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

Z

R ^=

06t61

^Z[−R,R]

W ^0 (s, t)^2 dµ (s)).

(2.110)

sup

Assumption 1 follows from the continuous mapping theorem applied with the functional

ΦR

(

) =

06t61

^Z[−R,R]

(2.111)

f

sup

f (s, t)^2 dµ (s) .

In order to show that assumptions 2 and 3 hold, it suffices to show that

R→+∞ n>1

^ZR\[−R,R] E

06t61

n

(s, t)^2

( )=0

(2.112)

lim

sup

sup

e

dµ

s

and

R→+∞ n>1

^ZR\[−R,R] E

06t61

(

)

(

) = 0

n

s, t

dµ

s

.

(2.113)

lim

sup

sup

e^0