Can anyone give me some advice about this problem? Thanks.(adsbygoogle = window.adsbygoogle || []).push({});

Let [tex]\lim_{n \to \infty}X_{n}=X[/tex] a.s.

And let [tex]Y=\sup_n|X_{n}-X|[/tex].

- Proove that [tex]Y<\infty[/tex] a.s.
- Let [tex]Q[/tex] a new probability measure defined as it follows:

[tex]\displaystyle Q(A)=\frac{1}{c} \mathbb{E}\!\left[1_{A} \frac{1}{1+Y}\right][/tex], where [tex]\displaystyle c=\mathbb{E}\!\left[\frac{1}{1+Y}\right][/tex].

Proove that [tex]X_{n} \rightarrow X[/tex] (in [tex]L_{1}(Q)[/tex]).

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# Problem about almost sure and L1 convergence

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