# 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 , 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 , 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 , the convergence of the 1-sample U-statistic defined by

1 X

(1.2)

n3/2
16i6=j6n
has been investigated.

• In , 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 .

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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) (t0 − t) 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)

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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}.

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