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

Proof of Theorem 1.3.
2.2.1. Hoeffding’s decomposition. Using the functions h_1,s, h_2,s and h_3,s defined by (2.1) and (2.2), we derive that

en^0 (s, t) = Wn^0

(s, t) + Rn^0

(s, t) ,

(2.74)

where

1

[nt]

1

n

Wn^0 (s, t) =

(n − [nt])

h1,s (Xi) +

[nt]

h2,s (Xj)

n3/2

n3/2

nt

i=1

−

X

(

− )

j=[^X]+1

(

− 1)

2,s ^(

j)

(2.75)

n3/2

n

1,s

(

i) +

[nt] (n

−

[nt]) 1

n−1

n

i

h

X

n

j

h

X

and

^X

X

2

i=1

j=2

^Rn^0 ^(^s, t^) := n3/2

[nt]

h3,s (Xi, Xj) −

n

h3,s (Xi, Xj) .

(2.76)

1

n

1

^X j=[^X]+1

6^X6

1

n

i=1

nt

2

i

2.2.2. Convergence of the finite dimensional distributions. We treat the convergence of the finite dimensional distributions.