Probab. Theory Relat. Fields (2010) 148:5–35
DOI 10.1007/s00440-009-0220-z
Necessary and sufficient conditions for the asymptotic
distribution of the largest entry of a sample correlation
matrix
Deli Li · Wei-Dong Liu · Andrew Rosalsky
Received: 6 March 2008 / Revised: 24 February 2009 / Published online: 6 May 2009
© Springer-Verlag 2009
Abstract Let {X k,i ; i ≥ 1, k ≥ 1} be a double array of nondegenerate i.i.d. random variables and let { pn ; n ≥ 1} be a sequence of positive integers such that n/pn
is bounded away from 0 and ∞. In this paper we give necessary and sufficient con(n)
ditions for the asymptotic distribution of the largest entry L n = max1≤i< j≤ pn |ρ̂i, j | of
the sample correlation matrix n = (ρ̂i,(n)j )1≤i, j≤ pn where ρ̂i,(n)j denotes the
Pearson correlation coefficient between (X 1,i , . . . , X n,i )
and (X 1, j , . . . , X n, j ) . Write
F(x) = P(|X 1,1 | ≤ x), x ≥ 0, Wc,n = max1≤i< j≤ pn | nk=1 (X k,i − c)(X k, j − c)|,
and Wn = W0,n , n ≥ 1, c ∈ (−∞, ∞). Under the assumption that E|X 1,1 |2+δ < ∞
for some δ > 0, we show that the following six statements are equivalent:
∞
(i)
lim n
2
n→∞
√
F n−1 (x) − F n−1
n log n
x
d F(x) = 0,
(n log n)1/4
(ii) nP
max |X 1,i X 1, j | ≥
1≤i< j≤n
n log n → 0 as n → ∞,
D. Li
Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON P7B 5E1, Canada
e-mail: [email protected]
W.-D. Liu
Department of Mathematics, Zhejiang University, 310027 Hangzhou, China
e-mail: [email protected]
A. Rosalsky (B)
Department of Statistics, University of Florida, Gainesville, FL 32611, USA
e-mail: [email protected]
123
6
D. Li et al.
Wμ,n
P
(iii) √
→ 2σ 2 ,
n log n
(iv)
n
log n
1/2
lim P
(v)
n→∞
P
L n → 2,
2
Wμ,n
nσ 4
− an ≤ t
1
= exp − √ e−t/2 , − ∞ < t < ∞,
8π
1
lim P n L 2n − an ≤ t = exp − √ e−t/2 , − ∞ < t < ∞
n→∞
8π
(vi)
where μ = EX 1,1 , σ 2 = E(X 1,1 − μ)2 , and an = 4 log pn − log log pn . The
2
< ∞. Weak
equivalences between (i), (ii), (iii), and (v) assume that only EX 1,1
laws of large numbers for Wn and L n , n ≥ 1, are also established and these are of the
P
P
form Wn /n α → 0 (α > 1/2) and n 1−α L n → 0 (1/2 < α ≤ 1), respectively. The
current work thus provides weak limit analogues of the strong limit theorems of Li and
Rosalsky as well as a necessary and sufficient condition for the asymptotic distribution
of L n obtained by Jiang. Some open problems are also posed.
Keywords Asymptotic distribution · Largest entries of sample correlation matrices ·
Weak law of large numbers · Weak law of the logarithm
Mathematics Subject Classification (2000)
62H99
Primary: 60F05 · 60F10; Secondary:
1 Introduction
In this paper we investigate the weak asymptotic behavior of the largest entry of a sample correlation matrix. Let n ≥ 2. Consider a p-variate population ( p ≥ 2) represented
by a random vector X = (X 1 , . . . , X p ) with unknown mean μn = (μ1 , . . . , μ p ),
unknown covariance matrix , and unknown correlation coefficient matrix R. Let
Mn, p = (X k,i )1≤k≤n,1≤i≤ p be an n × p matrix whose rows are an observed random
sample of size n from the X population; that is, the rows of Mn, p are independent
(n)
copies of X. Set X i = nk=1 X k,i /n, 1 ≤ i ≤ p. Write
n
(n)
ρ̂i, j = n
k=1
123
(n)
(n)
X k,i − X i
X k, j − X j
1/2 1/2 ,
(n) 2
(n) 2
n
− Xi
k=1 X k, j − X j
k=1
X k,i
(1.1)
Asymptotic distribution of largest entry
7
which is the Pearson correlation coefficient between the ith and jth columns of Mn, p .
Set
(n)
n = ρ̂i, j
1≤i, j≤ p
,
which is the p × p sample correlation matrix obtained from the p columns of Mn, p .
At the origin of the current investigation is the statistical hypothesis testing problem
studied by Jiang [4] based on the asymptotic distribution of the test statistic
Ln =
max ρ̂i,(n)j ,
(1.2)
1≤i< j≤ p
which is the largest entry of the sample correlation matrix n . When both n and p are
large, Jiang [4] considered the statistical test with null hypothesis H0 : R = I, where I
is the p × p identity matrix and obtained the asymptotic distribution of L n as n and p
both approach infinity. If we assume that the columns of Mn, p are independent, all the
(n)
ρ̂i, j , 1 ≤ i < j ≤ p should be close to 0. In other words, L n should be small. Thus
this null hypothesis asserts that the components of X = (X 1 , . . . , X p ) are uncorrelated whereas when X has a p-variate normal distribution, this null hypothesis asserts
that these components of X are independent. Jiang [4] established two limit theorems
concerning the test statistic L n when p = pn ∼ γ −1 n as n → ∞ (0 < γ < ∞)
and {X k,i ; i ≥ 1, k ≥ 1} is an array of independent and identically distributed (i.i.d.)
nondegenerate random variables. Write X i = X 1,i , i ≥ 1. In the first limit theorem,
assuming that
E |X 1 |r < ∞ for some r > 30,
(1.3)
Jiang [4] obtained the asymptotic distribution for L n . Specifically, Jiang [4] proved
that
1
lim P n L 2n − an ≤ t = exp − √ e−t/2 , −∞ < t < ∞,
n→∞
8π
(1.4)
where the centering constants an are given by an = 4 log pn − log log pn , n ≥ 2. The
limiting distribution in (1.4) is a type I extreme value distribution.
In the second limit theorem, under the assumption that
E |X 1 |r < ∞ for all 0 < r < 30,
Jiang [4] proved the following strong limit theorem which is referred to as the law of
the logarithm (LL) for L n , n ≥ 2:
lim
n→∞
n
log n
1/2
L n = 2 almost surely (a.s.).
(1.5)
123
8
D. Li et al.
Throughout this paper, we let { pn ; n ≥ 1} be a sequence of integers in [2, ∞) such
that n/pn is bounded away from 0 and ∞; this condition is of course less restrictive
than Jiang’s [4] condition limn→∞ pnn = γ ∈ (0, ∞).
Recently, Li and Rosalsky [7, Theorem 2.4] proved that (1.5) holds under the condition
∞
P
max |X i X j | ≥ n log n < ∞.
