On limiting distribution of U-statistics based on

arXiv:1411.3083v3 [] 19 Sep 2017

associated random variables
Mansi Garg and Isha Dewan
Indian Statistical Institute
New Delhi-110016 (India)
mansibirla@gmail.com and ishadewan@gmail.com
Abstract
Let {Xn , n ≥ 1} be a sequence of stationary associated random variables. We discuss another
set of conditions under which a central limit theorem for U-statistics based on {Xn , n ≥ 1} holds.
We look at U-statistics based on differentiable kernels of degree 2 and above. We also discuss some
applications.

Keywords: Associated random variables; U-statistics; Central limit theorem; Skewness; Kurtosis.

1

Introduction

In this paper, we obtain another set of assumptions under which a central limit theorem for
U-statistics based on stationary associated random variables holds. We look at non-degenerate
U-statistics based on differentiable (component-wise monotonic or non-monotonic) kernels of any
finite degree k ≥ 2. The proof requires relatively non-restrictive assumptions. We also illustrate
some applications of our results using examples. Apropos our discussion, we give the following.
Definition 1.1. (Esary et al. (1967)) A finite collection of random variables {Xj , 1 ≤ j ≤ n} is

said to be associated, if for any choice of component-wise nondecreasing functions h, g : Rn → R,
we have
Cov(h(X1 , . . . , Xn ), g(X1 , . . . , Xn )) ≥ 0,
whenever it exists. An infinite collection of random variables {Xj , j ≥ 1} is associated if every
finite sub-collection is associated.

1

Any set of independent random variables is associated. Nondecreasing functions of associated
random variables are associated (cf. Esary et al. (1967)). A detailed presentation of the asymptotic
results and examples relating to associated sequences can be found in Bulinski and Shashkin (2007),
Oliveira (2012) and Prakasa Rao (2012).
For the rest of the paper, assume that {Xn , n ≥ 1} is a sequence of stationary associated
random variables with F as the marginal distribution function of X1 . We next briefly discuss
existing results on central limit theorem of U-statistics based on {Xn , n ≥ 1}.
Dewan and Prakasa Rao (2001) gave a central limit theorem for degenerate and non-degenerate
U-statistics using an orthogonal expansion of the underlying kernel. Dewan and Prakasa Rao
(2002) and its corrigendum Dewan and Prakasa Rao (2015) obtained a central limit theorem for
U-statistics with differentiable kernels of degree 2, using Hoeffding’s decomposition. The limiting
distribution of U-statistics based on differentiable kernels can also be obtained using the results of
Beutner and Zähle (2012, 2014) and Garg and Dewan (2015). We have discussed the difference in
our assumptions with an example in section 4.
The paper is organized as following. In section 2, we state a few results and definitions which
will be required to prove our main results. Limiting distribution of non-degenerate U-statistics
based on {Xn , n ≥ 1} is given in section 3. In section 4, we apply our results to discuss the
asymptotic distribution of estimators of second, third and the fourth central moments. We also
discuss estimators of skewness and kurtosis, when the underlying sample is from {Xn , n ≥ 1}.

2

Preliminaries

In this section, we give results and definitions which will be needed to prove our main results given
in section 3.
Definition 2.1. Hoeffding’s decomposition for U-statistics based on a symmetric measurable function ρ : R2 → R. Define the U-statistic, Un , by
 −1 X
n
Un =
2

ρ(Xi , Xj ).

(2.1)

1≤i<j≤n

Let θ =

R

ρ(x1 , x2 ) dF (x1 )dF (x2 ) and ρ1 (x1 ) =

R2

R

ρ(x1 , x2 ) dF (x2 ). Further, let

R

h(1) (x1 ) = ρ1 (x1 ) − θ and h(2) (x1 , x2 ) = ρ(x1 , x2 ) − ρ1 (x1 ) − ρ1 (x2 ) + θ.

(2.2)

Then, the Hoeffding’s decomposition for Un is
Un = θ + 2Hn(1) + Hn(2) ,

2

(2.3)

(j)

where Hn is the U-statistic of degree j based on the kernel h(j) , j = 1, 2. When Xj , 1 ≤ j ≤ n are
i.i.d., E(Un ) = θ.
Remark 2.2. An extension of the Hoeffding’s decomposition for U-statistics of a finite degree
k > 2 can be found in Lee (1990).
Lemma 2.3. (Newman (1980)) Let X and Y be two associated random variables with E(X 2 ) < ∞
and E(Y 2 ) < ∞. Let f and g be differentiable functions with sup |f ′ (x)| < ∞ and sup |g′ (y)| < ∞.
x

y

Then,
Cov(f (X), g(Y )) =

Z

R2

f ′ (x)g′ (y)[P (X ≤ x, Y ≤ y) − P (X ≤ x)P (Y ≤ y)]dxdy

≤ sup f ′ (x) sup g′ (y) Cov(X, Y ).
x

y

Lemma 2.4. (Lebowitz (1972)) Define, for A and B, subsets of {1, 2, ..., n} and real xj ′ s,
HA,B (xj , j ∈ A ∪ B) = P [Xj > xj , j ∈ A ∪ B] − P [Xk > xk , k ∈ A]P [Xl > xl , l ∈ B].
If the random variables X1 , X2 , ..., Xn are associated, then
0 ≤ HA,B ≤

XX

H{i},{j} .

