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

n Cov

^Yn,i^,

Yn,i

[ntu0−1]−1 +

1

K

∞

^i In,u

i0

^In,u

0

k=0

j=k+1

∈^X

∈^X

X

X

which goes to zero in

view of assumption (A.3). Therefore,

2 .

lim

1

E

n

^Yn,i

2

= d+1

lim

1

E

^Yn,i

(2.19)

n→+∞ n

!

n→+∞ n

∈^X

X

X

i=1

u=1

^i In,u

By stationarity,

E

^Yn,i

2

= E

^[^ntu^]−^[^ntu−1^] Yn,i

2

(2.20)

X

∈^X

i

^In,u

i=1

and defining for 1 6 u 6 d + 1 the random variable

d

X

ak,` ((1 − tk)