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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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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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21 views

Uniform convergence of $I(x) = \int_{0}^{+\infty} {\frac{\cos{(xt)}}{\sqrt{t}}dt}$ on $\mathbb R$

Check if $I(x) = \int_{0}^{+\infty} {\frac{\cos{(xt)}}{\sqrt{t}}dt}$ is uniformly convergent function on $\mathbb R$. I think it's not uniformly convergent function on $\mathbb R$. Could you give ...
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1answer
32 views

Stopped random walk is not uniformly integrable

I know that in general Doob's Optional Stopping Theorem doesn't hold for unbounded stopping times, but that it does when the system up to the stopping time is uniformly integrable. One counter ...
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1answer
23 views

Uniform integrability of the following martingale

Let $A_{n,k}\in F$ where $1\leq k\leq 2^n$ such that $A_{n,k}$ are disjoint for fixed $n$ and different $k$, $A_{n,k}=A_{n+1,2k-1}\cup A_{n+1,2k}$. Let $F_n=\sigma(\{A_{n,k}:1\leq k\leq 2^n\})$ and ...
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35 views

Uniform integrability of negative part implied by convergence in L1 by “lower bound”?

Let $(X^n),(Y^n)$ be a random variables with $E[X^n]\leq x \, \forall n $ for some fixed $x\in \mathbb{R}$ and $X^n\geq Y^n\, \forall n$. Also assume that $Y^n\rightarrow Y$ in $L^1(P)$ as $n\...
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35 views

Is a continuous function of (integrable) Brownian motion always integrable?

Let $Z_{t}=e^{-W(t)}$ with $\{W(t):t\geq0\}$ a Brownian motion. Is $Z_{t}$ integrable, since it is a continuous function of Brownian motion? Furthermore, are all continuous functions of Brownian ...
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1answer
36 views

Show that if $p > 1$ and $\sup_{n \geq 1} |X_n| \in L^p$ then $\{X_n, n \geq 1\}$ is uniformly integrable.

Let $(\Omega,\mathcal{\Sigma,\mathbb{P}})$ be a complete probability space. I have to show that $$\lim_{\alpha \to \infty} \sup_{n \geq 1}\int_{\{|X_n| \geq \alpha\}}|X_n|\, d\mathbb{P} = 0$$ ...
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82 views

A martingale that does not converge in $L^1$

I am currently studying martingales and I am working on the following problem: Let $\Omega = \mathbb N^*$ and associated probability measure $$\forall\, k \in \mathbb N^*, P({\{k\}})=\frac{1}{...
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1answer
36 views

Extension of result to stopping times

Suppose I have a UI martingale $M_t$. Then by the martingale convergence theorem, I know that there is $M_\infty$ such that $\mathbb{E}[M_\infty\vert\mathcal{F}_t]=M_t$. I want to extend the result to ...
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68 views

Supremem of Uniformly Integrable Sequences

I'm doing an exercise about uniformly integrable random variables: Suppose $(\xi_n)$ is a sequence of uniformly integrable random variables, then \begin{equation} \lim_{n\rightarrow\infty}\mathbb{...
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1answer
73 views

Proving uniform integrability

Let $W_t, t \geq 0$ be Brownian motion. Now, fix $u \in \mathbb{R}$ and define $X_{(k)}^t := \sum_{n=0}^k \frac{(u \sqrt{t})^n}{n!} H_n \left( \frac{W_t}{\sqrt{t}} \right)$ $\textbf{Claim: $\{ X_{(...
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1answer
33 views

Can we interchange integral and limit if the sequence of function is uniformly integrable?

Assumptions: 1) $h(x,\omega)$ is defined on $\forall\omega\in\Omega, \forall x\in R\setminus\{x_0\}$ where $\omega$ is a random variable. 2) Let $h(x_0,\omega)=\lim\limits_{x\rightarrow x_0}h(x,\...
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63 views

Characterization of Martingale and Definition of Uniform Integrability

Let $M$ be progressively $\mathbb{F}$-adapted process, where index set is $[0,T]$. $(T<\infty)$ Claim If $E[M_{\tau}] = E[M_0]$ for all stopping times $\tau \in \mathcal{T}(\mathbb{F})$, then $M$...
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95 views

Necessity of uniform integrability in martingale convergence theorem

Statement of theorem: If $X_n$ is a uniformly integrable martingale, then $\lim_n X_n$ exists a.s. and in $L^1$, and $$X_n=E(\lim_{n \to \infty} X_n \mid \mathcal F_n) \quad \text{a.s.}$$ I can'...
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92 views

Is a uniformly integrable martingale stopped at every stopping time also uniformly integrable?