(2.4)

i∈A j∈B

Definition 2.5. (Newman (1984)) Let g and g̃ be two real-valued functions on Rm , for some
m ∈ N. g ≪ g̃ iff g̃ + g and g̃ − g are both coordinate-wise nondecreasing. If g ≪ g̃, then g̃ will be
a coordinate-wise nondecreasing function.
Lemma 2.6. (Newman (1984)) For each j, j ≥ 1, let Yj = f (Xj ) and Y˜j = f˜(Xj ). Suppose that
P
P∞
2
˜ ˜
f ≪ f˜. Define σ 2 = V ar(Y1 ) + 2 ∞
j=2 Cov(Y1 , Yj ). Let σ > 0 and
j=1 Cov(Y1 , Yj ) < ∞. Then,
n
1 X
L
√
(Yj − E(Yj )) −
→ N (0, 1) as n → ∞.
nσ

(2.5)

j=1

In the following, g ≪A g̃ if g ≪ g̃ and both g and g̃ depend only on x′j s with j ∈ A. A is a
finite subset of {k, k ≥ 1}.
Lemma 2.7. (Newman (1984)) Let g1 ≪A g̃1 and g2 ≪A g̃2 . Then,
|Cov(g1 (X1 , X2 , ...), g2 (X1 , X2 , ...))|≤ Cov(g̃1 (X1 , X2 , ...), g̃2 (X1 , X2 , ...)).

3

(2.6)

3

Asymptotic distribution of U-statistics

The main result of this section, Theorem 3.6, gives a central limit theorem for non-degenerate
U-statistics with differentiable kernels of degree 2 based on {Xn , n ≥ 1}. The extension of this
theorem to U-statistics with kernels of a general finite degree k > 2 is also discussed. Before
proceeding to the proof of the main theorem, we discuss the following lemma.
Assume all the expectations and derivatives defined in this section exist, and C is a generic
positive constant in the sequel.
Lemma 3.1. Let the random variables Z1 , Z2 , Z3 , Z4 be identically distributed with F as the
marginal distribution function of Z1 and E[Z12 ] < ∞. Define z(u, v, x, y) = h(2) (u, v)h(2) (x, y),

where the kernel h(2) (., .) is defined in (2.2). Then, for Z1′ , Z2′ , Z3′ , Z4′ i.i.d. random variables with
marginal distribution function F and independent of Z1 , Z2 , Z3 , Z4 ,
h
i
E z(Z1 , Z2 , Z3 , Z4 ) − z(Z1′ , Z2 , Z3 , Z4 )



hZ 
I(Z1 > u) − I(Z1′ > u) I(Z2 > v) − I(Z2′ > v) I(Z3 > x) − I(Z3′ > x)
=E
R4


i
I(Z4 > y) − I(Z4′ > y) z ′′′′ (u, v, x, y)dudvdxdy ,
(3.1)
 4

∂ z(t1 ,t2 ,t3 ,t4 )
where z ′′′′ (u, v, x, y) = ∂dt
.
|
t
=u,t
=v,t
=x,t
=y
1
2
3
4
1 ∂dt2 ∂dt3 ∂dt4

Proof. From the results discussed in Christofides and Vaggelatou (2004) we get
Z 

′
I(Z1 > u) − I(Z1′ > u) z ′ (u, Z2 , Z3 , Z4 )du,
z(Z1 , Z2 , Z3 , Z4 ) − z(Z1 , Z2 , Z3 , Z4 ) =

(3.2)

R

where z ′ (u, Z2 , Z3 , Z4 ) =
Then,



∂z(t,Z2 ,Z3 ,Z4 )
|t=u
∂dt


 2

1 ,t2 ,Z3 ,Z4 )
. Let z ′′ (u, v, Z3 , Z4 ) = ∂ z(t
|
t1 =u,t2 =v .
∂dt1 ∂dt2

z(Z1 , Z2 , Z3 , Z4 ) − z(Z1′ , Z2 , Z3 , Z4 )
Z 


I(Z1 > u) − I(Z1′ > u) z ′ (u, Z2 , Z3 , Z4 ) − z ′ (u, Z2′ , Z3 , Z4 ) du
=
R
Z 

I(Z1 > u) − I(Z1′ > u) z ′ (u, Z2′ , Z3 , Z4 )du
+
R
Z 


I(Z1 > u) − I(Z1′ > u) I(Z2 > v) − I(Z2′ > v) z ′′ (u, v, Z3 , Z4 )dvdu
=
R2

+ z(Z1 , Z2′ , Z3 , Z4 ) − z(Z1′ , Z2′ , Z3 , Z4 ).

