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


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(1 {g (Xi, Xj) 6 s} − P{g (Xi, Xj) 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 kjR 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 (Xi)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(AiBj) − 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 kpα (k) converges.



  • HEROLD DEHLING, DAVIDE GIRAUDO AND OLIMJON SHARIPOV

where the supremum runs over all the partitions (Ai)Ii=1 and (Bj)Jj=1 of Ω of elements of A and B respectively.



Given a sequence (Xi)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 Fuv, 1 6 u 6 v 6 +∞ is the σ-algebra generated by the random variables Xi, 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 (Xi)iZ be a strictly stationary sequence. Let en 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 f1,u +∞.

(A.2) For all v R, the random variable g (X1, v) has a density f2,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 supx,u∈R f1,u (x) <
and supx,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 (s0, t0)) = t (1 − t) (1 − t0) C1,1 (s, s0)




+ t (1 − t0) (t0t) C2,1 (s, s0) + tt0 (1 − t0) C2,2 (s, s0) ,

(1.10)

where for i, j = 1; 2 and s, s0 R,




X

(1.11)

Ci,j (s, s0) = 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 (s0, t0)) = t (1 − t0) (1 + 2t0 − 2t) C1,1 (s, s0) .

(1.14)

3
In practical cases, the probability P{g (Xi, Xj) 6 s} is unknown, and we only have the values of 1 {g (Xi, Xj) 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 (Xi, Xj) 6 s} is replaced by its empirical estimator



n −1 P16i6n 1 {g (Xi, Xj) 6 s}.

2


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