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

sup sup

max

R

s,

k

<

+

,

(2.121)

#

"

∞

n>1 s∈R ^E

16k6n

n

n

2

sup sup

max

R^0

s,

k

<

+

,

(2.122)

where Wn, Wn^0, Rn and Rn^0 are defined respectively by (2.5), (2.75), (2.4) and (2.76).

Let us show (2.119). Observe that

16k6n ^n

n

6

n^1^/^2 16k6n

k

1,s ^(

i) ^!

2

n1/2 16k6n

n

2,s

j

2

+ 2

k

2

1

1

i=1

j=k+1

max W

s,

2

max

X

X

max

^X

h

(X )

h

(2.123)

hence using Theorem A.3 two times gives (2.119).

Let us show (2.120). In view of (2.75), the equality

W ^0 s, t

W s, t [nt] (n − [nt]) 1

n−1

n i h X

n

j

X

X

n ( ) =

n ( ) −

n3/2

n

i=1

( − ) 1,s ( i) +

j=2

( −1)

2

holds hence it suffices to show that

sup sup

1

E

n−1

(n

i) h1,s (Xi) +

n

(j

1) h2,s (Xj)

2 < +

.

3

−

X

−

∞

n>1 s

R ^n

∈

X

i=1

j=2

h_2,s (X_j)
(2.124)

(2.125)

21
This follows from a rewriting of the sums in terms of partial sums of h_1,s (X_i) and h_2,s (X_i) and an application of Theorem A.3.

Let us show (2.121). Letting hi,j := h3,s (Xi, Xj), we get in view of (2.61) that

E

max Rn

s,

k

2

= E

2

max

^hi,`

2 + E

2

max

k n

^h`,j

2 .

3

3

"

16k6n

n

#

n

16k6n

i<`

k

n

16k6n

1

`=1 j=`+1

6^X6

X X

(2.

Boundedness follows from Lemma A.5.

126)

Let us show (2.122). Noticing that

R^0

(

s, t

) =

R

s, t

)

1

[nt] (n − [nt])

h

X , X

,

(2.127)

(

n

n

^− n3/2

n

3,s ^(

i

j^)

2

i

n

it suffices to show, in view of (2.121), that

16^X6

sup sup

1

E

^h3,s ^(^Xi

, Xj)

2 < +

∞

.

(2.128)

3

n>1 s

∈

R ^n

16^X6

i
This can be seen by an other use of Lemma A.5.