1≤i< j≤n
n=1
(1.6)
For c ∈ (−∞, ∞) write
Wc,n
n
X k,i − c X k, j − c and Wn = W0,n , n ≥ 1. (1.7)
= max 1≤i< j≤ pn k=1
Under the assumption that EX 14 < ∞, as in the proof of Theorem 2.4 of Li and
Rosalsky [7], we see that (1.5) is equivalent to
lim
n→∞
σ2
Wμ,n
= 2 a.s.,
√
n log n
where μ = EX 1 and σ 2 = E(X 1 − μ)2 , which by Theorem 2.3 of Li and
Rosalsky [7] and Lemma 4.1 of this paper is, in turn, equivalent to (1.6). Thus, under
the assumptions that X 1 is nondegenerate and EX 14 < ∞, (1.6) is actually a necessary
and sufficient condition for (1.5) to hold.
It is natural to seek necessary and sufficient conditions for (1.4) to hold. Since the
appearance of Jiang’s [4] paper, in subsequent papers by several authors, the moment
condition (1.3) has been gradually relaxed. Zhou [11, Theorem 1.1] showed that (1.4)
holds if
x 6 P (|X 1 X 2 | ≥ x) → 0 as x → ∞.
(1.8)
Another moment condition for (1.4) to hold has been obtained recently by Liu et al.
[8, Theorem 1.1] who showed that (1.4) holds under the condition
n 3 P |X 1 X 2 | ≥ n log n → 0 as n → ∞,
which is equivalent to
x6
P (|X 1 X 2 | ≥ x) → 0 as x → ∞.
log3 x
(1.9)
Clearly (1.8) holds if EX 16 < ∞ which is substantially weaker than (1.3), and (1.9) is
weaker than (1.8). Actually, following the proof of Theorem 3.1 of Liu et al. [8], one
123
Asymptotic distribution of largest entry
9
can see that (1.4) and
lim P
2
Wμ,n
nσ 4
n→∞
− an ≤ t
1
= exp − √ e−t/2 , −∞ < t < ∞
8π
(1.10)
hold if
nP
max |X i X j | ≥
1≤i< j≤n
n log n → 0 as n → ∞.
(1.11)
The details are provided in Sect. 4 in the proofs of Theorems 2.5 and 2.6. In this paper,
we will show that (1.11) is a necessary condition for (1.10) assuming only that X 1 is
nondegenerate with EX 12 < ∞; see Theorem 2.5. Moreover, we will show that (1.11) is
a necessary condition for (1.4) assuming that X 1 is nondegenerate with E|X 1 |2+δ < ∞
for some δ > 0; see Theorem 2.6. Consequently, for X 1 nondegenerate, if EX 12 < ∞,
then
(1.10) and (1.11) are equivalent
and if E|X 1 |2+δ < ∞ for some δ > 0, then
(1.4) and (1.11) are equivalent.
Since (1.11) ensures that E|X 1 |β < ∞ for all 0 < β < 6, the moment condition (1.9) is optimal for (1.4) to hold. While it is clear that (1.9) implies (1.11) (see
Remark 2.1 below), it remains open as to whether or not (1.9) and (1.11) are equivalent.
We conjecture that the answer is negative but with (1.9) being only slightly stronger
than (1.11). See Remark 2.5 below.
The Kolmogorov–Marcinkiewicz–Zygmund type strong laws of large numbers
1−α L = 0 a.s.
n
(SLLNs) (i) limn→∞ W
n
n α = 0 a.s. (α > 1/2) and (ii) lim n→∞ n
W
n 1/2
n
(1/2 < α ≤ 1) and the LLs (iii) limn→∞ √n log
=
2
a.s.
and
(iv)
lim
n→∞ ( log n )
n
L n = 2 a.s. have been established by Li and Rosalsky [7, Theorems 2.1–2.4] to hold
under optimal sets of conditions. For example, for α > 1/2, Li and Rosalsky [7,
Theorem 2.1] proved that
lim
n→∞
Wn
= 0 a.s.
nα
if and only if
∞
α
<∞
P
max |X i X j | ≥ n
n=1
1≤i< j≤n
and EX 1 = 0 whenever α ≤ 1.
123
10
D. Li et al.
In the current work, weak laws of large numbers (WLLNs) (Theorems 2.1 and
2.2) as well as weak laws of the logarithm (WLLs) (Theorems 2.3 and 2.4) for both
{Wn ; n ≥ 2} and {L n ; n ≥ 2} are established. In other words, we establish weak limit
theorem analogues of the strong limit theorems of Li and Rosalsky [7, Theorems 2.1–
2.4]. We then apply Theorem 2.3 to provide necessary and sufficient conditions for
Jiang’s [4] asymptotic distribution (1.4) of L n .
The plan of the paper is as follows. The main results, Theorems 2.1–2.6, will be
stated in Sect. 2. In Sect. 3 we prove Theorems 2.1–2.4 by using a few results from
classical probability theory and recent methods due to Li and Rosalsky [7] and Liu
et al. [8]. In Sect. 4, we prove Theorems 2.5 and 2.6 and provide some comments
on Jiang’s [4] work. The proof of Theorem 2.5 is based on Theorem 3.1 of Liu et al.
[8] and on Theorem 2.3 (necessity half) and Remark 2.4 of the current work plus an
additional lemma (Lemma 4.1). The proof of Theorem 2.6 is based on Theorem 2.5
and Remark 2.3 of the current work plus an additional lemma (Lemma 4.2) and on
Corollary 3.3 of Li and Rosalsky [7].
2 The main results
Throughout, let {X k,i ; i ≥ 1, k ≥ 1} be an array of nondegenerate i.i.d. square integrable random variables, let { pn ; n ≥ 1} be a sequence of integers in [2, ∞), and
(n)
for n ≥ 1, consider the n × pn matrix Mn, pn = (X k,i )1≤k≤n,1≤i≤ pn and let ρ̂i, j , the
Pearson correlation coefficient between the ith and jth columns of Mn, pn , be defined
as in (1.1), 1 ≤ i, j ≤ pn , n ≥ 1. Let {L n ; n ≥ 1} be as in (1.2) with p = pn and let
{Wn ; n ≥ 1} be as in (1.7).
The first and second theorems are WLLNs for {Wn ; n ≥ 1} and {L n ; n ≥ 1},
respectively.
Theorem 2.1 Suppose that n/pn is bounded away from 0 and ∞. Let α > 1/2. Then
Wn P
→0
nα
(2.1)
if and only if
nP
max |X i X j | ≥ n α
1≤i< j≤n
→ 0 as n → ∞
(2.2)
and EX 1 = 0 whenever α ≤ 1. Here and below X i = X 1,i , i ≥ 1.
Theorem 2.2 Suppose that n/pn is bounded away from 0 and ∞. Let 1/2 < α ≤ 1.
If X 1 is nondegenerate and (2.2) holds, then
P
n 1−α L n → 0.
123
(2.3)
Asymptotic distribution of largest entry
11
Remark 2.1 Let {bn ; n ≥ 1} be a sequence of positive constants such that bn ↑ ∞
and
nP
max |X i X j | ≥ bn
1≤i< j≤n
→ 0 as n → ∞.
(2.4)
Note that
P
max |X i X j | ≥ bn
1≤i< j≤n
≤ n 2 P (|X 1 X 2 | ≥ bn ) , n ≥ 1
and so (2.4) holds if
n 3 P (|X 1 X 2 | ≥ bn ) → 0 as n → ∞.
Also note that
P
max |X i X j | ≥ bn
1≤i< j≤n
= P (Z n:1 Z n:2 ≥ bn ) ,
where Z n:1 and Z n:2 are, respectively, the largest and the second largest of the random
variables |X 1 |, |X 2 |, . . . , |X n |. Thus (2.4) is equivalent to
nP (Z n:1 Z n:2 ≥ bn ) → 0 as n → ∞.
