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

Proposition 2.6. For all d > 1 and all s₁ < · · · < s_d and 0 6 t₁ < · · · < t_d 6 1, the vector (W_n⁰ (s_{`</sub>, t<sub>k</sub>))<sup>d</sup><sub>k,`=1} converges in distribution to (W ⁰ (s_{`</sub>, t<sub>k</sub>))<sup>d</sup><sub>k,`=1}, where W_n⁰ is defined by (2.75) and W is like in Theorem 1.3.
Here again, we will prove the convergence of linear combinations, that is, for all (a_{k,`</sub>)<sup>d</sup><sub>k,`=1}, the convergence in distribution

d	d
X	X
ak,`Wn^0 (s`, tk) →	ak,`W ^0 (s`, tk)	(2.77)
k,`=1	k,`=1

15

holds. To this aim, we will express

d

ak,`W ^0 (s`, tk) as a linear combination of a sum of

k,`=1

n

functions of Xi. Using the definition^Pof In,u given by (2.8) for 1 6 u 6 d + 1, we derive that

d

1

n

X

X

ak,`Wn^0 (s`, tk) =

√

^An,i^,

(2.78)

n

i=1

k,`=1

where, for i ∈ In,u,

1

d

X

^An,i ^=

ak,` (n − [ntk]) h1,s` (Xi) 1 {u 6 k}

n

k,`=1

1

1

d

1

d

X

X

ak,`[ntk]h2,s` (Xi) 1 {u > k}

d

− n

n2

(n − i)

^ak,`^[^ntk^] (^n − ^[^ntk^]) ^h1,s` ^(^Xi^) + n

k,`=1

k,`=1

1

1

X

ak,`[ntk] (n − [ntk]) h2,s` (Xi) .

− n

n2

(i − 1)

(2.79)

k,`=1