Convergence of the empirical two-sample -statistics with -mixing data

(1 {g (X_i, X_j) 6 s} − P{g (X_i, X_j) 6 s})
CONVERGENCE OF THE EMPIRICAL TWO-SAMPLE -STATISTICS
WITH -MIXING DATA
HEROLD DEHLING, DAVIDE GIRAUDO AND OLIMJON SHARIPOV
Abstract. We consider the empirical two-sample -statistic with strictly -mixing strictly stationary data and inverstigate its convergence in Skorohod spaces. We then provide an application of such convergence.
1. INTRODUCTION AND MAIN RESULTS
Let a strictly stationary mixing sequence of real valued random variables. In this paper, we will study the asymptotic behavior of the empirical -statistic defined by

_(1,1)

where kj→ R is a measurable function and for a real number x, [x] denotes the unique

integer k such that k 6 x < k + 1. The following is known:
• In [2], the convergence of the 1-sample U-statistic defined by
1

n3/2 X 16i6=j6n (1 {g (Xi, Xj) 6 s} - P {g (Xi, Xj) 6 s})

(1.2)

has been investigated.
• In [4], the convergence of the 2-sample U-statistic of kernel h defined by
Tn (t) := 1
n3/2
[nt]
Xi
=1
nX
j=[nt]+1
h (Xi, Xj) (1.3)
has been investigated.
In this paper, we focus on the case of mixing sequences.
Let (Ω, F, P) be a probability space. The α-mixing and β-mixing coefficients between two
sub-σ-algebras A and B of F are defined defined respectively by

α (A, B) = sup {|P (A ∩ B) - P (A) P (B)| , A ∈ A, B ∈ B} ; β (A, B) = 1 2 sup  IXi =1 JXj =1 |P (Ai ∩ Bj) - P (Ai) P (Bj)|  ,	(1.4) (1.5)
	

CONVERGENCE OF THE EMPIRICAL TWO-SAMPLE STATISTICS WITH β-MIXING DATA
HEROLD DEHLING, DAVIDE GIRAUDO AND OLIMJON SHARIPOV
Abstract. We consider the empirical two-sample U-statistic with strictly β-mixing strictly stationary data and inverstigate its convergence in Skorohod spaces. We then provide an application of such convergence.

1. 
   Introduction and main results
   

Let (X_i)_i>1 a strictly stationary mixing sequence of real valued random variables. In this paper, we will study the asymptotic behavior of the empirical U-statistic defined by

	1 [nt]		n
en (s, t) :=			(1 {g (Xi, Xj) 6 s} − P{g (Xi, Xj) 6 s}) , n > 1, 0 6 t 6 1, s ∈ R,
	n3/2	i=1
			nt
		X	j=[^X]+1

(1.1)
where g : R² → R is a measurable function and for a real number x, [x] denotes the unique integer k such that k 6 x < k + 1. The following is known:

• 
  In [2], the convergence of the 1-sample U-statistic defined by
  

1 X

(1.2)


_n3/2
16i6=j6n
has been investigated.

• 
  In [4], the convergence of the 2-sample U-statistic of kernel h defined by
  

Tn (t) :=			1 [nt]		n	h (Xi, Xj)	(1.3)
		n	3/2	i=1
					nt
				X	j=[^X]+1

has been investigated.
In this paper, we focus on the case of mixing sequences.
Let (Ω, F, P) be a probability space. The α-mixing and β-mixing coefficients between two sub-σ-algebras A and B of F are defined defined respectively by

α (A, B) = sup {|P(A ∩ B) − P(A) P(B)| , A ∈ A, B ∈ B} ;						(1.4)
	1		I J	|P(Ai ∩ Bj) − P(Ai) P(Bj)| ,
β(A,B) =			i=1 j=1		(1.5)
	2 sup
			X X

Date: April 3, 2020.
Key words and phrases. U-statistics, empirical process .

1

_k>1 k^pα (k) converges.

• 
  HEROLD DEHLING, DAVIDE GIRAUDO AND OLIMJON SHARIPOV
  

where the supremum runs over all the partitions (A_i)^I_i=1 and (B_j)^J_j=1 of Ω of elements of A and B respectively.

Given a sequence (X_i)_i>1, we associate its sequences of α and β-mixing coefficients by letting

α	(	k	) :=	sup α			^`,	∞	,	(1.6)
				`>1			F1	^F`+n
β	(	k	) :=	sup β			`,	∞	,	(1.7)
				`>1		F1		^F`+n

where F_u^v, 1 6 u 6 v 6 +∞ is the σ-algebra generated by the random variables X_i, u 6 i 6 v (u 6 i for v = +∞).
1.1. Convergence of the two-sample U-statistic in Skorohod spaces D ([−R, R] × [0, 1]).
Let us state one of the two main results of the paper.
Theorem 1.1. Let (X_i)_i∈Z be a strictly stationary sequence. Let e_n be the two-sample U-statistics empirical process with kernel g : R × R → R defined for n > 1, 0 6 t 6 1 and s ∈ R by

	1 [nt]		n
en (s, t) :=			(1 {g (Xi, Xj) 6 s} − P{g (Xi, Xj) 6 s}) .	(1.8)
	n3/2	i=1
			nt
		X	j=[^X]+1

Suppose that the following four conditions holds.
(A.1) For all u ∈ R, the random variable g (u, X₁) has a density f_1,u +∞.

(A.2) For all v ∈ R, the random variable g (X₁, v) has a density f_2,v
+∞.

P

(A.3) There exists a p > 2 such that

P

(A.4) The series _k>1 kβ (k) converges.
Then for all R,
^{and sup}x,u∈R ^f1,u ^{(x) <
and sup}x,v∈R ^f2,v ^{(x) <}

en (s, t) → W (s, t) in distribution in D ([−R, R] × [0, 1]) ,

(1.9)

where (W (s, t) , s ∈ R, t ∈ [0, 1]) is a centered Gaussian process, with covariance given for 0 6 t 6 t⁰ 6 1 and s, s⁰ ∈ R by the following formula:

Cov (W (s, t) , W (s^0, t^0)) = t (1 − t) (1 − t^0) C1,1 (s, s^0)
+ t (1 − t^0) (t^0 − t) C2,1 (s, s^0) + tt^0 (1 − t^0) C2,2 (s, s^0) ,	(1.10)
where for i, j = 1; 2 and s, s^0 ∈ R,
X	(1.11)
Ci,j (s, s^0) = E[hi,s (X0) hj,s0 (Xk)] ,
k∈Z
h1,s (u) = P{g (u, X1) 6 s} − P{g (X1, X2) 6 s} ,	(1.12)
h2,s (v) = P{g (X1, v) 6 s} − P{g (X1, X2) 6 s} .	(1.13)

Remark 1.2. We did not make a symmetry assumption on g. When g is symmetric, in the sense that g (u, v) = g (v, u) for all u and v ∈ R, the covariance of the limiting process W reads

Cov (W (s, t) , W (s^0, t^0)) = t (1 − t^0) (1 + 2t^0 − 2t) C1,1 (s, s^0) .

(1.14)

3
In practical cases, the probability P{g (X_i, X_j) 6 s} is unknown, and we only have the values of 1 {g (X_i, X_j) 6 s}, 1 6 i < j 6 n, at our disposal. This leads to an analoguous result as Theorem 1.1, where the quantity P{g (X_i, X_j) 6 s} is replaced by its empirical estimator

^{n −1 P}_16i6n 1 {g (X_i, X_j) 6 s}.

2