(2.5)
1/2
, n ≥ 1. Clearly,
Let tn = P |X 1 | ≥ bn
2
nP (Z n:1 Z n:2 ≥ bn ) ≥ nP Z n:2
≥ bn
1/2
= nP Z n:2 ≥ bn
= n 1 − (1 − tn )n − ntn (1 − tn )n−1 , n ≥ 1.
Since (2.5) implies that ntn = o(1), one can see using Taylor’s theorem that
nP (Z n:1 Z n:2 ≥ bn ) ≥ n 1 − (1 − tn )n − ntn (1 − tn )n−1
∼
n 3 tn2
as n → ∞
2
and hence (2.4) ensures that
1/2
n 3/2 P |X 1 | ≥ bn
→ 0 as n → ∞.
123
12
D. Li et al.
Remark 2.2 If bn = n α , n ≥ 1 where α > 1/2, then by Remark 2.1 a sufficient
condition for (2.2) to hold is
n 3 P |X 1 X 2 | ≥ n α → 0 as n → ∞,
(2.6)
which is equivalent to
x 3/α P (|X 1 X 2 | ≥ x) → 0 as x → ∞
and a necessary condition for (2.2) to hold is
n 3/2 P |X 1 | ≥ n α/2 → 0 as n → ∞,
which is equivalent to
x 3/α P (|X 1 | ≥ x) → 0 as x → ∞.
(2.7)
Using Fubini’s theorem and integration by parts, one can show that (2.6) holds if
x 3/α log xP (|X 1 | ≥ x) → 0 as x → ∞.
The details are omitted and are left to the reader.
The third and fourth theorems establish WLLs for {Wn ; n ≥ 1} and {L n ; n ≥ 1},
respectively.
Theorem 2.3 Suppose that n/pn is bounded away from 0 and ∞. Then
Wn
P
→2
n log n
(2.8)
EX 1 = 0, EX 12 = 1,
√
and nP max1≤i< j≤n |X i X j | ≥ n log n → 0 as n → ∞.
(2.9)
√
if and only if
Theorem 2.4 Suppose that n/pn is bounded away from 0 and ∞. If X 1 is nondegenerate and
nP
max |X i X j | ≥ n log n → 0 as n → ∞,
(2.10)
1≤i< j≤n
then
123
n
log n
1/2
P
L n → 2.
(2.11)
Asymptotic distribution of largest entry
13
√
Remark 2.3 If bn = n log n, n ≥ 2, then by Remark 2.1 a sufficient condition for
(2.10) (i.e., the last part of 2.9) to hold is
n 3 P |X 1 X 2 | ≥ n log n → 0 as n → ∞,
(2.12)
which is equivalent to
x6
P (|X 1 X 2 | ≥ x) → 0 as x → ∞
log3 x
(2.13)
and a necessary condition for (2.10) to hold is
n 3/2 P |X 1 | ≥ (n log n)1/4 → 0 as n → ∞,
which is equivalent to
x6
P (|X 1 | ≥ x) → 0 as x → ∞.
log3/2 x
(2.14)
We now show that (2.12) holds if
x6
P (|X 1 | ≥ x) → 0 as x → ∞.
log x
(2.15)
In fact, since X 1 and X 2 are i.i.d. random variables, it follows from Fubini’s theorem
and (2.15) that as x → ∞,
P (|X 1 X 2 | ≥ x) = P (|X 1 X 2 | ≥ x and |X 2 | ≤ e) + P (|X 1 X 2 | ≥ x and |X 2 | > x/e)
+ P (|X 1 X 2 | ≥ x and e < |X 2 | ≤ x/e)
x/e
≤ 2P (|X 1 | ≥ x/e) +
P (|t X 1 | ≥ x) dP(|X 1 | ≤ t)
e
⎛ x/e
⎞
log x − log t
log x
+ o⎝
=o
dP(|X 1 | ≤ t)⎠
x6
(x/t)6
log x
=o
x6
⎛
e
log x
+ o⎝
x6
x/e
⎞
t 6 dP(|X 1 | ≤ t)⎠ .
e
123
14
D. Li et al.
Now using integration by parts, (2.15) ensures that as x → ∞,
x/e
x/e
t=x/e
t dP(|X 1 | ≤ t) = −t P(|X 1 | > t)
+ 6 t 5 P(|X 1 | > t)dt
6
6
t=e
e
e
⎛ x/e
⎞
log
t
= O(1) + o ⎝ t 5 × 6 dt ⎠
t
e
= O(1) + o (log x)2
= o (log x)2 .
We thus have that as x → ∞,
P (|X 1 X 2 | ≥ x) = o
log x
x6
+o
(log x)3
x6
=o
(log x)3
x6
;
i.e., (2.13) holds.
Remark 2.4 Let Z n:1 and Z n:2 be, respectively, the largest and the second largest of
the random variables |X 1 |, |X 2 |, . . . , |X n |. Write F(x) = P(|X 1 | ≤ x), x ≥ 0. Then
the joint distribution function G(x, y) = P(Z n:1 ≤ x, Z n:2 ≤ y), x ≥ 0, y ≥ 0 of
Z n:1 and Z n:2 is given by
G(x, y) =
if 0 ≤ x < y,
F n (x),
F n (y) + n F n−1 (y) (F(x) − F(y)) , if 0 ≤ y ≤ x.
Using Fubini’s theorem we have that for t > 0,
P
max |X i X j | > t
1≤i< j≤n
= P (Z n:1 Z n:2 > t and Z n:2 ≤ Z n:1 )
=
dG(x, y)
x y>t
⎛and 0≤y≤x
∞
x
⎜
⎝
=
√
t
=n
123
⎟
n(n − 1)F n−2 (y)d F(y)⎠ d F(x)
t/x
∞ √
⎞
t
F n−1 (x) − F n−1 (t/x) d F(x).
Asymptotic distribution of largest entry
15
We thus see that (2.2) is equivalent to
lim n
2
∞ n→∞
F n−1 (x) − F n−1 (n α/x) d F(x) = 0
n α/2
and the last part of (2.9) (i.e., 2.10) is equivalent to
∞
F n−1 (x) − F n−1
lim n 2
n→∞
√
n log n
d F(x) = 0.
x
(2.16)
(n log n)1/4
Remark 2.5 By Remark 2.4, we have
∞ x x6
2x 6
|
d F(t).
≥
x)
=
F(t)
−
F
X
P
(|X
1
2
t
log3 x
log3 x √
x
Note that (2.14) ensures that
∞
x6
lim
(1 − F(t))d F(t) = 0
x→∞ log3 x
√
x
and
x6
lim
x→∞ log3 x
∞ F(t) − F
x t
d F(t) = 0.
x
√
n log n, we see that, under (2.14), (2.13) (i.e., 1.9) is equivalent to
So, letting x =
√
n log n
lim n
3
n→∞
√
n log n
1− F
d F(x) = 0.
x
(2.17)
(n log n)1/4
Clearly, (2.14) implies that
lim n 2 P |X 1 | ≥ n 1/3 (log n)3/2 = 0.
n→∞
Under (2.14), by using the Taylor expansion, one can check that
3/2
n 1/3 (log
n) lim n
2
F
n→∞
n−1
(x) − F
n−1
√
n log n
d F(x) = 0
x
(n log n)1/4
123
16
D. Li et al.
if and only if
√
n log n
d F(x) = 0.
x
3/2
n 1/3 (log
n) lim n
F(x) − F
3
n→∞
(n log n)1/4
Thus, by (2.16), we see that, under (2.14), (2.10) (i.e., 1.11) is equivalent to
√
3/2
n 1/3 (log
n) lim n
1− F
3
n→∞
n log n
x
d F(x) = 0.