The last equality follows by repeating the argument of (3.2). Taking expectations, we get
h
i
E z(Z1 , Z2 , Z3 , Z4 ) − z(Z1′ , Z2 , Z3 , Z4 )


i
hZ 
I(Z1 > u) − I(Z1′ > u) I(Z2 > v) − I(Z2′ > v) z ′′ (u, v, Z3 , Z4 )dvdu + 0.
=E
R2

4

(3.3)

(3.4)

R
As h(2) (., .) is defined in (2.2) is a degenerate kernel ( R h(2) (x, y)dF (y) = 0, for all x ∈ R),

E(h(2) (Z1′ , Z2 )h(2) (Z3 , Z4 )) = 0 and E(h(2) (Z1′ , Z2′ )h(2) (Z3 , Z4 )) = 0. Hence, (3.4) follows. Solving

similarly as (3.4), we get
h
i
E z(Z1 , Z2 , Z3 , Z4 ) − z(Z1′ , Z2 , Z3 , Z4 )



hZ 
I(Z1 > u) − I(Z1′ > u) I(Z2 > v) − I(Z2′ > v) I(Z3 > x) − I(Z3′ > x)
=E
R4


i
I(Z4 > y) − I(Z4′ > y) z ′′′′ (u, v, x, y)dudvdxdy .

In the following, assume {Xi′ , i ≥ 1} to be a sequence of random variables independent of

Xi , i ≥ 1 such that Xi′ , i ≥ 1 are i.i.d. with F as the marginal distribution function of X1′ .

Lemma 3.2. Let the functions h(2) (., .), z(., ., ., .) and z ′′′′ (., ., ., .) be as defined in Lemma 3.1.
Assume the following holds.

2
(i) E h(2) (X1 , X1+k ) < ∞, for all k ∈ N;

(ii) For any 0 < C1 < ∞,

sup

u,v,x,y∈[−C1 ,C1 ]

||z ′′′′ (u, v, x, y)|| = CC1b , for some b ∈ N ∪ {0};

(iii) For the chosen C1 , define functions f1 (.), and f2 (.) as follows.
f1 (x) = xI(|x|≤ C1 ) + C1 I(x > C1 ) − C1 I(x < −C1 ) and
f2 (x) = (x − C1 )I(x > C1 ) + (x + C1 )I(x < −C1 ).
Then, for all k1 , k3 ∈ N, k2 ∈ N ∪ {0}, and some δ > 0,
X

i,j,p,q=1,2
(i,j,p,q)6=(1,1,1,1)




C
E h(2) (fi (Z1 ), fj (Z1+k1 )) h(2) (fp (Z1+k2 ), fq (Z1+k3 )) ≤ δ ,
C1

′ , for m = 1, 1 + k , 1 + k , 1 + k ;
where Zm = Xm or Xm
1
2
3
P∞
δ/(3(p+δ)) < ∞ for p = b + 2, where b and δ are defined in (ii) and (iii),
(iv)
j=1 Cov(X1 , Xj )

respectively.

Then, as n → ∞,
X

X

|E(h(2) (Xi , Xj )h(2) (Xk , Xl ))|= o(n3 ).

(3.5)

1≤i<j≤n 1≤k<l≤n

Proof. Note that, due to (i),
X

|E(h(2) (Xi , Xj ))2 |= o(n3 ).

1≤i<j≤n

5

(3.6)

Since h(2) (x, y) is a degenerate kernel,
E(h(2) (Xi′ , Xj )h(2) (Xk , Xl )) = 0.

(3.7)

Using Lemma 3.1, for i, j, k, l all distinct,
|E(h(2) (Xi , Xj )h(2) (Xk , Xl ))|= |E(z(Xi , Xj , Xk , Xl ) − z(Xi′ , Xj , Xk , Xl ))|



hZ 
I(Xi > ui ) − I(Xi′ > ui ) I(Xj > uj ) − I(Xj′ > uj ) I(Xk > uk ) − I(Xk′ > uk )
= E
R4


i
I(Xl > ul ) − I(Xl′ > ul ) z ′′′′ (ui , uj , uk , ul )dui duj duk dul
hZ
 Y
i
Y 
z ′′′′ (ui , uj , uk , ul )
= E
I(Xt > ut ) − I(Xt′ > ut )
dut .
R4

t=i,j,k,l

t=i,j,k,l

Choose a C1 > 0. Define A = [−C1 , C1 ]4 .
|E(h(2) (Xi , Xj )h(2) (Xk , Xl ))|= |E(z(Xi , Xj , Xk , Xl ) − z(Xi′ , Xj , Xk , Xl ))|
hZ
 Y
i
Y 
≤ E
z ′′′′ (ui , uj , uk , ul )
I(Xt > ut ) − I(Xt′ > ut )
dut
A

t=i,j,k,l

+ E

hZ

Ac

t=i,j,k,l

 Y
i
Y 
z ′′′′ (ui , uj , uk , ul )
I(Xt > ut ) − I(Xt′ > ut )
dut
t=i,j,k,l

t=i,j,k,l

= |I1 |+|I2 |.
I1 =
≤C

Z

z ′′′′ (ui , uj , uk , ul )E
A

≤

t=i,j,k,l

sup
u,v,x,y∈[−C1 ,C1 ]

