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

Proposition 2.1. For all d > 1 and all s₁ < · · · < s_d and 0 6 t₁ < · · · < t_d 6 1, the vector (W_n (s_{`</sub>, t<sub>k</sub>))<sup>d</sup><sub>k,`=1} converges in distribution to (W (s_{`</sub>, t<sub>k</sub>))<sup>d</sup><sub>k,`=1}, where W_n is defined by (2.5) and W is like in Theorem 1.1.
Proof. We will use the Cramer-Wold device. Let (a_{k,`</sub>)<sup>d</sup><sub>k,`=1} be a family of real numbers. We have to prove that

d	d
X	X	(2.7)
ak,`Wn (s`, tk) →	ak,`W (s`, tk) in distribution.
k,`=1	k,`=1

^Pd

To this aim, we will express _{k,`=1</sub> a<sub>k,`}W_n (s_`, t_k) as a sum of linear combinations of indepen-dent random variables, and then apply a central limit theorem. Let

In,u = {i ∈ N | [ntu−1] + 1 6 i 6 [ntu]} , 2 6 u 6 d,

(2.8)

_n3/2

[nt_k

• 
  HEROLD DEHLING, DAVIDE GIRAUDO AND OLIMJON SHARIPOV
  

In,1 = {i ∈ N | 1 6 i 6 [nt1]} and In,d+1

= {i ∈ N | [ntd] + 1 6 i 6 n}.

Then the following

equality holds:

d

ak,`Wn (s`, tk)

k,`=1

X

d+1

[ntk] ^d^+1

d

a

n − [ntk]

1 {

u

k

h

X

1 {

u

k

h

X

,

=

6

}

1,s`

(

i) +

n3/2

>

+ 1}

2,s` ^(

i^)

k,`

n3/2

^i In,u

k,`=1

u=1

^i In,u

u=1

X

X

∈^X

X

∈^X

or in other words,

d

X

ak,`Wn (s`, tk)

k,`=1

d+1

d

k,`

n3/2

1 {

6

}

1,s` ^(

i) +

^= u=1 i In,u k,`=1

a

u

k

h

X

∈X X

n − [ntk]

X

(2.9)
^]