(2.18)
(n log n)1/4
Example 2.1 Let dn = exp (2n ) , n ≥ 1. Given α ∈ (−∞, ∞), let {X k,i ; i ≥ 1, k ≥ 1}
be an array of symmetric i.i.d. random variables such that
P (X 1 = −dn ) = P (X 1 = dn ) =
c logα d
n
2
dn6
=
c 2nα
2
dn6
, n ≥ 1,
where
c = cα =
∞
−1
logα dn
n=1
dn6
> 0.
Then EX 1 = 0 and
lim sup
x→∞
x6
P (|X 1 | ≥ x) = c > 0.
logα x
It follows that (2.15) holds if and only if α < 1. Note that (2.14) holds if α < 3/2.
For this example, using (2.17) one can check that (2.13) (i.e., 1.9) holds if and only if
α < 3/2. However using (2.18) one also can check that (2.10) (i.e., 1.11) holds if and
only if α < 3/2.
Example 2.2 Let {X k,i ; i ≥ 1, k ≥ 1} be an array of symmetric i.i.d. random variables
such that
P (|X 1 | ≥ x) =
log x
, x ≥ e.
x6
Clearly, for this example, (2.15) does not hold; i.e.,
x6
P (|X 1 | ≥ x) → 1 = 0 as x → ∞
log x
123
Asymptotic distribution of largest entry
17
but (2.14) does hold. For this example, we see that for any 1/4 < δ < 1/3
(n log n)δ
lim n
3
n→∞
√
n log n
F(x) − F
d F(x) > 0
x
(n log n)1/4
and hence that both (2.17) and (2.18) fail. Thus (1.9) and (1.11) fail.
We now state two unifying results, Theorems 2.5 and 2.6. Theorem 2.6 provides necessary and sufficient conditions for the asymptotic distribution (1.4) of L n obtained
by Jiang [4]. In Theorems 2.5 and 2.6, we suppose that X 1 is nondegenerate with
EX 12 < ∞ in Theorem 2.5 and E|X 1 |2+δ < ∞ for some δ > 0 in Theorem 2.6.
Consider the following six statements:
∞
(i)
lim n
n→∞
(ii) nP
(iii)
(iv)
(v)
(vi)
2
F
n−1
(x) − F
(n log n)1/4
max |X i X j | ≥
1≤i< j≤n
n−1
√
n log n
d F(x) = 0,
x
n log n → 0 as n → ∞,
Wμ,n
P
→ 2σ 2 ,
√
n log n
1/2
n
P
L n → 2,
log n
2
Wμ,n
1 −t/2
, −∞ < t < ∞,
−
a
≤
t
=
exp
−
lim P
e
√
n
n→∞
nσ 4
8π
1 −t/2
2
, −∞ < t < ∞,
lim P n L n − an ≤ t = exp − √ e
n→∞
8π
where F(x) = P (|X 1 | ≤ x) , x ≥ 0, μ = EX 1 , σ 2 = E(X 1 − μ)2 , and an =
4 log pn − log log pn , and Wμ,n is as in (1.7) with c = μ.
Theorem 2.5 Suppose that n/pn is bounded away from 0 and ∞. If X 1 is nondegenerate with EX 12 < ∞, then the four statements (i), (ii), (iii), and (v) are equivalent.
Theorem 2.6 Suppose that n/pn is bounded away from 0 and ∞. If X 1 is nondegenerate with E|X 1 |2+δ < ∞ for some δ > 0, then the six statements (i), (ii), (iii), (iv),
(v), and (vi) are equivalent.
Remark 2.6 It is an open problem as to whether or not Theorem 2.6 still holds if
EX 12 < ∞ but E|X 1 |2+δ = ∞ for all δ > 0. We conjecture specifically that the implications (iv) ⇒ (ii) and (iv) ⇒ (vi) can both fail if it is only assumed that EX 12 < ∞.
Remark 2.7 A Referee so kindly suggested that it may be possible to obtain alternative
proofs of some of the results in the current work concerning Wμ,n by applying some
123
18
D. Li et al.
results and/or ideas in the papers by Cuzick et al. [2] and Latała and Zinn [5]. Indeed,
the ideas and estimates in these two papers may help in answering the open problem
concerning the necessity of a 2 + δ moment for some δ > 0 mentioned in Remark 2.6
and, moreover, these two papers may be of key significance in further extensions.
Remark 2.8 Noting that the statements (iii) and (v) concern the non-normalized quantities Wμ,n and statements (iv) and (vi) concern the self-normalized quantities L n ,
the Editor so kindly suggested that it would be worthwhile to obtain necessary and
sufficient conditions for (vi) to hold, without forcing equivalence to (v) (that is, to
obtain necessary and sufficient conditions for (vi) to hold under the moment condition
EX 12 < ∞ or perhaps under an even weaker moment condition). The Editor pointed
out to us the article by Giné et al. [3] providing necessary and sufficient conditions for
asymptotic normality of self-normalized partial sums from a sequence of i.i.d. random
variables; this article may help to shed some light on solving the problem posed by
the Editor.
3 Proofs of Theorems 2.1–2.4
In this section, we give the proofs of Theorems 2.1–2.4.
Proof of Theorem 2.1 We prove the sufficiency part first. By Remark 2.2, as in the
proof of Lemma 3.1 of Li and Rosalsky [7], we can obtain from (2.2) that for every
c > 0,
nP
max
1≤i< j≤cn
|X i X j | ≥ εn α
→ 0 as n → ∞ for every ε > 0.
(3.1)
Moreover, by Remark 2.2, (2.2) implies (2.7) which, in turn, implies that
E|X 1 |β < ∞ for all 0 < β < 3/α.
(3.2)
Set
(n)
Yk,i, j = X k,i X k, j I {|X k,i X k, j | ≤ δn α }, 1 ≤ k ≤ n,
where δ > 0 is a small number which will be specified later. Then for every ε > 0,
P Wn ≥ εn
α
n
(n) α
≤
Yk,1,2 ≥ εn
k=1
+ nP
max |X i X j | ≥ δn α .
pn2 P
(3.3)
1≤i< j≤ pn
(n)
By hypothesis EX 1 = 0 whenever α ≤ 1. So it follows from (3.2) that n 1−α |EY1,1,2 | →
0 as n → ∞. Thus, for all large n, by Corollary 1.6 of Nagaev [9] which is a
123
Asymptotic distribution of largest entry
19
Fuk–Nagaev type inequality and letting ε/δ be sufficiently large,
n
n
(n)
(n)
(n)
Yk,1,2 ≥ εn α ≤ pn2 P (Yk,1,2 − EYk,1,2 ) ≥ εn α /2
pn2 P k=1
k=1
⎛ ⎛ ⎞ε/(8δ) ⎞
(n)
⎜ 2 ⎜ nE Y1,1,2
= O⎜
⎝ pn ⎝
n 2α
⎛
= O ⎝n 2
2
⎟
⎟
⎠
⎟
⎠
n 1+(2α−2+
α)∨0
n 2α
ε/(8δ) ⎞
⎠
→ 0 as n → ∞.