Let $\{X_n \}$ be a uniformly integrable martingale w.r.t. the natural filtration $\{ \mathcal{F}_n \}$. Is $\{ X_\tau : \tau \text{ is a stopping time w.r.t. } \{ \mathcal{F}_n \} \}$ uniformly ...
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1answer
55 views

show that $\text{ sup}_n \int{|f_n|^pdm<C}$ implies $\int_{E}{|f_n|dm}<\epsilon \text{ for large enough } n$.

My teacher claimed the following, without further explanation: Let $\{f_n \}$ be a sequence of integrable functions on $\Bbb R$ with lebesgue measure $m. \text{ Meaning, } \{f_n\}\subset L^1 \bigl( \...
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1answer
51 views

Non-negative uniformly integrable local martingale

I am not sure if the following is true or not: "A non-negative uniformly integrable continuous local martingale is a (true) martingale." Is this statement true? Or can someone give a counter-example?...
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260 views

Uniformly integrable local martingale

Can someone give me an example of a uniformly integrable local martingale that is not a martingale? Or are all U.I. local martingales true martingales (continuous, of course).
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1answer
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Optimal Stopping Theorem for UI Martingales

In my notes, I was given that: "Let $X_n$ be an $F_n$ (super,sub) martingale and $\tau$ an $F_n$ stopping time. Then $X_{n \wedge \tau}$ is a (super, sub) martingale wrt $F_n$ and $F_{n \wedge \tau}$"...
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66 views

Example: $\{X_n \}$ is uniformly integrable but $E[ f(X_n) ]=\infty$ with $f(x)=\omega(x)$ [closed]

I am looking for a counterexample where the sequence of random variables is $\{X_n \}$ is uniformly integrable, that is \begin{align} \lim_{ k \to \infty} \sup_{n \ge 0} E[ |X_{n}| 1_{ |X_n| \ge k} ]=...
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1answer
64 views

What's a condition that's weaker than uniform integrability and implies convergence of an integral to $0$?

Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let $(X_n)$, $(Y_n)$ be two sequences of nonnegative, integrable random variables. Suppose $X_n \leq 1$, $X_n \to 0$, and $Y_n \to Y$ almost ...
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1answer
84 views

Sufficient conditions for uniformly integrable

I have learned the definition of uniformly integrable and its sufficient and necessary condition: We call $\{X_n, n\in \mathbb{N}\}$ uniformly integrable, if $$ \lim_{x \to \infty} \sup_{n\in \...
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1answer
66 views

Continuous square integrable martingales and family of stopping paths

I have the following question and I'm not sure of the solution that I've been given: "Show that for an element $M\in\mathcal{M}_c^2$, the family $\{M_T\}_{T\in\mathcal{J}_a}$ is uniformly integrable ...
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2answers
49 views

Weak convergence of a sequence of probability measures implies integrability of the limiting probability measure

Let $(X_{n})_{n \in \mathbb{N}}$ be a sequence of uniformly integrable random vectors with values in some normed vector space $V$ with $\mathbb{E}[\|X_n\|] < \infty$. This means that $$ \lim_{C \to ...
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1answer
34 views

Different forms of uniform integrability proof from Schilling

This is part of a proof from Rene Schilling's Measures, Integrals, and Maringales. There is a part I don't understand here. (vii). (a) $\sup_{u \in F} \int |u|d\mu <\infty$; (b) $\lim_{n\to \infty}...
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1answer
57 views

If $\{f_n\}$ uniformly integral, then $\{ f_n - f \}$ uniformly integrable?

