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


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n

3/2

i=[nt ]+1







3/2




























n

i=[nt]+1







X0

X







Taking into account that t0



t

>

1/n, it follows that

























































































































































































































































































































[nt0] − [nt] 6 nt0nt + 1 6 2n (t0t)



































































(2.41)

hence







s) + 1/2




s)




n (

)

n (

) 6 2

1/2

(

s (

i)




( s

(

i)







ξ

s, t

ξ

s, t0

t0

t

[nt]

a

X

m

1

[nt0]

a

X

m

|



|

n

i=1



n




i=[nt]+1



X

X




n

[nt0]




+ 2

t0

t













b







X

i)




m










+

1







b

s (

X




m










.




(2.42)














































n

1/2

i=[nt ]+1

(

s

(









s)




n

1/2

(

i)



s)






















i=[nt]+1







X0

X







By stationarity,

E
















E n (

)

n (

)

6 p

1/2




( s ( i)

s)























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