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

X

Bounding n

−

[nt] and [nt] by n, we derive by

stationarity that

E ξn (s^0

, t)

−

ξn (s, t)

|

2p

6 Cpn^−^pE

[nt] (as

(Xi)

−

as0 (Xi) + ms0

−

ms)

2p

^h|

i

i=1

^X

n

[nt]

2p

−

(bs (Xi) bs0

(Xi) + bs0

bs)

(2.46)

+ Cpn−pE

−

−

i=1

^X

and taking into account the inequalities |a_s (x) − a_s0 (x)| 6 2RM |s − s⁰| and |b_s (x) − b_s0 (x)| 6 2RM |s − s⁰|, we derive that

h	i
E |ξn (s^0, t) − ξn (s, t)|^2^p	6 C (s^0 − s)^p ,	(2.47)

showing (2.36).
Now we show (2.37). Going back to the expression of ξ_n^∗ given by (2.34), we derive that

|ξn^∗ (s, t) − ξn^∗ (s^0, t)| 6

1

n

(|as (Xi) − as0 (Xi)| + |bs (Xi) − bs0 (Xi)|) + 2^√

|ms − ms0|

n

n3/2

i=1

6 8R^√

X

(2.48)

|s^0 − s|

n

hence

6n

n

n

n

−

n

n n

6

√n

06j1,j2

(2.49)

max

ξ∗

j1 + 1

, ^j2

ξ∗

^j1 , ^j2

8R ,

giving (2.37).
Finally, we show (2.38). For 0 6 t 6 t⁰ 6 1 and s ∈ [0, 1], denoting c_s following inequalities hold:

n ^(

)

n

(

) 6

3/2

s (

i)

3/2

s (

i) + 2

s

1/2

|

ξ∗

s, t

−

ξ∗

s, t^0

|

[nt^0

]

[nt^0] c

X

−

[nt]

[nt]

c

X

m

[nt^0

] − [nt]

n

i=1

n

i=1

n

X

X

:= a_s + b_s, the

(2.50)

[nt0] − [nt] ^[^nt^]

c

X

[nt^0]

[nt^0]

c

X

m

[nt^0] − [nt]

6

s ^(

i) +

s ( i) + 2

s

(2.51)

n

3/2

i=1

n

3/2

n

1/2

nt

X

i=[^X]+1

1

nt^0

nt

nt

nt^0

[nt^0] − [nt]

6

^4 n3/2 ^([

] − [

]) ([

] + [

])+8

(2.52)

n1/2

6

8

(

n

(

t^0

t

)+1)+8

n (t^0 − t) + 1

(2.53)

−

n3/2

n1/2

6 16n^1^/^2 (t^0 − t) + 16n^−1^/^2.

(2.54)

12 HEROLD DEHLING, DAVIDE GIRAUDO AND OLIMJON SHARIPOV

As a consequence, we derive that

06j1	,j2	6n	n		n		n	−	n	n n
max				ξ^∗		^j1 , ^j2	+ 1		ξ^∗	^j1 , ^j2

and (2.38) follows. This ends the proof of Proposition 2.2.

2.1.4. Negligibility of the degenerated part.

_{6 32n}^−1/2_{, (2.55)}