# 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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### Stopped version of UI martingale is UI

Suppose $(X_n)_{n\geq0}$ is a uniformly integrable martingale. Let $T$ be any stopping time. Show that the stopped process $(X_n^T)_{n\geq0}=(X_{n\wedge T})_{n\geq0}$ is uniformly integrable. My only ...
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### 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 ...
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### Integrability of a family of functions, parameterized by the points of the compact

For any $\theta \in \Theta$, let $g(t;\theta)$ be integrable in $t$ over, say, the half line $\mathbb{R}_+$. Then suppose $\Theta$ is a compact set, then can we say $\sup_{\Theta}g(t;\theta)$ is also ...
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### How to prove that $(UM_t)_{t \geq 0}$ is a Continuous Local Martingale?

Consider a filtered probability space $(\Omega, \mathcal{F},(\mathcal{F}_t)_{t \geq 0},P)$ and $(M_t)_{t \geq 0}$ a continuous local martingale (CLM) w.r.t $(\mathcal{F}_t)_{t \geq 0}$. Let $U$ denote ...
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### Limits of an Integral: $\lim_{n\to \infty} \int x^2 \sin\left(f_n(x)\right)\mathrm{d}x=0$

I have a problem with the following exercise. I think by the Lebesgue dominated convergence theorem it can be solved but i don't know that $\sin\left(f_n(x)\right) x^2$ dominated by ? It it true to ...
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### If $\sup_n$ $E|X_n|^{1+\sigma} \lt \infty$ for $\sigma \gt$0, then $\{|X_n|\}$ is uniformly integrable

If $\sup_n$ $E|X_n|^{1+\sigma} \lt \infty$ for $\sigma \gt$0, then $\{|X_n|\}$ is uniformly integrable. I've seen a similar problem without the exponent on $|X_n|^{1+\sigma}$ and tried to apply it ...
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### Does Uniform Integrability of a product martingale $M$ imply integrability of $\sup|M_n|$?

Let $(M_n)$ be a non-negative uniformly integrable product martingale - i.e. $M_n=\prod_{j=1}^{n}X_j$ for independent non-negative r.v. $X_n$. The problem asks if there exists a random variable such ...
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Suppose $(X, \mathscr{F})$ is a martingale. Show that $(X_{\tau \wedge n}, \mathscr{F})$ is a uniformly integrable for any finite stopping time $\tau$ such that $\{X_n\}$ is uniformly integrable. My ...
Let $\mu_n$ be a sequence of probability measures on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ such that $\mu_n \to^d \mu$. Let $\{f_a\}_{a\in\mathbb{R}}$ be a colection of bounded real functions, each ...