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Let $\{X_n\}$ be a sequence of real random variables, such that $$\displaystyle\liminf_{n\to \infty}E({X_n}^2)<\infty$$ Then show there exists an integrable random variable $X$ and a subsequence $\{n_k\}$ such that $$X_{n_k}\implies X\quad \text{as }k\to \infty$$ and $$\lim_{k\to\infty}E(X_{n_k})=E(X)$$ where $\implies$ denotes weak convergence, or convergence in distribution.

I'm completely stuck in this problem. How does one begin with looking for such a subsequence. I tried to consider the subsequence $\{n_k\}$ such that $$\lim_{k\to \infty}E({X_{n_k}}^2)=\ell<\infty$$ but I couldn't proceed anywhere from here.

shadow10
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    The subsequence you have chosen is bounded in $L^2$. By the Banach-Alaoglu theorem we get that this subsequence has a weakly (in $L^2$) convergent subsequence. – Severin Schraven Feb 23 '19 at 19:44

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The subsequence you have chosen is bounded in $\mathbb L^2$. This sequence is tight, because $$\mathbb P\left(\left\lvert X_{n_k}\right\rvert \gt R\right)\leqslant R^{-2}\mathbb E\left[X_{n_k}^2\right]\leqslant R^{-2}\sup_{i\geqslant 1}\mathbb E\left[X_{n_i}^2\right].$$ This means that (from Prokhorov's theorem) we can extract from $\left(X_{n_k}\right)_{k\geqslant 1}$ a subsequence which converges in distribution to some random variable $X$. This subsequence is also uniformly integrable, which guarantees the converges of the expectations.

Davide Giraudo
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  • So we consider variables like $|X_{n_k}|\mathbb{I}{{|X{n_k}|\le r_k}}$ where $r_k,-r_k$ are both continuity points of $X$ and $r_k\to \infty$. Then we can apply DCT and force the convergence of $E(X_{n_k})$. But proceeding like this we can also get the same result for second moment. Also, DCT actually gives $L_1$ convergence. So we are actually getting something stronger. Am I on the right path? – shadow10 Feb 24 '19 at 11:48
  • A problem is that you may not have the pointwise convergence hence it is not so clear how you apply the dominated convergence theorem. See the link. – Davide Giraudo Feb 25 '19 at 11:37
  • I see, you're right. I was thinking of applying Skorohod's representation and then using DCT, but it's getting messy and possibly incorrect. Thank you for your answer. – shadow10 Feb 26 '19 at 10:33