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

• 
  
  

X

(b_s (X_i) − m_s) . (2.39)

i=[nt]+1

10

HEROLD DEHLING, DAVIDE GIRAUDO AND OLIMJON SHARIPOV

Decomposing the sum

[nt^0]

[nt]

[nt^0]

n

[nt^0]

n

^Pi=1

as ^Pi=1 + ^Pi=[nt]+1 and the sum ^Pi=[nt]+1 as ^Pi=[nt]+1 + ^Pi=[nt0]+1,

we derive that

n ^(

)

n ^(

) 6

3/2

(

s ^(

i^)

s)

+

3/2

(

s ^(

i^)

s)

ξ

s, t

ξ

s, t^0

[nt^0] −

[nt]

[nt]

a

X

m

n

−

[nt^0]

[nt^0]

a

X

m

|

−

|

i=1

−

i=[nt]+1

−

n

n

^X

X

[nt^0] − [nt]

n

b

X

m

[nt]

[nt^0]

b

X

m

.

+

(

s ^(

i^)

−

s^)

(

s ^(

i^)

−

s^)

(2.40)