≤ CC1b C12
CC1p

Z

h Y 
i Y
I(Xt > ut ) − I(Xt′ > ut )
dut

Z

R2

Z
|z ′′′′ (u, v, x, y)|

[−C1 ,C1 ]2

h X

t=j,k,l

h X

t=j,k,l

A

t=i,j,k,l

h Y 
i
E
I(Xt > ut ) − I(Xt′ > ut )
t=i,j,k,l

Y

t=i,j,k,l

P (Xi > ui , Xt > ut ) − P (Xi > ui )P (Xt > ut )dui dut

P (Xi > ui , Xt > ut ) − P (Xi > ui )P (Xt > ut )dui dut

dut

i

(3.8)

i

= CC1p[Cov(Xi , Xj ) + Cov(Xi , Xk ) + Cov(Xi , Xl )].

(3.9)

The inequality in (3.8) follows from Lebowitz’s inequality, Lemma 2.4. The equality in (3.9) follows
using the Newman’s inequality, Lemma 2.3. Hence,
|I1 |≤ CC1p [Cov(Xi , Xj ) + Cov(Xi , Xk ) + Cov(Xi , Xl )] = CC1p [

X

Cov(Xi , Xt1 )].

(3.10)

|I1 |≤ CC1p(Cov(Xj , Xi ) + Cov(Xj , Xk ) + Cov(Xj , Xl )) = CC1p[

X

Cov(Xj , Xt2 )].

(3.11)

|I1 |≤ CC1p(Cov(Xk , Xi ) + Cov(Xk , Xj ) + Cov(Xk , Xl )) = CC1p [

X

Cov(Xk , Xt3 )].

(3.12)

t1 =j,k,l

Similarly, it can be shown

t2 =i,k,l

t3 =i,j,l

6

Combining (3.10) - (3.12), we get the following.
|I1 |≤ CC1p T 1/3 ,

(3.13)

P
P
P
where T = [ t1 =j,k,l Cov(Xi , Xt1 )] × [ t2 =i,k,l Cov(Xj , Xt2 )] × [ t3 =i,j,l Cov(Xk , Xt3 )]. Next,
I2 =E

"Z

Ac

′′′′

z (ui , uj , uk , ul )

Y 

t=i,j,k,l

I(Xt > ut ) −

I(Xt′

#
 Y
> ut )
dut .
t=i,j,k,l

There are several combinations possible under Ac . We solve the integral for a possible combination
under Ac , say B = {|ui |> C1 , |uj |≤ C1 , |uk |≤ C1 , |ul |≤ C1 }. Calculations for other combinations
will follow similarly.
For a differentiable function g(.) with derivative g′ (.), and random variables X and X ′ , we
have
Z

|v|≤C1

Z

|v|>C1

Using this, we have
"Z



I(X > v) − I(X ′ > v) g′ (v)dv = g(f1 (X)) − g(f1 (X ′ ))



I(X > v) − I(X ′ > v) g′ (v)dv = g(f2 (X)) − g(f2 (X ′ )).

 Y
Y 
I(Xt > ut ) − I(Xt′ > ut )
z (ui , uj , uk , ul )
dut
′′′′

E

B

"

t=i,j,k,l

t=i,j,k,l

(3.14)

#



= E h(2) (f2 (Xi ), f1 (Xj )) − h(2) (f2 (Xi ), f1 (Xj′ )) − h(2) (f2 (Xi′ ), f1 (Xj )) + h(2) (f2 (Xi′ ), f1 (Xj′ )) ×

#


′
(2)
′
(2)
′
(2)
(2)
′
h (f1 (Xk ), f1 (Xl )) − h (f1 (Xk ), f1 (Xl )) − h (f1 (Xk ), f1 (Xl )) + h (f1 (Xk ), f1 (Xl )) .

From (iii), we get
"Z
E

B

 Y
Y 
z ′′′′ (ui , uj , uk , ul )
I(Xt > ut ) − I(Xt′ > ut )
dut
t=i,j,k,l

t=i,j,k,l

#

≤

C
.
C1δ

Similarly, solving for other combinations under Ac , we get
|I2 |≤

C
.
C1δ

(3.15)

From (3.13) and (3.15), for i, j, k, l all distinct, we have
|E(h(2) (Xi , Xj )h(2) (Xk , Xl ))|≤ CC1p T 1/3 +

C
.
C1δ

(3.16)

Choosing C1 = T −1/(3(p+δ)) , we get,
|E(h(2) (Xi , Xj )h(2) (Xk , Xl ))|≤ CT δ/(3(p+δ)) .