This together with (3.3) and (3.1) proves (2.1).
We now prove the necessity part. Let {X k,i ; i ≥ 1, k ≥ 1} be an independent
p2 − p
copy of {X k,i ; i ≥ 1, k ≥ 1}. Let · n be the maximal norm in the n 2 n -dimensional space Bn = {x = (xi, j ; 1 ≤ i < j ≤ pn ) : xi, j ∈ R, 1 ≤ i < j ≤ pn}.
Write Dn,k = X k,i X k, j ; 1 ≤ i < j ≤ pn , Dn,k = X k,i X k, j ; 1 ≤ i < j ≤ pn ,
Dsn,k = Dn,k − Dn,k , k = 1, 2, . . . , n, and
Wns
n
n
s
= max (X k,i X k, j − X k,i X k, j ) =: max (X k,i X k, j ) .
1≤i< j≤ pn 1≤i< j≤ pn k=1
k=1
Then Dsn,k , k = 1, 2, . . . , n are symmetric Bn -valued i.i.d. random variables, Wns =
n
k=1 Dsn,k , and it follows from (2.1) that
n
P
Wns /n α → 0.
By Lévy’s inequality in a Banach space setting (see, e.g., [1, p. 102] or [6, p. 47]), we
have that for every ε > 0,
P
max
max
1≤k≤n 1≤i< j≤ pn
|(X k,i X k, j )s | ≥ εn α
max Dsn,k n ≥ εn α
1≤k≤n
s
≤ 2P Wn ≥ εn α → 0 as n → ∞.
(3.4)
=P
123
20
D. Li et al.
Since max1≤i< j≤ pn |(X k,i X k, j )s |, k = 1, 2, . . . , n are i.i.d. random variables, it follows from (3.4) that for every ε > 0,
nP
|(X 1,i X 1, j ) | ≥ εn
s
max
1≤i< j≤ pn
α
= nP Dn,1 − Dn,1 ≥ εn α
n
(3.5)
→ 0 as n → ∞.
As in the proof of Lemma 3.1 in Li and Rosalsky [7], we can see that (3.5) implies
that for every ε > 0,
nP
max |(X 1,i X 1, j ) | ≥ εn
s
1≤i< j≤n
α
→ 0 as n → ∞.
(3.6)
Note that
n n P
max |(X 1,i X 1, j )s | ≥ εn α ∧ P |(X 1,1 X 1,2 )s | ≥ εn α
1≤i< j≤n
2
≥ nP max (X 1,i X 1,[n/2]+i )s ≥ εn α
1≤i≤n/2
n ≥ n 1 − exp − P |(X 1,1 X 1,2 )s | ≥ εn α
, n≥2
2
and hence since 1 − e−x ∼ x as x → 0, (3.6) implies that for every ε > 0,
n 2 P |X 1,1 X 1,2 − X 1,1 X 1,2 | ≥ εn α = n 2 P |(X 1,1 X 1,2 )s | ≥ εn α
→ 0 as n → ∞.
Thus for every ε > 0,
n 2 P |X 1,1 X 1,2 | ≥ εn α → 0 as n → ∞.
(3.7)
Since n/pn is bounded away from 0, (3.7) implies that for every ε > 0,
α
α
max |X 1,i X 1, j | ≥ εn
P Dn,1 ≥ εn = P
1≤i< j≤ pn
n
≤ pn2 P |X 1,1 X 1,2 | ≥ εn α → 0 as n → ∞.
(3.8)
Hence by Ottaviani’s inequality, it follows from (3.5) and (3.8) that for every ε > 0,
nP
max
1≤i< j≤ pn
|X 1,i X 1, j | ≥ εn α
= nP Dn,1 n ≥ εn α → 0 as n → ∞,
which, again as in the proof of Lemma 3.1 in Li and Rosalsky [7], is equivalent to
(2.2).
123
Asymptotic distribution of largest entry
21
When α ≤ 1, one can see that (2.1) and (3.2) (which follows from (2.2) as was
noted above) ensure that
n
k=1
lim
n→∞
X k,1 X k,2
= 0 a.s.
n
and hence by the converse of the Kolmogorov SLLN, E(X k,1 X k,2 ) = (EX 1 )2 = 0;
i.e., EX 1 = 0. The proof of Theorem 2.1 is therefore complete.
Proof of Theorem 2.2 Recall that (2.2) implies (3.1) and (3.2). Let μ = EX 1 . It is
easy to see that
α
max |(X i − μ)(X j − μ)| ≥ n
α
α
≤ nP
max |X i X j | ≥ n /4 + 2nP |μ| max |X i | ≥ n /4
1≤i< j≤ pn
1≤i≤ pn
2
α
+ nP μ ≥ n /4
nP
1≤i< j≤ pn
→ 0 as n → ∞.
Without loss of generality, we can assume that μ = 0. Since X 1 is nondegenerate and
1/2 < α ≤ 1, it follows from (3.2) that 0 < EX 12 ≤ 1 + E|X 1 |2/α < ∞. Thus, by
Corollary 3.3 (i) of Li and Rosalsky [7], we have that
lim n
n→∞
1−α
max
1≤i≤ pn
(n)
|X i |
max1≤i≤ pn nk=1 X k,i = lim
= 0 a.s.
n→∞
nα
(3.9)
and, from the proof of Theorem 2.2 of Li and Rosalsky [7], we see that
n
lim inf min
n→∞ 1≤i≤ pn
k=1 (X k,i
(n)
− X i )2
n
≥ EX 12 > 0 a.s.
(3.10)
Note that
n
n
(n)
(n)
(n) (n)
(X k,i − X i )(X k, j − X j ) =
X k,i X k, j − n X i X j .
k=1
(3.11)
k=1
So, combining our Theorem 2.1, (3.9), and (3.10), the conclusion (2.3) is established.
Proof of Theorem 2.3 (Sufficiency) By Remark 2.3, (2.9) ensures that (2.14) holds
which implies that
E|X 1 |β < ∞ for all 0 < β < 6.
(3.12)
123
22
D. Li et al.
For i ≥ 1, j ≥ 1, put
Yn,k,i, j = X k,i X k, j I |X k,i X k, j | ≤ δ n log n ,
√
Yn,k,i, j = X k,i X k, j I |X k,i X k, j | ≤ n/(log n)4 ,
√
Yn,k,i, j = X k,i X k, j I
n/(log n)4 < |X k,i X k, j | ≤ δ n log n , 1 ≤ k ≤ n,
where δ > 0 is a small number which will be specified later. We have for every ε > 0,
P Wn ≥ (2 + 4ε) n log n
n
Yn,k,1,2 ≥ (2 + 4ε) n log n
≤ pn2 P k=1
+ nP
max |X i X j | ≥ δ n log n .
1≤i< j≤ pn
(3.13)
Moreover,
n
2
pn P Yn,k,1,2 ≥ (2 + 4ε) n log n
k=1
n
2
≤ pn P Yn,k,1,2 ≥ (2 + 3ε) n log n
k=1
n
2
+ pn P Yn,k,1,2 ≥ ε n log n
=:
In(1)
+
k=1
In(2) .
