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

^j_n¹ , ^j_n<sup>2


j</sup>_n¹ , ^j_n²

→ 0 in probability;

→ 0 in probability.
(2.37)

(2.38)

In order to prove (2.35) and (2.36), we need to control the differences in s and t.

Let us show (2.35). We first control |ξ_n (s, t) − ξ_n (s, t⁰)| for t⁰ − t > 1/n. By definition of ξ_n,

ξn (s, t)

ξn (s, t^0) 6

n −3[/2

0

]

(as (Xi)

ms)

−3/[

2

]

(as (Xi)

ms)

|

−

|

nt

[nt^0]

−

−

n

nt

[nt]

−

n

i=1

n

i=1

X

X

+ [ 3/^02]		(bs (Xi)	ms)	^[ 3/2^]
n		n	−	− n
		i=[nt ]+1
	nt	^X0		nt