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

In view of the decompositions (2.3) and (2.74) and due to the fact that the dependence in t lies in [nt], which is an integer between 0 and n, it suffices to show that

"

#

∞

n>1 s∈R ^E

16k6n

n

n

2

sup sup

max

W

s,

k

<

+

,

(2.119)

#

"

∞

n>1 s∈R ^E

16k6n

n

n

2

sup sup

max

W ^0

s,

k

<

+

,

(2.120)

#

"

∞

n>1 s∈R ^E

16k6n

n

n

2