We now estimate In(1) . Note that, from (2.9) and (3.12),
√
n EYn,1,1,2 nE|X 1,1 X 1,2 |I {|X 1,1 X 1,2 | > n/(log n)4 }
≤
√
√
n log n
n log n
2
23/2
(log n)
EX 14
≤
n
→ 0 as n → ∞,
n
2
E Yn,k,1,2 − EYn,k,1,2 ∼ n as n → ∞,
k=1
√
max Yn,k,1,2 − EYn,k,1,2 ≤ 2 n/(log n)4 , n ≥ 2,
1≤k≤n
123
(3.14)
Asymptotic distribution of largest entry
23
and
√
√
(2 + 2ε) n log n 2 n/(log n)4
= 0.
lim
n→∞
2n
Now applying one of the classical Kolmogorov exponential bound inequalities (see,
e.g., [10, Lemma 7.1]), we have that for all large n,
In(1)
n
≤
(Yn,k,1,2 − EYn,k,1,2 ) ≥ (2 + 2ε) n log n
k=1
= O n 2 exp {−(2 + ε) log n}
= O n −ε .
pn2 P
(3.15)
(2)
As for In , since (3.12) implies that
2 (log n)8 2
EX 14 ,
E Yn,1,1,2 ≤
n
applying Corollary 1.6 of Nagaev [9] and letting δ be sufficiently small, we have that
n
(2)
2
In ≤ pn P (Yn,k,1,2 − EYn,k,1,2 ) ≥ ε n log n/2
k=1
⎛ ⎞
ε/(8δ)
nE(Y1,1,2 )2
2
⎠
= O ⎝n
n log n
7 ε/(8δ)
(log
n)
= O n2
n
→ 0 as n → ∞.
(3.16)
As in the proof of Lemma 3.1 of Li and Rosalsky [7], the third condition of (2.9)
ensures that
nP
max |X i X j | ≥ δ n log n → 0 as n → ∞.
1≤i< j≤ pn
Combining (3.13)–(3.16), we get
P Wn ≥ (2 + ε) n log n → 0 as n → ∞ for every ε > 0.
(3.17)
It remains for us to prove that
P Wn ≥ (2 − ε) n log n → 1 as n → ∞ for every ε > 0.
(3.18)
123
24
D. Li et al.
To see this, let b > 0 be sufficiently large and, for i ≥ 1, k ≥ 1, set
Uk,i (b) = X k,i I {|X k,i | ≤ b} − EX k,i I {|X k,i | ≤ b},
Vk,i (b) = X k,i I {|X k,i | > b} − EX k,i I {|X k,i | > b}.
Then
P Wn ≥ (2 − 4ε) n log n
n
Uk,i (b)Uk, j (b) ≥ (2 − ε) n log n
≥P
max 1≤i< j≤ pn k=1
n
−2P
max Uk,i (b)Vk, j (b) ≥ ε n log n
1≤i< j≤ pn k=1
n
−P
max Vk,i (b)Vk, j (b) ≥ ε n log n .
1≤i< j≤ pn k=1
2 (b) → 1 as b → ∞, by Theorem 2.3 of Li and Rosalsky [7], we see that
Since EU1,1
the first term on the right-hand side of the inequality above tends to 1 for any fixed
2 (b) → 0 as
ε > 0 and all sufficiently large b > 0. On the other hand, since EV1,1
b → ∞, proceeding using the same arguments as in the proof of (3.17), the other two
terms tend to zero as n → ∞ for any fixed ε > 0 and all sufficiently large b > 0.
Hence (3.18) holds and the proof of the sufficiency part of Theorem 2.3 is complete.
(Necessity) Let {X k,i ; i ≥ 1, k ≥ 1}, · n , Bn , Dn,k , Dn,k , Dsn,k , k = 1, 2, . . . , n,
and Wns , n ≥ 1 be as in the proof of the necessity half of Theorem 2.1. Then it follows
from (2.8) that
Ws
lim P √ n
≥ 5 = 0.
n→∞
n log n
Then by Lévy’s inequality in a Banach space setting, we have that
P
max max |(X k,i X k, j )s | ≥ 5 n log n
1≤k≤n 1≤i< j≤ pn
= P max Dsn,k n ≥ 5 n log n
1≤k≤n
≤ 2P Wns ≥ 5 n log n
→ 0 as n → ∞,
which, in turn, implies that
nP
123
max
1≤i< j≤ pn
|(X 1,i X 1, j )s | ≥ 5 n log n → 0 as n → ∞.
(3.19)
Asymptotic distribution of largest entry
25
As in the proof of Lemma 3.1 of Li and Rosalsky [7], we see that (3.19) implies that
for every ε > 0,
nP
max |(X 1,i X 1, j )s | ≥ ε n log n → 0 as n → ∞.
1≤i< j≤n
(3.20)
Then the first and third conditions of (2.9) hold by arguing in a similar manner as in
the end of the proof of the necessity half of Theorem 2.1. Since the third condition of
(2.9) implies (2.14) (see Remark 2.3), we clearly have 0 < EX 12 < ∞. Now by the
same argument as in the proof of the necessity half of Theorem 2.1, (3.20) yields for
all ε > 0
nP
max |X 1,i X 1, j | ≥ ε n log n → 0 as n → ∞.
1≤i< j≤n
(3.21)
By taking ε = EX 12 in (3.21) and applying the sufficiency half of Theorem 2.3 we get
n X k,i X k, j max1≤i< j≤ pn k=1
·
E X 12
E X 12 P
→ 2,
√
n log n
which when combined with (2.8) yields EX 12 = 1.
Proof of Theorem 2.4 We can of course assume that EX 1 = 0 and EX 12 = 1. By
Remark 2.3, (2.10) implies (2.14) and hence (3.12) holds. Proceeding as in the proof
of Theorem 2.4 of Li and Rosalsky [7], we can then get that
n (X − X (n) )2
k=1 k,i
i
− 1 = 0 a.s.
lim max n→∞ 1≤i≤ pn n
(3.22)
and
(n)
(n)
n max1≤i, j≤ pn |X i X j |
= 0 a.s.
√
n→∞
n log n
lim
(3.23)
Then, in view of (3.11), the conclusion (2.11) follows immediately from Theorem 2.3,
(3.22), and (3.23).
4 Proofs of Theorems 2.5 and 2.6 and some comments on Jiang’s [4] theorems
We begin by stating and proving two lemmas. Lemmas 4.1 and 4.2 are used in the
proofs of Theorems 2.5 and 2.6, respectively.
123
26
D. Li et al.
Lemma 4.1 Let c ∈ (−∞, ∞) be a constant. Then
nP
max (X i − c)(X j − c) ≥ n log n → 0 as n → ∞
1≤i< j≤n
(4.1)
if and only if (ii) holds. Similarly,
∞
P
max (X i − c)(X j − c) ≥ n log n < ∞
1≤i< j≤n
n=1
if and only if
∞
P
max X i X j ≥ n log n < ∞.
n=1
1≤i< j≤n
Proof Since X i = (X i − c) − (−c), i ≥ 1, we only need to show that (ii) follows
from (4.1). As in the proof of Lemma 3.1 of Li and Rosalsky [7], (4.1) implies that
nP
max (X i − c)(X j − c) ≥ ε n log n → 0 as n → ∞ for every ε > 0,
1≤i< j≤n
which, by Remark 2.3, ensures that
E |X 1 − c|β < ∞ for all 0 < β < 6.
It now is easy to see that (4.1) implies that
max |X i X j | ≥ n log n
≤ nP
max (X i − c)(X j − c) ≥ n log n/4
1≤i< j≤n
+2nP |c| max |X i − c| ≥ n log n/4 + nP c2 ≥ n log n/4
nP
1≤i< j≤n
1≤i≤n
→ 0 as n → ∞;
i.e., (ii) holds. This completes the proof of Lemma 4.1.
