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

Then the sequence (Y_n)_n>1 converges in distribution to Z.

In order to prove (1.28) (respectively (1.29)), we apply Proposition 2.8 to

n

=

06t61

^Z[−R,R]

n

(

)

(

)

n ^=

06t61

ZR

n ^(

)

^2 dµ

(

)

(2.107)

Y (R)

sup

e

s, t ^2 dµ

s

, Y

sup

e

s, t

s

Z

R ^=

06t61

^Z[−R,R]

(

)

(

s

)

(2.108)

sup

W

s, t ^2 dµ

(respectively

^Z[−R,R]

ZR

n

06t61

n

(s, t)^2 dµ (s)

n =

06t61

n

(

)

^2 dµ

(

)

(2.109)

Y(R) =

sup

e^0

, Y

sup

e^0

s, t

s

,