# 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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### Convergence in $\mathscr{L}^1$ implies uniform integrability?

Quoting from Rogers and Williams, p. 116, Theorem 21.2 (A necessary and sufficient condition for $\mathscr{L}^1$ convergence): Let $(X_n)$ be a sequence in $\mathscr{L}^1$, and let $X\in\mathscr{L}^1$....
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### KL-divergence, total variation distance and weak convergence against unbounded functions

Let $p_n,p$ be absolutely continuous probability measures on $\mathbb R^d$. Suppose they are close w.r.t. Kullback-Leibler divergence: $$D_{KL}(p_n|p) \leq \frac{C}{n} \quad\forall\,n\geq1\,.$$ As a ...
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### In a finite measure space, uniform integrability implies $L^p$ bounded?

In the probability ambient we work with a measure space $(\Omega, \mathscr{F}, \mathbb{P})$ that is of finite measure (i.e. $\mathbb{P}(\Omega)<+\infty$). I'm interest to understad all the relation ...
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### Redundancy In Definition of Uniform Integrability?

In his notes here on modes of convergence for measurable functions $f_n : X \to \mathbb{C}$, Terry Tao provides the following definition of uniform integrability: Let $(X,\mathcal{B},\mu)$ be a ...
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