# Questions tagged [uniform-integrability]

For questions about families of uniformly integrable random variables. Use the tags (measure-theory) or (probablity-theory).

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### If $f$ us periodic and even, what I can conclude about of $\int f \;dx$?

Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be a periodic, even and differentiable function. If $L>0$ is the minimal period of $f$, what can I conclude about $$I :=\int_{0}^{L} f(x)\; dx?$$ By ...
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### using DCT to prove Uniform Integrable

If X$_n$ is dominated by some Y in L$^1$ or if it is identically distributed with finite mean, then prove that X$_n$ is uniformly integrable. Approach: Start with dominated convergence theorem but ...
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### $\sup L^1$ space with uniform integrability

I have a following question. For $t\in[0,T]$, let $f_t:\mathbb{R}\rightarrow\mathbb{R}$ be uniformly integrable family i.e. $\{f_t,\,t\in[0,T]\}$- uniformly integrable. We consider the space of such ...
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### Uniform integrability of product of random variables

Let $\mathcal{Y}$ be a collection of random variables on a probability space $(\Omega,\mathcal{F},P)$ that is bounded in $L^p$-norm, $p \in (1,\infty)$. Fix $X \in L^q$ where $L^q$ is the dual space ...
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### Equivalent Definition of Uniformly Integrable

This question is about the equivalent definition of uniformly integrable, which A possible uncountable collection of random variables $\{X_{\alpha}, \alpha\in I\}$ is said to be uniformly integrable ...
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### Product of uniformly integrable functions

Let $S$ be a set of measurable functions $E \to [- \infty, \infty]$. We say that $S$ is uniformly integrable iff for every $\epsilon>0$ there exists a $\delta >0$ such that, for any $f \in S$ ...
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According to Wikipedia, A class $\mathcal{C}$ of random variables is $\textbf{uniformly integrable}$ if given $\epsilon > 0$, there exists $K \in [0. \infty)$ such that $\textbf{E}(|X|I_{|X| \... 2answers 35 views ### A question about a UI martingale with 2 stopping times If$L$and$M$are stopping times with$L\leq M$and$Y_{n\wedge M}$is a uniformly integrable submartingale, then$EY_L\leq EY_M$and$Y_L\leq E(Y_M|\mathcal{F}_L)$. Since$Y_{n\wedge M}$is UI,$Y_{...
I have got a statement which seems really trivial to me, but I was not able to prove it, so maybe someone here can help me out. We consider a Family of random variables $(X_n)_{n\in\mathbb{N}}$ such ...