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

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

i already increased the offer
Answer
 The questioner was satisfied and accepted the answer, or
 The answer was disputed, but the judge evaluated it as 100% correct.

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

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!

Your answer wasn't fully helpful but you made a good job thank you
 answered
 109 views
 $25.00
Related Questions
 Statistics Probability
 Probability that a pump will fail during its design life
 Assess team win rate by combining individual win rates?
 Statistics Probability, Hypotheses , Standard Error
 Help with probability proofs and matrices proofs (5 problems)
 foundations in probability
 Find the maximum likelihood estimator
 Pdf/cdf Probability