Lemma 4.2 Suppose that n/ pn is bounded away from 0 and ∞. If X 1 is nondegenerate and E|X 1 |5 < ∞, then
lim
n→∞
123
n L 2n
−
2
Wμ,n
nσ 4
= 0 a.s.
(4.2)
Asymptotic distribution of largest entry
27
Proof We first show that
Wμ,n
= 0 a.s.
lim n 3/5 L n −
n→∞
nσ 2
(4.3)
To see this, write
n
2
σ̂n,i
=
k=1 (X k,i
n
(n)
− X i )2
n
k=1 (X k,i
=
n
− μ)2
(n)
2
− X i − μ , 1 ≤ i ≤ pn .
By Corollary 3.3 (ii) of Li and Rosalsky [7], the moment condition E|X 1 |5 < ∞
implies that
lim sup
n→∞
n
log n
1/2
(n)
max X i − μ ≤ 2σ a.s.
1≤i≤ pn
and we thus have that
lim sup
n→∞
(n)
2
n
max X i − μ ≤ 4σ 2 a.s.
log n 1≤i≤ pn
(4.4)
2/α
Since, for α = 4/5 (∈ (1/2, 1]), E (X 1 − μ)2 = E|X 1 − μ|5 < ∞, by Corollary
3.3 (i) of Li and Rosalsky [7] we have that
n
(X k,i − μ)2
− σ 2 lim n
max k=1
n→∞
1≤i≤ pn
n
n 2
2 max1≤i≤ pn
k=1 (X k,i − μ) − σ
= lim
n→∞
nα
= 0 a.s.
1/5
(4.5)
Combining (4.4) and (4.5), we get
2
lim n 1/5 max σ̂n,i
− σ 2 = 0 a.s.,
n→∞
1≤i≤ pn
which clearly ensures that
2
lim n 1/5 max σ̂n,i
− σ 2 = 0 a.s.
n→∞
1≤i≤ pn
2
and lim n 1/5 min σ̂n,i
− σ 2 = 0 a.s.
n→∞
1≤i≤ pn
(4.6)
123
28
D. Li et al.
Note that
1
(n)
(n) X k, j − X j max1≤i< j≤ pn nk=1 X k,i − X i
2
max1≤i≤ pn σ̂n,i
≤ Ln
≤
and
n
1
(n)
(n) X k, j − X j max1≤i< j≤ pn nk=1 X k,i − X i
2
min1≤i≤ pn σ̂n,i
n
(4.7)
1
1 = 2 1 + o n −1/5 . So it follows from (4.6) that
σ
σ 2 + o n −1/5
(n)
(n) max1≤i< j≤ pn nk=1 X k,i − X i
X k, j − X j L n = 1 + o n −1/5
a.s.
nσ 2
Now note that
n (n)
X k,i − X i
(n)
X k, j − X j
k=1
=
n
(n)
(n)
X k,i − μ X k, j − μ − n X i − μ X j − μ .
k=1
It is easy to see that
(n)
(n) n
max
X k, j − X j W 1≤i< j≤ pn k=1 X k,i − X i
μ,n −
nσ 2
nσ 2 (n)
2
max1≤i≤ pn X i − μ
≤
2
σ
log n
a.s.
=O
n
by (4.4). We thus have that
W
log n
μ,n
L n = 1 + o n −1/5
+
O
nσ 2
n
Wμ,n
Wμ,n
log n
−1/5
a.s.
+
O
=
+
×
o
n
nσ 2
nσ 2
n
123
(4.8)
Asymptotic distribution of largest entry
29
By Theorem 2.1 and Remark 2.1 of Li and Rosalsky [7], the moment condition
E |X 1 − μ|3/(3/5) = E |X 1 − μ|5 < ∞ ensures that
lim
n→∞
Wμ,n
= 0 a.s.
n 3/5
(4.9)
Combining (4.8) and (4.9), (4.3) follows. Finally, it follows from (4.3) and (4.9) that
n L 2n
−
2
Wμ,n
nσ 4
2 W 2
Wμ,n
μ,n
−3/5
=n
+o n
−
nσ 2
nσ 4
2
−3/5
−1/5
W
+
o
n
=
×
o
n
μ,n
σ2
= o(1) a.s.;
i.e., (4.2) holds.
Proof of Theorem 2.5 By Remark 2.4, (i) and (ii) are equivalent.
We now show how we can obtain the implication (ii) ⇒ (v) from the proof of
Theorem 3.1 of Liu et al. [8] as was noted above in Sect. 1. Actually, as in the proof
of Theorem 3.1 of Liu et al. [8], (ii) implies that for any 0 < ε ≤ 10−4 there exists a
finite constant C = C(ε) > 0 such that
2
Wμ,n
pn2 − pn 2
P N ≥ an + t − an ≤ t − exp −
sup P
4
nσ
2
−∞<t<∞
2
√
log5/2 n −1+20 ε
≤ C pn
E|X 1 |3
+ C 1/2
n
+ CnP
max (X i − μ)(X j − μ) ≥ n log n ,
1≤i< j≤n
where N is a standard normal random variable. Then by Lemma 4.1 we get that
2
Wμ,n
pn2 − pn 2
P N ≥ an + t = 0.
lim
sup P
− an ≤ t − exp −
4
n→∞ −∞<t<∞ nσ
2
It is easy to check that
pn2 − pn 2
1 −t/2
lim exp −
, −∞ < t < ∞.
P N ≥ an + t
= exp − √ e
n→∞
2
8π
Thus (v) follows from (ii).
123
30
D. Li et al.
Next, since n/pn is bounded away from 0 and ∞, it is easy to see that (v) implies
that
2
Wμ,n
n log n
P
→ 4σ 4 ,
which implies (iii).
It remains to show that (iii) implies (ii). By Theorem 2.3, we see that (iii) implies
that
nP
max (X i − μ)(X j − μ) ≥ σ 2 n log n → 0 as n → ∞,
1≤i< j≤n
which, as in the proof of Lemma 3.1 of Li and Rosalsky [7], implies that
nP
max (X i − μ)(X j − μ) ≥ ε n log n → 0 as n → ∞ for every ε > 0.
1≤i< j≤n
Thus, by Lemma 4.1, (ii) follows. We have thus shown that (iii) implies (ii) thereby
completing the proof of Theorem 2.5.
Proof of Theorem 2.6 Assume that E|X 1 |2+δ < ∞ for some δ > 0. In view of
Theorem 2.5, it suffices to establish the implications (ii) ⇒ (vi), (vi) ⇒ (iv), and
(iv) ⇒ (ii).
By Remark 2.3, (ii) ensures that E|X 1 |5 < ∞, and then by Lemma 4.2, (vi) follows
from (ii) by way of (v) (which holds by Theorem 2.5).
Next, since n/pn is bounded away from 0 and ∞, it is easy to see that (vi) implies
that
n
P
L 2 → 4,
log n n
which is (iv).
To show that (iv) implies (ii), suppose initially that δ = 2. Now as in the proof of
Lemma 4.2 above, by Corollary 3.3 (i) of Li and Rosalsky [7], we have that
(n)
lim max X i − μ = 0 a.s.
n→∞ 1≤i≤ pn
and
2
lim max σ̂n,i
− σ 2 = 0 a.s.
n→∞ 1≤i≤ pn
Hence it follows from (4.7) that (iv) is equivalent to
(n)
(n) max1≤i< j≤ pn nk=1 X k,i − X i
X k, j − X j P
→ 2σ 2 .
