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We all know that 'converge in distribution' is weak, and thus can not directly achieve the 'Lp convergence' for the random variables.

My question is: Is there any theorem that can achieve it, equipped with additional assumptions?

zolan eric
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  • Any sequence $(X_n)$ where all the $X_n$'s have the same distribution converges in distribution. It is too much to ask for $L^{p}$ convergence from this. – geetha290krm Jul 22 '22 at 08:12
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    Do you have any more context to offer about the random variables you're studying? In general, it looks very unlikely that this can be done. – Sarvesh Ravichandran Iyer Jul 22 '22 at 08:30
  • In simple terms, the central limit theorem, for example, is a classic case of convergence by distribution, and I wonder if it contains a strengthened form of Lp convergence well, even if by adding some more assumptions. – zolan eric Jul 22 '22 at 09:55
  • My question is very similar to this \urf{https://math.stackexchange.com/questions/2465820/does-a-clt-imply-other-forms-of-convergence}. I wonder if there is any theorem, even if by adding more assumptions, to enhance the CLT to the Lp-convergence result. Thank you everyone! – zolan eric Jul 22 '22 at 10:03
  • Your link is about convergence to a constant. Convergence to a non-constant random variable ($L^p$ or in probability or almost sure) is much more restrictive than convergence in distribution – Henry Jul 22 '22 at 14:02

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I'm not sure this is what you're looking for, but you could put together the two results that guarantee the reverse of convergence in probability and in $L^p$.

This is what I mean: we know that if a succession of r.v. converges in distriution to a constant then it converges also in Probability to that constant. Moreover if a succession converges in probability to a r.v. X and the succession is bounded, then it also converges in $L^p$.

Hence we could state this theorem: If $ X_n \overset{\mathcal{D}}{\to} c, |X_n| < M \in \mathbb{R} \Rightarrow X_n \overset{P}{\to} c$.

Hope this helps.

finch
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  • Thanks for your advice! First, In your 2-3 paragraphs, if these two statements are combined, we can only get the convergence on the constant $c$. It will be more exciting if it can converge to a random variable $X$. Second, the latter statement is like the dominated converging theorem in the 'convergence in probability form'? The most popular one is the 'convergence a.s.'. Could you give a source for formal proof? (I found an informal proof \url{https://math.stackexchange.com/questions/4273085/dominated-convergence-theorem-for-convergence-in-probability}) – zolan eric Jul 22 '22 at 09:43
  • Perhaps my question itself is a fantasy, and I am curious if there is a relevant theory that can support such a transition from weak to strong. In simple terms, the central limit theorem, for example, is a classic case of convergence by distribution, and I wonder if it contains a strengthened form of Lp convergence well, even if by adding some more assumptions. – zolan eric Jul 22 '22 at 09:51
  • @zolaneric For source material you can look up almost everywhere you like topics such as convergence of random variables and it usually gives you also the proof for the reverse. I like the proof given in Probability Essentials by j. Jacod and P. Protter (p. 146). As to whether you can find a th. for X instead of c, I tend for the no. If it were possible it should mean that I could prove that conv in law implies conv in probability for X (I go to conv in $L^p$ and then to conv in prob) which is not true. Hope I was clear enough – finch Jul 22 '22 at 10:42
  • Thanks. It's clear! – zolan eric Jul 22 '22 at 11:16