ≥ un

max

1≤i1 <i2 <···<im ≤n h=1

= 0 and then

nm P



m

∏ |X1,h| ≥ un

h=1

m

h=1

P

∑
n∈N\A

nm P







(A.8)

= 0, ∀n ∈ A.

m


|X
|
≥
u
∏ 1,h n .

(A.9)

h=1

Hence, from (A.8) and (A.9), it suffices to prove that if


m

m
m
max
∑ P
∏ |X1,ih | ≥ un < ∞ then ∑ n P ∏ |X1,h| ≥ un < ∞. Hence,
n∈N\A

1≤i1 <i2 <···<im ≤n h=1

n∈N\A

13

h=1

in the sequel, we suppose without loss of generality that P
∞

n = 1, 2, . . . Thus, if

∑P

n=1



lim P

n→∞

∏ |X1,ih | ≥ un

1≤i1 <i2 <···<im ≤n h=1





m

∏ |X1,ih | ≥ un

max

m


|X
|
≥
u
> 0 for all
n
1,h
∏

h=1

m

max



1≤i1 <i2 <···<im ≤n h=1

< ∞, we have



= 0.

(A.10)

Then, it suffices to prove that
m!P



m


 m

m
|X
|
≥
u
∼
n
P
|X
|
≥
u
∏ 1,ih n
∏ 1,h n ,

max

1≤i1 <i2 <···<im ≤n h=1

h=1

as n → ∞.

For convenience X1,i is written as Xi .
First, observe that log x ≤ x − 1, ∀x > 0. Then,
log P



m


 m

|X
|
≤
u
≤
P
|X
|
≤
u
∏ h n
∏ h n − 1 = −P

h=1

h=1

m

∏ |Xh| > un

h=1



,

and then,
n!
log P
(n − m)!m!

m

∏ |Xh| ≤ un

h=1

!

≤−

m
n!
P( ∏ |Xh| > un ).
(n − m)!m! h=1

Taking the exponential both side, we have
h
P

m

∏ |Xh| ≤ un

h=1

n!
i (n − m)!m!

h
≤ exp −

i
m
n!
P( ∏ |Xh| > un ) .
(n − m)!m! h=1

Thus,
P



m

∏ |X1,ih | ≤ un

max

1≤i1 <i2 <···<im ≤n h=1



i
m
n!
≤ exp −
P( ∏ |Xh | > un ) ,
(n − m)!m! h=1
h

and then
P



m

max

∏ |X1,ih | > un

1≤i1 <i2 <···<im ≤n h=1



"

n!
P
≥ 1 − exp −
(n − m)!m!

m

∏ |Xh| > un

h=1

!#

> 0.
(A.11)

14

Then, by combining (A.10) and (A.11), we get

lim P

n→∞



m

∏ |X1,ih | > un
1≤i1 <i2 <···<im ≤n
max

h=1



m
n!
P( ∏ |Xh| > un ) = 0. (A.12)
n→∞ (n − m)!m!
h=1

= lim

By combining (A.11) and (A.12) along with the fact that 1 − e−x ∼ x as x → 0, we get
P
lim

n→∞



m

∏ |X1,ih | > un

max

1≤i1 <i2 <···<im ≤n h=1

m
n!
P( ∏ |Xh| > un )
(n − m)!m! h=1



≥ 1.

(A.13)

Further, by sub-additivity, we have


m
m
n!
P( ∏ |Xh| > un ) > P
max
|X
|
>
u
,
n
1,i
∏ h
(n − m)!m! h=1
1≤i1 <i2 <···<im ≤n h=1
and then,
P
lim

n→∞

m


|X
|
>
u
∏ 1,ih n
1≤i1 <i2 <···<im ≤nh=1
! 6 1.
m
n!
P ∏ |Xh| > un
(n − m)!m!
h=1



max

(A.14)

Hence, by combining (A.13) and (A.14), we have
m

n!
P
(n − m)!m!

∏ |Xh| > un

h=1

!

∼P



m


∏ |X1,ih | > un .

max

1≤i1 <i2 <···<im ≤n h=1

Therefore,
nm
P
m!

m

∏ |Xh| > un

h=1

!

∼P



m

max


|X
|
>
u
,
n
∏ 1,ih

1≤i1 <i2 <···<im ≤n h=1

this completes the proof.
∞

Proof of Corollary 2.6. By Lemma 2.4,

m

∑P

n=1

if and only if

∞

m

n=3

h=1

∑ nmP ∏ |X1,h| ≥

p

n ln(n)

!

15

max

∏ |X1,ih | ≥

1≤i1 <i2 <···<im ≤n h=1

p

< ∞, and this is equivalent to

!

n ln(n) < ∞

∞

∑n

m

n=3

m

P

∏ |X1,h|

2

h=1

≥ n ln(n)

!

< ∞. Then, by taking f (x) =

x√
,
ln(e+ x)

x > 3, one can ver-



ify that f (n ln(n)) n > 1/2 for all n > 3 and lim f (n ln(n)) n = 2 and this implies that
n→∞


f (n ln(n)) n > 1/2 is bounded away from 0 and from infinity. Therefore, the proof fol-

lows from Lemma 2.5 by taking β = 1.

B Theorems of Baum and Katz (1965) and Lai (1974) used
For the convenience of the reader, we recall in this Appendix two results from the
celebrated theorems of Baum and Katz (1965), and Lai (1974) which are used in this paper.
These results can also been found in Li and Rosalsky (2006).
Theorem B.1 (Baum and Katz (1965)). Let {Sn , n > 1} be a sequence of partial sums as
defined in Section 3. Let β > 0 and α > 1/2, and suppose that E(Y1 ) = 0 if α 6 1. Then,
the following are equivalent:
∞

∑n

2β −1

n=1
∞

∑n

2β −1

n=1





|Sn |
P
>ε
nα



< ∞

for all ε > 0,


|Sm |
P sup α > ε
< ∞ for all ε > 0,
m>n m


E |Y1 |(2β +1)/α
< ∞.

Theorem B.2 (Lai (1974)). Let β > 0, let Sn be as in Theorem B.1 and suppose that
!
β
+2
4
|Y1 |
< ∞.
E(Y1 ) = 0, E(Y12 ) = 1, and E
(ln (e + |Y1 |))2β +1
Then,
∞

∑ n2β −1 P

n=1

p

|Sn |

n ln(n)

>λ

!
16

< ∞

p
for all λ > 2 β .

References
[1] K. L. Chung (1974). A Course in Probability Theory, Third Edition, Academic Press.
[2] P. Erdös (1949). On a theorem of Hsu and Robbins, Ann. Math. Statist. 20, 286–291.
[3] L. E. Baum, and M. Katz (1965). Convergence rates in the law of large numbers.,
Trans. Amer. Math. Soc., 120, 108–123.
[4] M. Katz (1963). The probability in the tail of a distribution, Ann. Math. Statist. 34,
312–318.
[5] T. Jiang (2004). The asymptotic distributions of the largest entries of sample correlation matrices, Ann. Appl. Probab., 14, 865–880.
[6] D. Li, and A. Rosalsky (2006). Some strong limit theorems for the largest entries of
sample correlation matrices, Ann. App. Prob., 16, 1, 423–447.
[7] T. L. Lai (1974). Limit theorems for delayed sums, Ann. Probab., 2, 3, 432–440.

17

