# Convergence in probability and uniform integrability implies convergence in Lp

Let $\{X_n\}$ be a sequence of random variables such that $X_n\overset{P}{\to}X$ for some random variable $X$. Suppose that for some $p\geq1$,  $$\lim\limits_{a\to\infty}\mathbb{E}\left[ |X_n|^p\mathbb{1}_{\{|X_n|>a\}}\right]=0,$$ where this limit is uniform in $n$. (This is the original redaction from my teacher I suspect this means that the sequence $\{|X_n|^p: n \in \mathbb{N}\}$ is uniformly integrable).

Show that $X_n\overset{L_p}{\to} X$.

• Daniel90

This is not my area of expertise, but I guess your offer is low for such a high level question.

• Interwebff

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• Interwebff

Thank you. I can't assume that X_n \in Lp (I have to prove it) and Vitali convergence theorem needs that help :(

• Dynkin

Vitali is the only theorem I know that uses "uniformly integrable", do you have any theorem that uses this property? As for X_n \in L^p : Note that Integral ( |X_n|^p ) = Integral ( |X_n|^p over the set |X_n|≤a) + Integral (|X_n|^p over the set |X_n|>a ). The first integral is ≤ a^p P(everything) = a^p because we are dealing with a probability space - the second integral is finite else the expression E( |X|^p 1_,,,) in your condition cannot tend to 0!

• Interwebff