# 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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### 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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### Uniform integrability and converging process

Let Y = (Y2)n∈N0 such that P({Yo=2})= 1/2 We consider the fY adapted process X = (X2)n∈N0 = (∏j=0nYj)n∈N0 Show that there is X∞ ∈ L1(P) such that Xn -> X∞, but X does not converge to X∞ in L1(P) ...
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### Confusion on Uniform Integrability of Random Variables

We have the definition that a random process, $X_n$, is (1st power) uniformly integrable if $$\lim_{M\to\infty}\sup_n\mathbb{E}(|X_n| ; |X_n|\geq M)=0$$. My question is whether the following four ...
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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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### Optional Stopping Theorem and Uniform Integrability

Here's a problem related to the Optional Stopping Theorem: Let $S_n$ be a symmetric simple random walk on $\mathbb{Z}$ (as in, at each step, the random walk moves a distance of 1 in a uniformly ...
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### Minimal stopping time of brownian motion

Suppose $W$ is a Brownian motion, let $H_B$ be the hitting of $B \in \mathbb{R}$ and let $\tau$ be another stopping time that is taken to be minimal, i.e $(W_{t\wedge \tau})_{t \geq 0}$ is uniformly ...
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### Sufficient Condition for Uniform Integrability $\mathcal{L}^1$-boundedness

I saw the following proof showing necessary and sufficient conditions for Uniform Integrability: And was just wondering does uniform integrability still hold if we ONLY have $\mathcal{L}^1$-...
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### Uniform convergence of exp integral

I need to investigate a uniform convergence of the integral $I$ on E, where $$I = \int_0^{+\infty} x^2 e^{-a x^4} \, \text{d} x, \, E = (0, \infty).$$ It's obvious for me that it converges uniformly ...
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### Uniformly Integrability of $M_{t\land \tau}$

I want to show the uniformly integrability of a solution to the Skorokhod embedding Problem. i.e. given a centered probability measure $\mu$ with finite first moment, we want to construct a stopping ...
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### Strengthening Central Limit Theorem to Convergence of Mean for Specific Case

The problem is as follows: Given two players, each flips a coin that produces heads with probability $p$. This is repeated $n$ times. What is the asymptotic behavior of the expected absolute total ...
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### Stopped process not uniformly integrable

I need to construct a counter example such that the process $\{X_n\}_{n \ge 1}$ is uniformly integrable; however, the stopped process $X_{\tau \wedge n}$ where $\tau$ is a stopping time, is NOT ...