7

(3.17)

Next, consider a case when there are 3 distinct indices in i, j, k, l. Let j = k, then solving as in
(3.10) and (3.15)
|E(h(2) (Xi , Xj )h(2) (Xj , Xl ))|≤ CC1p[Cov(Xi , Xj ) + Cov(Xi , Xl )] +

C
C1δ

(3.18)

Choosing C1 = T −1/(p+δ) , we get,
|E(h(2) (Xi , Xj )h(2) (Xj , Xl ))|≤ C[Cov(Xi , Xj )δ/(δ+p) + Cov(Xi , Xl )δ/(δ+p) ]

(3.19)

Using (3.6), (3.17), (3.19), stationarity of Xj′ s and (iv), (3.5) follows.
Remark 3.3. The condition (iii) of Lemma 3.2 may seem cumbersome, but in general can be
shown to be true under restrictions on the moments of X1 , as seen in Theorem 4.1.
If the random variables Xn , n ≥ 1 are uniformly bounded (as is often seen in applications in
the reliability and survival analysis) then we can use the following lemma. The assumptions on
the covariance structure become less restrictive.
Lemma 3.4. Let the functions h(2) (., .), z(., ., ., .) and z ′′′′ (., ., ., .) be as defined in Lemma 3.1.
Assume the following holds.
(a) Xn , n ≥ 1 are uniformly bounded, i.e. P (|X1 |< C1 ) = 1, for some C1 > 0;
P∞
1/3 < ∞;
(b)
j=1 Cov(X1 , Xj )

(c) |h(2) (x, y)| is bounded for all x, y ∈ [−C1 , C1 ], where h(2) (x, y) is as defined in (2.2);

(d)

sup

||z ′′′′ (u, v, x, y)|| is bounded.

u,v,x,y∈[−C1 ,C1 ]

Then, as n → ∞,
X

X

|E(h(2) (Xi , Xj )h(2) (Xk , Xl ))|= o(n3 ).

(3.20)

1≤i<j≤n 1≤k<l≤n

Proof. The proof follows similarly as Lemma 3.2. The second term

C
,
C1δ

for some δ > 0 in the upper

bounds obtained in inequalities (3.16) and (3.18) would not be needed.
Lemma 3.5. Let Un be the U-statistic based on a symmetric kernel ρ(x, y), such that the corresponding kernel h(2) (., .) (defined in (2.2)) satisfies the conditions of Lemma 3.2 or Lemma 3.4.
P
Let 0 < σ12 = V ar(ρ1 (X1 )) < ∞. Define σ1j = Cov(ρ1 (X1 ), ρ1 (X1+j )). Assume ∞
j=1 |σ1j |< ∞.

Then,

∞

V ar(Un ) =

1
X
4σU2
σ1j .
, where σU2 = σ12 + 2
+o
n
n
j=1

8

(3.21)

Proof. The proof follows similarly as Lemma 3.2 of Garg and Dewan (2015). By Hoeffding’s
P
(1)
(2)
(1)
(2)
decomposition, V ar(Un ) = 4V ar(Hn ) + V ar(Hn ) + 4Cov(Hn , Hn ). As ∞
j=1 |σ1j |< ∞,
∞

V

ar(Hn(1) )

1
X
1
.
σ1j ) + o
= (σ12 + 2
n
n

(3.22)

j=1

(2)

(2)

Also V ar(Hn ) ≤ E(Hn )2 = o(1/n) (from Lemma 3.2 or Lemma 3.4). Using Cauchy-Schwarz
(1)

(2)

inequality we have, |Cov(Hn , Hn )|≤ o(1/n). Hence, we get (3.21).
The following gives the central limit theorem for a non-degenerate U-statistic based on a
stationary sequence of associated observations with a kernel, ρ, of degree 2.
Theorem 3.6. Suppose the conditions of Lemma 3.5 hold. Further, assume σU2 > 0, where σU2 is
defined by (3.21). If there exists a function ρ̃1 (·) such that ρ1 ≪ ρ̃1 and
∞
X
j=1

then

√

Cov(ρ̃1 (X1 ), ρ̃1 (Xj )) < ∞,

n(Un − θ) L
−
→ N (0, 1) as n → ∞.
2σU

Proof. Using Hoeffding’s decomposition for Un ,
√
√
n
(2)
X
h(1) (Xj )
n(Un − θ)
nHn
= n−1/2
+
.
2σU
σU
σU

(3.23)

(3.24)

(3.25)

j=1

(2)

In addition, nE(Hn )2 −
→ 0 as n → ∞, from Lemma 3.2 or Lemma 3.4. Hence,
√
(2)
nHn p
−
→ 0 as n → ∞.
σU

(3.26)

From Lemma 2.6, we get that,
n−1/2

n
X
h(1) (Xj )
j=1

σU

L

−
→ N (0, 1) as n → ∞.

