This message has been cross posted to the following eGroups: Young Professionals Group and Statistical Consulting Section .
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Hi all,
I heve a question regarding the convergence in distribition (see the attached file for a better format):
Let $\{X_1,X_2,\ldots\}$ and $\{Y_1,Y_2,\ldots\}$ be two sequences of uniformly integrable random variables with distributions $F_n(x)$ and $G_n(x)$, respectively. If
$$'F_n(x)-G_n(x)'\leq \mathcal{O}(n^{-1}),$$
and $\lim_{n\to\infty}G_n(x)= G(x)$, for all $x\geq 0$, is it true we say that
$$\Big'\int h\, d(F_n)-\int h\, d(G)\Big'\leq \mathcal{O}(n^{-1}),$$
for all bounded continuous increasing function $h(x)$?
Best regards
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Rasool
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