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

Lemma 2.7. Let (c_j)_j∈Z be an absolutely summable sequence of real numbers such that c_j = c_−j for all j and let 0 6 a < b 6 1. Then

lim

1

bbnc

i

c

=

b^2 − a^2

c ;

(2.87)

n→+∞ n

^Xbc

n

j−i

2

X

j

i,j= an +1

j∈Z

bbnc

lim ^{1 X}


n→+∞ n
i,j=banc+1

i j c				= b^3 − a^3		c .	(2.88)
			j−i			X
n		n			3	j
						j∈Z

Proof. For the first convergence, we split the sum according to the values of j − i (between
N_n − 1 and −N_n + 1 where N_n = bbnc − banc):

bbnc i

Nn−1

bbnc

i

bbnc

^Xbc

^cj−i ^=

X

b^Xc

b^Xc

1

{j − i = k} .

(2.89)

ck

n

n

i,j= an +1

k=−Nn+1

i= an +1

j= an +1

Pbbnc

The sum _j=banc+1 1 {j − i = k} is 1 if banc + 1 6 i + k 6 bbnc and zero otherwise; for k > 0, this constraint means banc+ 1 6 i 6 bbnc−k and for k < 0, this means banc+ 1−k 6 i 6 bbnc hence

1

bbnc

i

X

1

bbnc−k i

^Xbc

^cj−i ^=

1 {0 6 k 6 Nn − 1} ck

b^Xc

^n i,j= an

+1

n

k∈Z

n

i= an

+1 ^n

1

X

b

bbnc

−

i

+

1 {−Nn + 1 6 k

6 −1} ck

Xc

. (2.90)

n

k∈Z

i= an +1

k

n

For a fixed k, the convergences

1

1

{0

6

k

6

N

n

−

1}

c

k

bbnc−k i

→

c

k

b^2 − a^2

(2.91)

n

n

2

X
i=banc+1

• 
  X
  

• 
  k∈Z
  

17

N

k

c

bbnc

i

c b^2 − a^2

1 {− n + 1 6

6 −1}

=b

Xc

−

(2.92)

k

n

→ k 2

i

an +1

k