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I have the following problem. Let $f: \mathbb{R} \to ]0,+ \infty[$ continuous, integrable function and $X_n$ a sequence of real random variables. If $\int_{\mathbb{R}}|F_n(x)-F(x)|f(x) dx \to 0$ then $X_n \to X$ in distribution.

I have proved by paradox. Indeed exists $x_0$ (point of continuity of $F$), $\varepsilon_0>0$ and $F_{n_k}$ such that $|F_{n_k}(x_0)-F(x_0|> \varepsilon_0 \quad \forall k$.

Than this produce an absurd since $0< \int_{\mathbb{R}}|F_{n_k}(x)-F(x)|f(x) dx \to 0$.

I would like to know if there is a another way to prove this fact.

Thanks

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    Ah yes, sorry. It does indeed.2012-12-20

0 Answers 0