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

In order to get rid of terms of the form bnt_kc and reduce the dependence in n, we will define for i ∈ I_n,u the random variable

d

X

^Yn,i ^=

ak,` (1 − tk) h1,s` (Xi) 1 {u 6 k}

k,`=1

2

d

d

X

X

−

(n − i)

ak,`tk (1 − tk) h1,s` (Xi) +

ak,`tkh2,s` (Xi) 1 {u > k}

n

k,`=1

k,`=1

2

d

−

i

X

ak,`tk (1 − tk) h2,s` (Xi) . (2.80)

n

k,`=1

Since |An,i − Yn,i| 6 1/n, it suffices to show that n^−1^/^2

n

d

i=1 Yn,i =: n^−1^/^2Sn →

k,`=1 ^ak,`^W ^0 ^(^s`^, tk^)

in distribution. Here again, will use Theorem A.1.

The first condition can be easily checked

P

P

by bounding |h_q,s` (X_i)| by 2 and the terms i/n by 1. For the third condition, we also use a_k = α (k), since each random variable Y_ni is a function of X_i. It remains to compute the limit of the sequence n⁻¹ Var (S_n) _n>1.
By similar argument as those who gave (2.13), this reduces to compute for all 1 6 u 6 d + 1 the limit

2

lim

1

∈^X

(2.81)

E

^Yn,i

.

n→+∞ n

i In,u

For convenience, we write

	d
Zi :=	ak,` (1 − tk) h1,s` (Xi) 1 {u 6 k}
	k,`=1
	X	d	d
		X	X
	+	ak,`tkh2,s` (Xi) 1 {u > k} − 2	ak,`tk (1 − tk) h1,s` (Xi) ; (2.82)
		k,`=1	k,`=1

16

HEROLD DEHLING, DAVIDE GIRAUDO AND OLIMJON SHARIPOV

2

d

X

Zn,i^0

=

i

ak,`tk (1 − tk) (h1,s` (Xi) − h2,s` (Xi)) ,

(2.83)

n

k,`=1

d

Zi^00

:= 2

X

ak,`tk (1 − tk) (h1,s` (Xi) − h2,s` (Xi)) .

(2.84)

k,`=1

Then Yn,i = Zi + Zi^0 and consequently

Zn,i^0

+ E

Zn,i^0

E

^Yn,i

2 = E

Zi

2

+ 2E

Zi

2

X

X

X

X

X

^i∈^In,u

^i∈^In,u

^i∈^In,u

i^0∈In,u

^i∈^In,u

(2.

85)

For the first term, we get, by stationarity, that

2

lim

1

X

X

= ( u

0 i^)

(2.86)

n +

n E

i

− u−1) Cov (

→ ∞

i∈Z

i∈In,u

The second and third term cannot be treated similarly because of the terms i/n. Neverthe-less, we can use the following lemma.