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

20 HEROLD DEHLING, DAVIDE GIRAUDO AND OLIMJON SHARIPOV

Using the weak convergence of (e_n (s, t))_n>1 to W (s, t) on D ([0, 1]) for a fixed s and then Fatou’s lemma, we derive that

E[|

R −

|

]

^6 n→+∞

^ZR\[−R,R] E

06t61

n

(2.116)

Z

Z

lim inf

sup e

(s, t)^2 dµ (s) .

Moreover,

6

^ZR\[−R,R] E

6

Z

R\[−R,R] ^E 06t61

E

n ^−

n

i

06t61

n

n>1

n

h

Y (R)

Y

sup

e

(s, t)^2 dµ (s)

sup

sup

e

(s, t)^2 dµ (s) ,

(2.117)

and similar inequalities hold where e_n and W are replaced by e⁰_n and W .

Now, using the fact that µ is a finite measure, it suffices to find uniform bounds in n and s

for E^hsup06t61 en (s, t)^2i

and E^hsup06t61 en^0 (s, t)^2i, namely,

n>1 s∈R

E

06t61

n

(s, t)^2

E

06t61

n

∞

(2.118)

sup sup

sup

e

+

sup

e^0 (s, t)^2

< +

.