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

holds and these sequenes are bounded in absolute value by |c_k| hence (2.87) follows by domi-nated convergence.

We use the same strategy to show (2.88). We start from

bbnc i j

Nn−1

bbnc

i

bbnc

j

^Xbc

^cj−i ^=

X

b^Xc

b^Xc

1

{j − i = k} .

(2.93)

ck

n

n

n

n

i,j= an +1

k=−Nn+1

i= an +1

j= an +1

and
1

n

P

bbnc

j

j=banc+1

1

{j − i = k} is (i + k) /n if banc+ 1

6 i + k 6 bbnc and zero otherwise hence

n

bbnc

i j

1

bbnc−k i i + k

^Xbc

X

b^Xc

+1 n n

i,j= an +1

n n ^cj−i ^= k∈Z n ^1 {^0 ^6 ^k ^6 ^Nn − ^1} ^ck

i= an

1

bbnc

i i + k

X

b

Xc

−

. (2.94)

^+ n k∈Z

^1 ^{−^Nn ^+ 1 ^6 ^k ^6 ^−1} ^ck i= an +1

k

n n