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

Cov (W (s, t) , W (s⁰, t⁰)) = t (1 − t) (1 − t⁰) C_1,1 (s, s⁰) − 2t (1 − t) t⁰ (1 − t⁰) C₁^a (s, s⁰)

• 
  2t² (1 − t) (1 − t⁰) C₁^a (s⁰, s) + 4t² (1 − t) t⁰ (1 − t⁰) C^a (s, s⁰)
  

• 
  (t⁰ − t) t (1 − t⁰) C_2,1 (s, s⁰) − 2 (t⁰ − t) tt⁰ (1 − t⁰) C₂^a (s, s⁰)
  

• 
  2 (t⁰ − t) t (1 − t) (1 − t⁰) C₁^a (s⁰, s) + 4 (t⁰ − t) t (1 − t) t⁰ (1 − t⁰) C^a (s, s⁰)
  

• 
  (1 − t⁰) tt⁰C_2,2 (s, s⁰) − 2 (1 − t⁰) tt⁰ (1 − t⁰) C₂^a (s, s⁰)
  

• 
  2 (1 − t⁰) t (1 − t) t⁰C₂^a (s⁰, s) + 4 (1 − t⁰) t (1 − t) t⁰ (1 − t⁰) C^a (s, s⁰)
  

• 
  2t² (1 − t) t⁰ (1 − t⁰) C₁^a (s, s⁰) − 2 (1 − t⁰) t²tt⁰ (1 − t⁰) C₂^a (s, s⁰)
  

+ 4t (1 − t) t^0 (1 − t^0) C^a (s, s^0) +	4	t (1	− t) t^0 (1 − t^0) Cs,s^a0. (2.96)
	3

Simplifying this expression leads to the covariance mentioned in (1.17).

18 HEROLD DEHLING, DAVIDE GIRAUDO AND OLIMJON SHARIPOV

2.2.3. Convergence of the linear part. We also use Theorem 2.3. The convergence of (W_n⁰)_n>1 also hold to a process having continuous paths. Define

ξn (s, t) := Wn^0 (−R + 2Rs, t) , s, t ∈ [0, 1],

(2.97)

where Wn^0 is defined by (2.75). Observe that

ξn (s, t) = Wn (−R + 2Rs, t)

(

− )

1,−R+2Rs ^( i^) −

(

− 1)

2,−R+2Rs ^(

j)

(2.98)

+

n3/2

n

[nt] (n − [nt]) 1

n−1

n

i h

X

n

j

h

X

.

^X

X

2

i=1

j=2

Now, it suffices to prove that

ξ

s, t

[nt] (n − [nt])

1

n−1

n i h

X

and

X

n,1

(

) :=

n

− )

1,−R+2Rs ^(

i^)

(2.99)

n3/2

(

2

i=1

ξ

s, t

[nt] (n − [nt]) 1

n

j

h

X

X

n,2 ^(

) :=

− 1)

2,−R+2Rs ^(

j)

(2.100)

n3/2

n

(

2

j=2

satisfy the conditions 2, 3 and 4. Since the treatment of ξ_n,1 is completely analoguous to that of ξ_n,2, we will do it only for the latter. By writing [nt] (n − [nt]) and h_2,−R+2Rs (X_j) as a difference of non-decreasing function, we can see that 2 is satisfied with

[nt]

1

n

[nt]^2 1

n

X

X

where

ξn,^∗2

(s, t) = n1/2

n

j=2

(j − 1) bs (Xj) +

n3/2 ^n

(j − 1) θ2,s (Xj) ,

(2.101)

2

2

j=2

bs (v) = P{g (X1, v) 6 −R + 2Rs} ; θs = P{g (X1, X2) 6 −R + 2Rs} .

(2.102)

To do, so rewrite (j − 1) θ_2,s (X_j) in terms of partial sums of θ_2,s (X_j), use triangle inequality for the L^2p-norm, then we apply Proposition A.4.

Similar estimates as those who led to (2.55) give

06j1,j26n

n,2

n

n

−

n,2

n n

6

32n−1/2.

(2.103)

max

ξ^∗

^j1 , ^j2

+ 1

ξ^∗

^j1 , ^j2

2.2.4. Negligibility of the degenerated part. In view of (2.4) and (2.76), it suffices to prove that

[nt] (n − [nt]) n

−1

h

X

.

s

[

R,R] 06t61

2

1

i

→

−

n

∈

3/2

6^X6

3,s ^(

i^)

0 in probability

(2.104)

sup sup

[nt](n [nt])

n

1

23/22

Bounding

n3^−/2

2

^− by 2n ^−

− , it

suffices to show that for all positive ε,

P

sup

^h3,s

(Xi)

> ε

→

0.

(2.105)

s

3/2

1

∈ −

1

6

6

X

This is done in the same way as in the proof of Proposition 2.4: we cut the interval [−R, R] into intervals of length δ_n = n^−3/4 and do the same estimate. This ends the proof of Theo-rem 1.3.

19
2.3.