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

By application of Proposition A.4 to each of the four terms of the right hand side of (2.43) and using boundedness of a_s (X_i)−m_s and b_s (X_i)−m_s by 2, we derive that for all s, t, t⁰ ∈ [0, 1] such that t⁰ − t > 1/n,

E

^h| n (

) −

n ^(

)|

i

6

(

(

))k>1 (

−

p

ξ

s, t

ξ

s, t^0

2p

C p,

α

k

t0

t)^2^p np

+

C p,

(

α

(

k

^))k>1

[nt^0] − [nt]

+

C p,

(

α

(

k

^))k>1

(

t^0

−

t

)

2p .

(2.44)

n