# 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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### Uniform integrability and $L^1$ convergence of $(1/X_n)$

Let $X_n \to c > 0$ almost surely, where $c$ is a constant and $X_n > 0$ for all $n$. Also, let $(X_n)$ be uniformly integrable, so in particular $X_n \to c$ in $L^1$. Question: Do we have ...
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### Stopped cadlag submartingale is integrable

I'm trying to understand, why a stopped submartingale is again a submartingale. In the lecture notes to my lecture this is just stated as a corollary of Doob's Optional Sampling Theorem but I don't ...
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### Show $(\mathbb{E}[Z\mid\mathcal{F}_t],\mathcal{F_t})_{t \geq0}$ is an *uniformly integrable* martingale.

Suppose $Z \in \mathcal{L}^1(P)$. I want to show that $(\mathbb{E}[Z\mid\mathcal{F}_t],\mathcal{F_t})_{t \geq0}$ is an uniformly integrable martingale. I have managed to show to it is a martingale but ...
Assume that $X_1, X_2, ...$ are independent and identically distributed random variables with mean $\mu$ and variance $1$, then let $\bar{X}_n=n^{-1}\sum_{i=1}^n X_i$ be the sample mean. We all know ...