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


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Proof of Corollary 1.5. We would like to use directly the continuous mapping theorem. However, Theorems 1.1 and 1.3 only give the convergence in distribution on D ([−R, R] × [0, 1]) which leads to the converge nce in (1.28) and (1.29) where the integral over R are replaced by integrals over [−R, R]. Then we show that the contribution of the integrals on R \ [−R, R] is negligible.


More formally, we will use the Theorem 4.2 in [1].


Proposition

2.8. Let Yn(R)

n,R>1

be

a family of random variable and let (Yn)n>1 and

(ZR)R>1 be a sequence of random variables such that

for all R




1, the sequence Y

(R)

converges in distribution to a random variable

(1) ZR;




>







n

n>1

  1. the sequence (ZR)R>1 converges in distribution to a random variable Z and




  1. for all positive ε,

lim lim sup

n

Y (R)
















o




(2.106)

P



Y

n




> ε

= 0

.

R→+∞ n→+∞

n














































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