Let $\{f_n\}$ be uniformly integrable, on space $(\Omega, F, P)$. $f_n \rightarrow f$ in measure. The notes I was reading then stated two facts i do not follow: By Fatou's $\int_\Omega |f| ...
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38 views

Claim the collection $\left\{\mathcal{E}(aL)_{T}\right\}$ ($T$ stopping time) is uniformly integrable in the proof of Novikov's Condition

Here I have a question in the proof of Novikov's condition from Le Gall's Book Brownian Motion and Stochastic Calculus Page 137: The proof is very long, and I only have problem in the following part: ...
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483 views

Prove that a martingale is uniformly integrable.

Let's consider the following stochastic process: $$ M_t = e^{\theta X_t - \psi(\theta)t}, $$ where: $\quad X_t = (r-\frac{\sigma^2}{2})t + \sigma W_t \quad$ and $\quad \psi(\theta) = (r-\frac{\sigma^2}...
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1answer
84 views

Uniform integrability, convergence in probability and weak convergence.

Suppose that $T_n$ is a sequence of random variables defined on the probability space $(\Omega,\mathcal{F},\mathbb{P})$ and suppose that $T_n$ weakly converges to $1$, that is $$ \mathbb{E}[T_n\,I(E)...
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1answer
175 views

Is a sequence of bounded random variables uniformly integrable?

It seems to me that a sequence of uniformly bounded random variables $\{X_n\}_{n=1}^\infty$ on a probability space $(\Omega_n, \mathcal{F}_n, P)$ such that $|X_n(\omega)| \leq M$ for all $\omega \in \...
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1answer
53 views

Bounded weighted supremum norm and uniform integrability

Let $g:\mathbf{X}\to\mathbb{R}_+$ be a positive continuous function where $\mathbf{X}\subseteq\mathbb{R}^d$. Let $f_n:\mathbf{X}\to\mathbb{R}$ where $n\in\mathbb{N}$ be a sequence of functions. Define ...
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1answer
51 views

How to get this convergence of mean?

Given a probability space $(\Omega,\mathcal F,\mathbf P)$. Let $\{X_n\}, \{Y_n\}$ be two sequences of real-valued random variables on $(\Omega,\mathcal F,\mathbf P)$, with $X_n\to X$ in distribution ...
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93 views

Equivalence of asymptotic uniform integrability and convergence of mean

I was reading van der Vaart's text on asymptotic statistics and found the following theorem. The text only provides a proof for the direction where we assume asymptotic uniform integrability and want ...
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106 views

Prove that stopped discrete time nonnegative supermartingales are uniformly integrable

I came across this note online. On page 40 theorem 2.2.18, it seems to suggest that any stopped discrete time nonnegative supermartingale is uniformly integrable and I cannot figure out why. To be ...
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1answer
188 views

Are all martingales uniformly integrable

There is this problem on my sheet: Let $X = (X)_{n≥0}$ be a martingale and $T$ be a finite stopping time. Suppose $X_T$ is integrable. Show that $$E [X_T] = E [X_0]$$ if and only if $$\lim_{n→∞} E[...
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48 views

Uniform integrabiliy of submartinales

Let $\mathcal X=(X_i)_{i\in I}$ be a family of integrable, real valued random variables on $(\Omega,\mathcal F,\mathbb P)$. One easily sees that $\mathcal X$ is uniformly integrable iff both $\mathcal ...
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1answer
312 views

Uniform integrability of stopped submartingales

The following is well-known and useful : Lemma. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probabiliy space, and let $\mathcal{X}=(X_i)_{i\in I}\in L^1(\mathbb{P})^I$ be a uniformly integrable ...
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1answer
64 views

Non uniformly integrable sequence

I am looking for a sequence of random variable $X_1, X_2, \cdots$ on $([0,1], \mathbb B, \lambda)$ such that $$X_n\to 0 \quad \text{a.s.}$$ $$EX_n\to 0$$ but such that $(X_n)_n$ is not uniformly ...
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1answer
200 views

How can I show whether or not the Dirichlet function is integrable using the definition of upper and lower integrals?

