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

Proposition 2.2. For all R > 0, the sequence (W_n (s, t) , s ∈ [ −R, R], t ∈ [0, 1])_n>1 converges in distribution in D ([−R, R] × [0, 1]) to (W (s, t) , s ∈ [−R, R], t ∈ [0, 1]).
In order to prove Proposition 2.2, we will use the following convergence criterion in D ([0, 1] × [0, 1]), which is Corollary 1 in [3].
Theorem 2.3. Let ξ_n, n > 1 be stochastic processes defined on [0, 1]², taking values in R, and whose paths are in the space D [0, 1]² almost surely. We make the following assumptions:

1. 
   the finite-dimensional distributions of ξ_n converges to the corresponding ones of a pro-cess ξ having continuous paths;
   
2. 
   the process ξ_n can be written as the difference to two coordinate-wise non decreasing processes ξ_n^◦ and ξ_n^∗.
   

3. 
   the exists constants γ > β > 2, c ∈ (0, ∞) such that for all n > 1, E[|ξ_n (0, 0)|^γ] 6 c and
   

E |ξ_n (s, t) − ξ_n (s⁰, t⁰)|^γ 6 c k(s, t) − (s⁰, t⁰)k^β_∞ whenever k(s, t) − (s⁰, t⁰)k_∞ > n⁻¹. (2.28)

(4) the following convergence in probability holds:

16j1,j26n

n

n n

−

n

n

n

+

n

n n

−

n

n

n

→

0, (2.29)

max

ξ^∗

^j1 , ^j2

ξ^∗

j1

− 1 ,

j2

ξ^∗

^j1 , ^j2

ξ^∗

^j1 ,

j2 − 1

2

Then (ξn)n

>

converges weakly to

ξ in

1

D

[0, 1]

.

In order to prove Proposition 2.1, we will use Theorem 2.3 in the following setting:

ξn (s, t) := Wn (−R + 2Rs, t) , s, t ∈ [0, 1].

(2.30)