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

By applying Lemma 2.7 and the convergences (2.87) to c_i := Cov (Z₀, Z_i⁰⁰) and (2.88) to c_i := Cov (Z₀⁰⁰, Z_i⁰⁰) we finally obtain that

2

lim

1 E

Y

=

d+1

t t

Z , Z

^d^+1 tu2

− tu^2−1

Z ,Z^00

X

X

X

X

X

n→+∞ n

^i∈^In,u

n,i

u=1

^( u − u−1^) i∈Z

Cov (

0

i^)+u=1

2

i∈Z ^Cov ( ^0 i ^)

d+1

3

3

X

X

^tu ^− ^tu−1

, Z^00) =: σ^2. (2.95)

+

Cov (Z^00

u=1

3

0

i

i∈Z

If σ = 0, we get a weak convergence to 0, otherwise by Theorem

A.1, we get the weak con-

such that

P

6

0

k,` k^0,`^0

P

a

`

k

`0

k0

)^d

is a Gaussian vector

vergence of

d

a

W ^0 (s , t

k

) to

d

k,ell

N

k,`

, where (N

k,`

k,`=1

k,`

n

`

k,`=1

k,`=1

for k

k , Cov (N

, N

) = Cov (W (s , t ) , W (s

, t

)), where