# Two questions about uniform integrability

I have two questions about the uniform integrability.

The definition I am using is that a class of random variables $$\mathbb{\chi}$$ is uniformly integrable if given an $$\epsilon >0$$, there exists a $$k$$ such that for any $$x$$ in $$\chi$$ we have $$\mathbb{E}[|x| \mathbb{I}\{x \geq k\}] < \epsilon$$.

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 $$(x_n)_{n\geq 1}$$ (with varying density function w.r.t n). For the two functions $$f,g$$, I know that $$f(x) \leq g(x)$$ for all $$x$$. Does the uniform integrability of the sequence $$(g(x_n))_{n \geq 1}$$ imply the uniform integrability of $$(f(x_n))_{n \geq 1}$$?

• There are varying definitions of uniform integrability, context and author dependent. It would be good to edit in the definition and context you are using Dec 27, 2021 at 16:48

A family of random variables $$(X_{\alpha})_{\alpha\in A}$$ is u.i. if $$\sup_{\alpha\in A}\mathsf{E}|X_{\alpha}|1\{|X_{\alpha}|>M\}\to 0$$ as $$M\to\infty$$. Note that $$X_{\alpha}$$'s may have different distributions (densities if exist). If $$(Y_{\alpha})_{\alpha\in A}$$ is a u.i. family of r.v.s. satisfying $$|X_{\alpha}|\le |Y_{\alpha}|$$ a.s. for all $$\alpha\in A$$, then $$(X_{\alpha})_{\alpha\in A}$$ is u.i. as well because for any $$\alpha\in A$$, $$\mathsf{E}|X_{\alpha}|1\{|X_{\alpha}|>M\}\le \mathsf{E}|Y_{\alpha}|1\{|Y_{\alpha}|>M\}.$$
• @Rostam22 That's because $|X_{\alpha}|1\{|X_{\alpha}|>M\}\le |Y_{\alpha}|1\{|Y_{\alpha}|>M\}$ a.s. without resorting to "the integral representation".