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

The convergence of the finite dimensional distributions to those of (W (−R + 2Rs, t) , s, t ∈ [0, 1]) is guaranted by Proposition 2.1. The covariance function of this process is Lipschitz continuous hence this process has continuous paths.

We will now express ξ_n as the difference of two coordinatewise non-decreasing process. To
ease the notations, we will write a_s (u) := P{g (u, X₁) 6 −R + 2Rs}, b_s (u) := P{g (X₁, v) 6 −R + 2Rs} and m_s := P{g (X₁, X₂) 6 −R + 2Rs}. These three functions are non-negative and non-decreasing. The following equalities take place:

ξ

n (

s, t

n − [nt]

[nt]

a

X

i) −

m

s) +

[nt]

n

b

s (

X

j)

−

m

s)

n3/2

) =

n3/2

i=1

(

s (

nt

(

[nt]

X

j=[^X]+1

=

n − [nt]

a

X

i) +

[nt]

n

b

X

−

m

n − [nt]

[

nt

] −

m

[nt]

(

n

− [

nt

])

,

(2.31)

n3/2

^s n3/2

n3/2

i=1

s ^(

nt

s

( j)

^s n3/2

X

j=[^X]+1