# Tag Info

### Prove that a martingale is uniformly integrable.

The martingale $M_t$ is not uniformly integrable. If it were, then by standard martingale facts, it would converge a.s. and in $L^1$ to some $M_\infty$. (An $L^1$-bounded martingale always converges ...
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### Why $\lim_{n\rightarrow \infty}\int_0^1 e^{x^n} = 1$

To show $\lim\limits_{n\rightarrow\infty}{\int\limits_{0}^{1}{e^{x^n}\,dx}} = 1$, it suffices to show that $\lim\limits_{n\rightarrow\infty}{\int\limits_{0}^{1}{(e^{x^n}-1)\,dx}} = 0$. Clearly, we ...
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### If $\mathbb{E}(\sup_n |X_n|)< \infty$ then $(X_n)_n$ is uniformly integrable

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables. If there exists an integrable random variable $Y$ such that $|X_n| \leq Y$ for all $n \in \mathbb{N}$, then $(X_n)_{n \in \mathbb{N}}$ ...
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### Limit and conditional expectation commute in a uniformly integrable sequence

The proposition in the post does not hold. The problem arises from the implicit assumption that the a.s. limit of the conditional expectations exists. The example described in the link [1] is correct, ...
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### Counterexample for uniform integrability of an $\mathbb{L}^1$-bounded sequence

Let $\Omega=[0,1]$ with Lebesgue measure, and $X_n=n1_{[0,\frac{1}{n}]}$. Then the $X_n$ are bounded in $L^1$ but not uniformly integrable.
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### Uniform Integrability of Random Variables

You meant $\{X_n\}$ are uniformly integrable if $$\lim_{M\to\infty} \sup_n E[ |X_n|\chi_{\{|X_n|>M\}}]=0$$ (e.g. https://en.wikipedia.org/wiki/Uniform_integrability) Now for your question. ...
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### Limit of an integral: Possible issue with uniform integrability?

Let $y = nx$. Then $$\int_0^1 f_n(x) \, dx = \int_0^n \frac{e^{y/n} \cos(y/n)}{1 + y^2}\, dy = \int_0^\infty g_n(y)\,dy$$ with $$g_n(y)=\frac{e^{y/n} \cos(y/n)}{1 + y^2} 1_{[0,n]}(y).$$ We have for ...
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### Stopped version of UI martingale is UI

Any uniformly integrable martingale can be written in the form $(E(X|\mathcal F_n))$ with $E|X| <\infty$. Optional Sampling Theorem (SeeTheorem 9.3.3, p.324 of K L Chung's "A Course in ...
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### $\lim X_n = 0$ iff $b > 0$
First, rewrite the exponenent as \begin{align*} \exp \left( n\left( a\frac{S_n}{n} - b \right) \right) \end{align*} By the Law of Large Numbers, we have $a S_n/n- b \to a \mathbb{E}(\xi_1) - b$ a.s.. ...