(3.27)

Relations (3.25), (3.26) and (3.27) prove the theorem.
Remark 3.7. The above results can be easily extended to a U-statistic Un based on a symmetric
kernel ρ(x1 , . . . , xk ) which is of a finite degree k > 2. Let {Xi′ , i ≥ 1} be a sequence of random
variables as defined earlier. Assume the following.
(i) E(ρ(Xi1 , · · · , Xik ))2 < ∞, for all ij ∈ N, j = 1, · · · , k, such that i1 < · · · < ik .
(ii) For any 0 < C1 < ∞, z(x1 , · · · , xk , x′1 , · · · , x′k ) = ρ(x1 , . . . , xk )ρ(x′1 , . . . , x′k ) has a bounded
derivative

2k
Qk ∂ z
′
i=1 ∂xi ∂xi

for all xi , x′i ∈ [−C1 , C1 ], i = 1, 2, · · · , k, and the bound is CC1b , for some

b ∈ N ∪ {0};

9

(iii) For some δ > 0 and all jp ∈ N, p = 1, · · · , 2k, such that j1 < · · · < jk and jk+1 < · · · < j2k ,
X

i1 ,···,i2k =1,2
(i1 ,···,i2k )6=(1,···,1)

E z(fi1 (Zj1 ), fi2 (Zj2 ), · · · , fik (Zjk ), fik+1 (Zjk+1 ), · · · , fi2k (Zj2k )) ≤

C
,
C1δ

′ , m = j ,j ,···,j ;
where Zm = Xm or Xm
1 2
2k
P∞
δ/(3(p+δ)) < ∞ for p = b + 2k − 2, where b and δ are respectively defined in
(iv)
j=1 Cov(X1 , Xj )

(ii) and (iii) respectively;
P
2
(v) 0 < σ12 < ∞ and ∞
j=1 |σ1j |< ∞, with σU > 0; and

(vi) There exists a function ρ̃1 , such that ρ1 ≪ ρ̃1 and (3.23) holds.
Then,

√

n(Un − θ) L
−
→ N (0, 1) as n → ∞.
kσU

(3.28)

Remark 3.8. Using Lemma 2.7, we get σ12 ≤ V ar(ρ̃1 (X1 ))and |σ1j |≤ C Cov(ρ̃1 (X1 ), ρ̃1 (Xj )). If
ρ1 is monotonic, then {ρ1 (Xn ), n ≥ 1} is a sequence of stationary associated random variables and
one can take ρ̃1 ≡ ρ1 .
Remark 3.9. If ρ̃1 has a bounded derivative, then using Lemma 2.3, if
then (3.23) holds.

P∞

j=1 Cov(X1 , Xj )

< ∞,

Remark 3.10. Let ρ̃1 (x) = cx for some constant c > 0 for all x ∈ R. If ρ1 ≪ ρ̃1 then ρ1 is a
Lipschitz function. A sufficient condition for ρ1 to be a Lipschitz function is that it should have a
bounded derivative.
Remark 3.11. If ρ1 (x) is a function of bounded variation, then there exist two increasing functions
U1 (x) and U2 (x) such that ρ1 (x) = U1 (x) − U2 (x) for all x ∈ R. ρ1 ≪ ρ̃1 by taking ρ̃1 (x) =
U1 (x) + U2 (x).

4

Applications

We use the results derived in section 3 to get the limiting distribution of estimators of second,
third and fourth central moments, when the underlying sample is from Xn , n ≥ 1. As given earlier,
{Xn , n ≥ 1} is a sequence of stationary associated random variables, with F as the marginal
distribution function of X1 . We also discuss estimators of skewness and kurtosis based on the
estimators of moments.
Define, µ = E(X1 ), µ′k = E(X1k ) and µk = E((X1 − µ)k ), k ≥ 2. Assume that µk and µ′k exist
for all k = 2, · · · , 8, and C is a generic positive constant in the sequel.

10

4.1

Estimator of the variance/ second central moment

A measure of variability is the variance µ2 = E((X1 − µ)2 ).
Given the sample Xj , 1 ≤ j ≤ n from F , a U-statistic, µ̂2n , estimating µ2 is
µ̂2n =

X
2
1
(Xi − X̄n )2 =
(n − 1)
n(n − 1)
1≤i≤n

where X̄n =

Pn

i=1

n

Xi

and the kernel ρ(x, y) =

(x−y)2
,
2

X

ρ(Xi , Xj ),

(4.1)

1≤i<j≤n

with ρ1 (x) = ((x − µ)2 + µ2 )/2.

We next discuss the limiting distribution of µ̂2n , when the sample is from {Xn , n ≥ 1}.
Theorem 4.1. Let E|X1γ |< ∞, where γ = (4 + δ) for some δ > 0. Further, if
∞
X
j=1

then

δ

Cov(X1 , Xj ) 3(2+δ) < ∞,

√

where σµ̂2n

n(µ̂2n − µ2 ) L
−
→ N (0, 1) as n → ∞,
2σµ̂2n
P
2
= V ar(ρ1 (X1 )) + 2 ∞
j=2 Cov(ρ1 (X1 ), ρ1 (Xj )) and V ar(ρ1 (X1 )) = µ4 − (µ2 ) .

(4.2)

(4.3)

If Xn , n ≥ 1 are uniformly bounded, then (4.3) holds under
∞
X
j=1

Cov(X1 , Xj )1/3 < ∞.

