Problem about almost sure and L1 convergence

[yavanna]

Can anyone give me some advice about this problem? Thanks.

Let $$\lim_{n \to \infty}X_{n}=X$$ a.s.

And let $$Y=\sup_n|X_{n}-X|$$.

• Proove that $$Y<\infty$$ a.s.
• Let $$Q$$ a new probability measure defined as it follows:
  $$\displaystyle Q(A)=\frac{1}{c} \mathbb{E}\!\left[1_{A} \frac{1}{1+Y}\right]$$, where $$\displaystyle c=\mathbb{E}\!\left[\frac{1}{1+Y}\right]$$.
  Proove that $$X_{n} \rightarrow X$$ (in $$L_{1}(Q)$$).

 

[Valkarie]

1. To prove that Y < ∞ a.s., note that the limit of Xn as n → ∞ is finite (X < ∞ a.s.) and the sequence of Xn is bounded, so Y must also be finite.2. To prove convergence in L1(Q), start by noting that |Xn - X| ≤ Y for all n, so that |Xn - X| ≤ Y ≤ 1 + Y Taking expectations with respect to Q we get \mathbb{E}_Q[|X_n - X|] \leq \frac{1}{c}\mathbb{E}[1+Y]=1 Thus, Xn converges to X in L1(Q).