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

n

3/2

i=[nt ]+1

3/2

n

i=[nt]+1

X0

X

Taking into account that t^0

−

t

>

1/n, it follows that

[nt^0] − [nt] 6 nt^0 − nt + 1 6 2n (t^0 − t)

(2.41)

hence

s) + 1/2

s)

n ^(

)

n ^(

) 6 2

1/2

(

s ^(

i^)

( s

(

i^)

ξ

s, t

ξ

s, t^0

t^0

− t

[nt]

a

X

m

1

[nt^0]

a

X

m

|

−

|

n

i=1

−

n

i=[nt]+1

−

^X

X

n

[nt^0]

+ 2

t0

− t

b

X

i)

m

+

1

b

s (

X

m

.

(2.42)

n

1/2

i=[nt ]+1

(

s

(

−

s)

n

1/2

(

i)

−

s)

i=[nt]+1

^X0

X

By stationarity,

E

E n (

)

n ^(

)

6 p

1/2

( s ( i)

s^)