If $f$ is a version of the Dirichlet function: $f(x) = 1$ when $x$ is an irrational in $[0,1]$ and $f(x) = 0$ otherwise How can I determine whether or not $f$ is integrable using the definition of ...
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1answer
89 views

If $\mathbb{E}(\sup_n |X_n|)< \infty$ then $(X_n)_n$ is uniformly integrable

Let $\{X_n: n\ge 1\}$ be a sequence of random variables satisfying $E [ \sup | X_n|] < \infty $. Show that $\{X_n\}$ is uniformly integrable. I know this is a basic question from Financial ...
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67 views

When does Moment Convergence imply convergence in $L^p$?

Under which condition does $E[X_n^2]\rightarrow E[X^2]$ impliy that $X_n\xrightarrow{L^2}X$. I think it is sufficient that $X_n$ is uniformly integrable but can't remember a proof at the moment. So is ...
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1answer
75 views

$L^{1}$ bounded but non uniformly integrable family

Let $X_{k}$ be a collection of independent R.V.'s with $E(X_{k})=1$ and $X_{k}>0$ for all $k$. Define $M_{n}=\Pi_{k=1}^{n} X_{k}$ and $a_{k}=E(\sqrt{X_{k}})$. Then 1) If $\Pi_{k=1}^{n}a_{k}\...
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191 views

Uniform integrability of continuous function of conditional expectations

Let $X$ be an integrable real valued random variable. Let $\sigma_n$ be a sub-sigma-algebra such that $\sigma(X) = \sigma(\cup_{n\in\mathbb{N}} \sigma_n)$. Suppose $f(X)$ is integrable where function $...
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1answer
76 views

Uniform intgrability and compact convergence

Suppose we have a family of uniformly integrable random variables $\{X_n:n\in\mathbb{N}\}$. Suppose also we have a sequence of continuous functions $f_n:\mathbb{R}\to\mathbb{R}$ converging to $f$ and ...
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2answers
275 views

Conditions for uniform integrability of sequence of integrable random variables converging to integrable random variable?

Let $X_n:\Omega\to \mathbb{R}$ be an integrable random variable converging to integrable $X$ almost surely. I know that this does not mean $\{X_n:n\in\mathbb{N}\}$ is uniformly integrable. But is ...
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1answer
258 views

Submartingale inequality

Let $X_n$ be a $F_n$ sub martingale. Suppose it exists an integrable random variable $X'$ such that $X_n \le \mathbb E(X' \mid F_n) \,\text{}a.s. \quad \forall n\in \mathbb N_0$. First I want to ...
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1answer
66 views

If $\{X_n\}_{n\in\mathbb{N}}$ uniformly integrable, is $\{X_n(w)\}_{n\in\mathbb{N}}$ a bounded set for sample path $w$?

My question is: If $\{X_n\}_{n\in\mathbb{N}}$ represents a class of uniformly integrable random variables, is $\{X_n(w)\}_{n\in\mathbb{N}}$ a bounded set for a fixed sample path $w$? My thoughts: ...
2
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1answer
45 views

The limit of a quotient of integral moments

I have the following quotient $$ \frac{\int_k^\infty x^1 \ f(x) dx}{\int_k^\infty x^{1+\delta} \ f(x) dx} $$ Where $f(x)$ is bounded and $\delta>0$. I am trying to prove that as $k\to\infty$ the ...
3
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1answer
140 views

$X_k$ uniformly integrable iff $\lim_{n \to \infty}\limsup_{k\to\infty}\mathbb E(\vert X_k \vert \mathbb 1_{\vert X_k \vert \ge n})=0$

A sequence $X_k$ of real random variables on $(\Omega,F,\mathbb P)$ is uniformly integrable iff all $X_k$ are integrable and $$\lim_{n \to \infty}\limsup_{k\to\infty}\mathbb E(\vert X_k \vert \mathbb ...
2
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2answers
37 views

Uniform integrability when $\forall g \in \mathcal{L}^1$ there is always an $|f_n|>|g|$

Suppose we have a sequence of measurable $f_n:X\rightarrow \mathbb{C}$ with the property that for every $g\in \mathcal{L}^1$, there is an $N$ such that $|f_N|>|g|$. Can such a sequence be ...