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

n

1/2

the processes ξ_n^o and ξ_n^∗ are both coordinatewise non-decreasing and ξ_n = ξ_n^o − ξ_n^∗.

Now, let p be such that Assumption (A.3) is satisfied and let γ := 2p and β := p (since p > 2, the condition β > 2 is fulfiled). In view of Theorem 2.3, the proof of Proposition 2.2 will be complete once we show the following:

h

i

6 C |t − t^0|^p ;

sup

sup

sup

E |ξn (s, t) − ξn (s, t^0)|^2^p

n>1 s∈[0,1] t,t^0∈[0,1]

|t^0−t|>n^−1

sup

sup

sup

ξ

(s, t)

ξ (s^0

, t)

2p

C

s

s^0

p

;

n>1

s,s^0∈[0,1]

1

t∈[0,1]

E^h|

n

−

n

|

i

6

|

−

|

|s^0−s|>n^−

(2.35)

(2.36)

_{max ξn}^∗

06j₁,j₂6n

_{max ξn}^∗

06j₁,j₂6n

	j1 + 1	,	j2			− ξn^∗
	n			n

			j2 + 1
	j1	,			− ξn^∗
	n		n