(4.4)

Proof. Assume Xn , n ≥ 1 are uniformly bounded, i.e. P (|X1 |< C1 ) = 1, for some C1 > 0. As

z(x, y, x′ , y ′ ) = h(2) (x, y)h(2) (x′ , y ′ ) = (x − µ)(y − µ)(x′ − µ)(y ′ − µ) and z ′′′′ (x, y, x′ , y ′ ) = 1, the
conditions of Lemma 3.4 are true under (4.4). As ρ1 (x) ≪ Cx, for all x ∈ [−C1 , C1 ], conditions of
Theorem 3.6 are satisfied, and we get (4.3).
The above can be easily extended to random variables which are not uniformly bounded,
under the conditions of Lemma 3.2. Without loss of generality assume µ = 0. The conditions
(i) − (iv) of Lemma 3.2 are satisfied under the given assumptions, as discussed in the following.

(i) Assuming E|X1 |4+δ < ∞, (i) of Lemma 3.2 is true. As h(2) (x, y) = xy,

E(h(2) (X1 , X1+k ))2 = E(X1 X1+k )2 ≤ CE(X1 )4 ≤ C(E|X1 |4+δ )4/(4+δ) < ∞, for all k ∈ N.
(ii) b = 0, as z ′′′′ (x, y, x′ , y ′ ) = 1, for all x, y, x′ , y ′ ∈ R.
(iii) Assuming E|X1 |4+δ < ∞, (iii) of Lemma 3.2 is true. Consider




E f2 (Xi )f2 (Xj )f1 (Xk )f1 (Xl ) ≤ C12 E |Xi ||Xj |I(|Xi |> C1 )I(|Xj |> C1 )

C
|Xi |1+δ/2 |Xj |1+δ/2 
≤ δ.
≤ C12 E |Xi ||Xj | 1+δ/2
1+δ/2
C1
C1
C1
Similarly, the bounds on the other terms can be obtained.

11

(iv) From (ii), p = 2 as b = 0. The restriction put on the covariance structure is therefore
δ
P∞
3(2+δ) < ∞.
j=1 Cov(X1 , Xj )
Define

ρ̃1 (x) =

x2 I(x ≥ 0) − x2 I(x < 0)
,
2

where I(.) denotes the indicator function. Then ρ1 (x) ≪ ρ̃1 (x). It can be shown that for any
0 < C3 < ∞,
Cov(ρ̃1 (Xi ), ρ̃1 (Xj )) ≤

C
+ CC32 Cov(Xi , Xj ).
C3δ

(4.5)

Choosing, C3 = Cov(Xi , Xj )−1/(2+δ) in (4.5),
Cov(ρ̃1 (Xi ), ρ̃1 (Xj )) ≤ CCov(Xi , Xj )δ/(2+δ) .
Using

P∞

j=2 Cov(X1 , Xj )

δ
2+δ

< ∞ (which follows from (4.2)) and Theorem 3.6, we get (4.3).

Remark 4.2. The limiting distribution of the variance estimator, µ̂2,n , when the underlying sample
is from Xn , n ≥ 1, can also be obtained under different set of assumptions.
(i) To apply the techniques of Beutner and Zähle (2012, 2014) to obtain a central limit theorem
for µ̂2,n , a result on weak convergence of weighted empirical distribution process of associated
sequences is needed. This result can be found in Shao and Yu (1996). Our results are valid for
non-degenerate U-statistics only, while the technique of Beutner and Zähle (2014) can also be
used when the U-statistics considered are degenerate.
The following is given in Example 3.8 of Beutner and Zähle (2012).
Let {Xi , i ≥ 1} be a stationary, associated sequence with the marginal distribution function F
√
and Cov(X1 , Xn ) = O(n−ν−ǫ ) for some ν ≥ (3+ 33)/2 ≈ 4.372281 and ǫ > 0. Then, whenever
F has a finite γ-moment for some γ >

4ν
ν−3 ,

a central limit theorem for µ̂2,n holds.

(ii) The kernel ρ(x1 , x2 ) = (x1 − x2 )2 /2 is a function of bounded Hardy-Krause variation on any
bounded rectangle [−C1 , C1 ]2 where 0 < C1 < ∞. This can be shown as following. We can write

(x1 −x2 )2 /2 as a difference of two component-wise nondecreasing bounded functions g1 (x1 , x2 ) =

((x1 + C1 )2 + (x2 + C1 )2 )/2 and g2 (x1 , x2 ) = (x1 + C1 )(x2 + C1 ), where for x1 ≤ y1 and x2 ≤ y2 ,

gi (x1 , x2 ) − gi (x1 , y2 ) − gi (y1 , x2 ) + gi (y1 , y2 ) ≥ 0, for i = 1, 2. Using the results of Garg and
P
ν
Dewan (2015), a central limit theorem for the variance estimator holds if ∞
j=1 Cov(X1 , Xj ) <

∞, for some 0 < ν < 1/6, when {Xn , n ∈ N} is uniformly bounded. The covariance structure
assumed in Theorem 4.1 is less restrictive. Also, the condition of differentiability is relatively
easier to verify.

