# Uniform integrability of reciprocal of random variables

Let $\{X_n, n\geq 0 \}$ be a sequence of positive random variables that are uniformly integrable.

Assume that $\frac{1}{X_n}$ is integrable. Then, is it true that $\left\{\frac{1}{X_n}, n \geq 0\right\}$ is uniformly integrable?

• Uniformly integrable deals with the tail of the distributions. So, when you take reciprocals, the tail deals with the behavior of the original distribution near zero. So, can you come up with a counter-example? For example, taking any uniformly integrable random variables $\{X_n\}$, adn then slightly changing their behavior when they take values in the range $[0,1]$, maintains uniform integrability. – Michael Dec 7 '15 at 16:04

No: consider the sequence $X_n(\omega):=1/n$ for each $n\in\mathbf N^*$. The sequence $\left(X_n\right)_{n\geqslant 1}$ is uniformly integrable but the sequence $\left(1/X_n\right)_{n\geqslant 1}$ is not even bounded in $\mathbb L^1$.