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

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

2. 
   the sequence (Z_R)_R>1 converges in distribution to a random variable Z and
   

2. 
   for all positive ε,
   

lim lim sup	n	Y (R)						o		(2.106)
	P		−	Y	n		> ε	= 0	.
R→+∞ n→+∞		n