√
n log n
123
(4.10)
Asymptotic distribution of largest entry
31
Since
n (n)
X k,i − X i
(n)
X k, j − X j
k=1
=
n
(n)
(n)
X k,i − μ X k, j − μ − n X i − μ X j − μ
k=1
and since, by Corollary 3.3 (i) of Li and Rosalsky [7],
(n)
(n)
n max1≤i, j≤ pn X i − μ X j − μ lim
= 0 a.s.,
√
n→∞
n log n
we see that (4.10) and (iii) are equivalent. Since (iii) and (ii) are equivalent by
Theorem 2.5, we have established the implication (iv) ⇒ (ii) when δ = 2.
It remains to show that (iv) implies (ii) for general δ > 0. In view of the EX 14 < ∞
case already established, without loss of generality we may assume that 0 < δ < 2
and it suffices to show that E|X 1 |2+δ < ∞ and (iv) imply that E|X 1 |β < ∞ for some
β > 4. By Corollary 3.3 (i) of Li and Rosalsky [7], we have that
lim
max1≤i≤ pn
n→∞
n
k=1 (X k,i
4/(2+δ)
n
− μ)2
= 0 a.s.
and
max1≤i≤ pn nk=1 X k,i − μ lim
= 0 a.s.
n→∞
n 2/(2+δ)
⎛
⎞
(n)
max1≤i≤ pn X i − μ
⎝i.e., lim
= 0 a.s.⎠ .
n→∞
n 2/(2+δ)−1
We hence have that
lim
max1≤i< j≤ pn
n
k=1
(n) 2 n
k=1 X k, j
X k,i − X i
n 8/(2+δ)
n→∞
(n) 2
− Xj
= 0 a.s.
Thus, it follows from (iv) that
(n)
(n) max1≤i< j≤ pn nk=1 X k,i − X i
X k, j − X j P
→ 0.
√
n (6−δ)/(4+2δ) log n
123
32
D. Li et al.
Note that
n (n)
X k,i − X i
k=1
(n)
X k, j − X j
=
n
X k,i − μ X k, j − μ
k=1
(n)
(n)
−n X i − μ X j − μ .
We thus have that
max1≤i< j≤ pn nk=1 (X k,i − μ)(X k, j − μ) P
→ 0.
√
n (6−δ)/(4+2δ) log n
Then it follows as in the proof of the necessity part of Theorem 2.1 that
nP
max
1≤i< j≤ pn
(X i − μ)(X j − μ) ≥ n (6−δ)/(4+2δ) log n → 0 as n → ∞,
√
which, by Remark 2.1 (with bn = n (6−δ)/(4+2δ) log n, n ≥ 1), implies that
E|X 1 |(12+6δ)/(6−δ)−ε < ∞ for any 0 < ε < (12 + 6δ)/(6 − δ).
Since 0 < δ < 2, it is easy to check that (12 + 6δ)/(6 − δ) > 2 + (7/6)δ. We thus
get that
E|X 1 |2+(7/6)δ < ∞.
Let n 0 be the smallest positive integer such that (7/6)n 0 δ > 2. Then by repeating the
above argument n 0 − 1 additional times we get
E|X 1 |β < ∞ for β = 2 + (7/6)n 0 δ > 4.
The proof is complete.
Remark 4.1 The implications (ii) ⇒ (vi) and (vi) ⇒ (iv) in Theorem 2.6 did not use
the assumption that E|X 1 |2+δ < ∞ for some δ > 0.
Remark 4.2 From Theorem 2.6 we see that when X 1 is nondegenerate with E|X 1 |2+δ <
∞ for some δ > 0, the assertion (1.5) (i.e., the LL for L n , n ≥ 2) always ensures (1.4)
(i.e., the asymptotic distribution of L n ), but the converse is not necessarily true; see
Example 4.1 below.
We now provide some further comments on Jiang’s [4] theorems. Suppose that n/pn
is bounded away from 0 and ∞, X 1 is nondegenerate, and E|X 1 |2+δ < ∞ for some
δ > 0. As we see from the proof of the necessity half of Theorem 2.3, (ii) follows
123
Asymptotic distribution of largest entry
33
from any of the following weak statements:
Wμ,n
(iii) lim P √
≥ λ = 0 for some λ ∈ (0, ∞),
n→∞
n log n
1/2
n
lim P
L n ≥ λ = 0 for some λ ∈ (0, ∞).
(iv)
n→∞
log n
Similarly, we can see from Theorems 2.3 and 2.4 and Lemma 3.1 of Li and
Rosalsky [7] and Remark 2.4 and Lemma 4.1 above that the following six statements
are equivalent:
(vii)
∞
∞
n=1
(viii)
F n−1 (x) − F n−1
n
∞
n=1
√
n log n
d F(x) < ∞,
x
(n log n)1/4
P
max |X i X j | ≥
1≤i< j≤n
n log n < ∞,
Wμ,n
lim √
= 2σ 2 a.s.,
n log n
Wμ,n
(ix) lim sup √
< ∞ a.s.,
n log n
n→∞
1/2
n
L n = 2 a.s.,
(x) lim
n→∞ log n
1/2
n
(x) lim sup
L n < ∞ a.s.
log n
n→∞
(ix)
n→∞
Clearly, each of (vii)–(x) implies (i)–(vi) but the converse is not true as will be seen
from the following example.
Example 4.1 Let {X k,i ; i ≥ 1, k ≥ 1} be an array of symmetric i.i.d. random variables
such that
P (X 1 ≤ −x) = P (X 1 ≥ x) =
where c =
c log x
2x 6 log log log x
, x ≥ 727,
7276 log log log 727
and again X 1 = X 1,1 . Then EX 1 = 0 and
log 727
P (|X 1 | ≥ x) =
c log x
x 6 log log log x
, x ≥ 727,
which clearly implies (2.15); i.e.,
x6
P (|X 1 | ≥ x) → 0 as x → ∞.
log x
123
34
D. Li et al.
Then from Theorem 4.1 and Remark 2.3 we can see that (i)–(vi) hold for this example.
However,
P |X 1 | ≥ (n log n)1/4 ∼
c
4n 3/2 (log n)1/2 log log log n
as n → ∞
and so
∞
2
n 2 P(|X 1 | ≥ (n log n)1/4 ) = ∞.
n=1
Thus by Remark 2.3 of Li and Rosalsky [7], (viii) does not hold which, in turn, implies
that
max1≤i< j≤ pn nk=1 X k,i X k, j lim sup
= ∞ a.s.
√
n log n
n→∞
and
n
lim sup
log n
n→∞
1/2
L n = ∞ a.s.
Acknowledgments The authors are grateful to the Referees and Editor for their constructive, perceptive,
and substantial comments and suggestions which enabled us to greatly improve the paper. In particular, we
were able to obtain Theorems 2.5 and 2.6 under moment conditions which are considerably weaker than
those of the initial version of this paper. Some points raised by the Referees and Editor which we were
not able to resolve are posed as open problems. The research of Deli Li was supported by a grant from
the Natural Sciences and Engineering Research Council of Canada and the research of Wei-Dong Liu was
supported by the National Natural Science Foundation of China (10571159, 10671176) and the Specialized
Research Fund for the Doctoral Program of Higher Education (20060335032).
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