ddcon4r

2022-07-01

I have two questions about the uniform integrability.
The definition I am using is that a class of random variables $\chi$ is uniformly integrable if given an $ϵ>0$, there exists a $k$ such that for any $x$ in $\chi$ we have $\mathbb{E}\left[|x|\mathbb{I}\left\{x\ge k\right\}\right]<ϵ$.
First, is that in the definition of uniform integrability, can the density function which we compute the expectation with respect to it, vary with $n$?
Second, suppose I have a sequence of random variables $\left({x}_{n}{\right)}_{n\ge 1}$ (with varying density function w.r.t n). For the two functions $f,g$, I know that $f\left(x\right)\le g\left(x\right)$ for all $x$. Does the uniform integrability of the sequence $\left(g\left({x}_{n}\right){\right)}_{n\ge 1}$ imply the uniform integrability of $\left(f\left({x}_{n}\right){\right)}_{n\ge 1}$?

Ordettyreomqu

Expert

A family of random variables $\left({X}_{\alpha }{\right)}_{\alpha \in A}$ is u.i. if
$\underset{\alpha \in A}{sup}\mathsf{E}|{X}_{\alpha }|1\left\{|{X}_{\alpha }|>M\right\}\to 0$
as $M\to \mathrm{\infty }$. Note that ${X}_{\alpha }$'s may have different distributions (densities if exist). If $\left({Y}_{\alpha }{\right)}_{\alpha \in A}$ is a u.i. family of r.v.s. satisfying $|{X}_{\alpha }|\le |{Y}_{\alpha }|$ a.s. for all $\alpha \in A$, then $\left({X}_{\alpha }{\right)}_{\alpha \in A}$ is u.i. as well because for any $\alpha \in A$,
$\mathsf{E}|{X}_{\alpha }|1\left\{|{X}_{\alpha }|>M\right\}\le \mathsf{E}|{Y}_{\alpha }|1\left\{|{Y}_{\alpha }|>M\right\}.$

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