12

Remark 4.3. The results of Dewan and Prakasa Rao (2002) and its corrigendum Dewan and
Prakasa Rao (2015) cannot be used to obtain the central limit theorem for the variance estimator.
They only consider the case when the differentiable kernels are component-wise monotonic. Moreover, their results are based on a very restrictive set of conditions. They assume that there exists
P
a non-negative function r(k) satisfying ∞
k=0 r(k) < ∞, such that for all i, j, k, l,
|Cov(h(2) (Xi , Xj ), h(2) (Xk , Xl ))|≤ r(max(|i − k|, |j − l|)),

They do not discuss examples of functions r(k) that satisfy the above inequality. The cases when
kernels have a degree greater than 2 are also not considered.

4.2

Estimator for the third central moment

Let the sample Xj , 1 ≤ j ≤ n be from F . A U-statistic, µ̂3n , based on Xj , 1 ≤ j ≤ n estimating
the third central moment µ3 is
µ̂3n =

1
n
3

u(x1 , x2 , x3 ) =

X

3
X
x3
i

i=1

u(Xi , Xj , Xk ), where

1≤i<j<k≤n

3

−

X

(i,j,k)∈π(1,2,3)

x2i (xj + xk )
+ 2[x1 x2 x3 ],
2

and π(1, 2, 3) = ((1, 2, 3), (2, 1, 3), (3, 1, 2)). Here, u1 (x) = (2µ3 + (x − µ)3 )/3 − µ2 (x − µ).
Theorem 4.4. Let E|X1γ |< ∞, where γ = (6 + δ) for some δ > 0. Further, if
δ
P∞
3(4+δ) < ∞, then
j=1 Cov(X1 , Xj )
√
n(µ̂3n − µ3 ) L
−
→ N (0, 1) as n → ∞,
(4.6)
3σµ̂3
P
2
where σµ̂2 3 = V ar(u1 (X1 )) + 2 ∞
j=2 Cov(u1 (X1 ), u1 (Xj )) and V ar(u1 (X1 )) = µ6 − µ3 .
P
1/3 < ∞.
If Xn , n ≥ 1 are uniformly bounded, then (4.6) is true under ∞
j=1 Cov(X1 , Xj )

Proof. Proof follows similarly as the proof of Theorem 4.1. The conditions are obtained by putting
k = 3, b = 0 and p = 4 in Remark 3.7.

4.3

Estimator for the fourth central moment

Let the sample Xj , 1 ≤ j ≤ n be from F . A U-statistic, µ̂4n , based on this sample estimating the
fourth central moment µ4 is
µ̂4n =

1
n
4

X

v(Xi , Xj , Xk , Xl ), where

1≤i<j<k<l≤n

13

v(x1 , x2 , x3 , x4 ) =

4
X
x4
i

4

i=1

−

+

X

(i,j,k,l)∈π(1,2,3,4)

X

(i,j,k,l)∈π(1,2,3,4)

x2i (xj xk + xk xl + xj xl )
2

x3i (xj + xk + xl )
− 3[x1 x2 x3 x4 ],
3

and π(1, 2, 3, 4) = ((1, 2, 3, 4), (2, 1, 3, 4), (3, 1, 2, 4), (4, 1, 2, 3)).
Here, v1 (x) = (3µ4 + (x − µ)4 )/4 − µ3 (x − µ).
Theorem 4.5. Let E|X1γ |< ∞, where γ = (8 + δ) for some δ > 0. Further, if
δ
P∞
3(6+δ) < ∞, then
j=1 Cov(X1 , Xj )
√
n(µ̂4n − µ4 ) L
−
→ N (0, 1) as n → ∞,
(4.7)
4σµ̂4
P
2
where σµ̂2 4 = V ar(v1 (X1 )) + 2 ∞
j=2 Cov(v1 (X1 ), v1 (Xj )) and V ar(v1 (X1 )) = µ8 − (µ4 ) .
P
1/3 < ∞.
If Xn , n ≥ 1 are uniformly bounded, then (4.7) is true under ∞
j=1 Cov(X1 , Xj )

Proof. Proof follows similarly as the proof of Theorem 4.1. The conditions are obtained by putting
k = 4, b = 0 and p = 6 in Remark 3.7.
Remark 4.6. The limiting distributions of U-statistic estimators of higher order central moments can be obtained similarly. In general, the r th central moment, r ∈ N/{1}, is estimated
by

Pn

r
j=1 (Xj −X̄n )

n

. This and the U-statistic estimator for the r th central moment will have the same

limiting distribution.

4.4

Estimation of skewness and kurtosis.

The coefficient of skewness and kurtosis for the distribution function F are defined as
τ=

µ3
(µ2 )3/2

and κ =

µ4
(µ2 )2 ,

respectively. We can use
τ̂n =

µ̂3n
µ̂4n
and κ̂n =
3/2
(µ̂2n )2
(µ̂2n )

(4.8)

as estimators of τ and κ, respectively.
Assume that {Xn , n ∈ N} is a sequence of stationary associated random variables, with F as
the marginal distribution function of X1 . Then, under the conditions of Theorems 4.1, 4.4 and 4.5,
p

p

it can be shown that τ̂n −
→ τ and κ̂n −
→ κ as n → ∞.

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