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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66 views

If $f$ us periodic and even, what I can conclude about of $\int f \;dx$?

Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be a periodic, even and differentiable function. If $L>0$ is the minimal period of $f$, what can I conclude about $$I :=\int_{0}^{L} f(x)\; dx?$$ By ...
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46 views

$f_\infty(t)=g_\infty(t)$ a.e?

Let $(E,\mathcal{A},\mu)$ be a finite measure space. Let $\{f_n\}$ and $\{g_n\}$ two uniformly integrable sequences such that: $$ \qquad\forall n\geq 1~:~ f_n(t)=g_n(t)~~ \text{ a.e.}\tag1 $$ $$ \...
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1answer
18 views

There exists a nonnull subset $A\in \mathcal{A}$ such that : $ \{f_n\}, \{g_n\}\text{ and }\{h_n\} \text{ are uniformly integrable on }A $?

Let $(E,\mathcal{A},\mu)$ be probability space. Lemma. Suppose $\{f_n\}$ is a sequence in $\mathcal{L}^1_\mathbb{R}$ such that $$ \sup_n\int_{E}|f_n|d\mu<+\infty. $$ Then there exists a ...
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Can we say that $\{f_n\}\text{ is uniformly integrable over }E\setminus (\cap_p B_p)$?

Let $(E,\mathcal{A},\mu)$ be probability space and $\{f_n\}$ be sequence of functions such that $$ \sup_n\int_{E}|f_n|d\mu<+\infty. $$ Let $\{B_p\}$ be a sequence non-increasing in $\mathcal{A}$...
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2answers
65 views

Showing $\{X_n\}$ is uniformly integrable when $\sup _{n} \mathbb{E}\left[X_{n}^{2}\right]<\infty$

I got a question that show that a family of rvs $\left\{X_{n}\right\}$ is uniformly integrable when $\sup _{n} \mathbb{E}\left[X_{n}^{2}\right]<\infty$ What I have tried: $$\sup_n\mathbb{E}[|X|] =...
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16 views

Why without loss of generality $\{f_n\}$ et $\{g_n\}$ are uniformly integrable over $A$?

Let $(E,\mathcal{A},\mu)$ be a probability space. We have the following lemma: Lemma : (biting lemma) Suppose $\{\phi_n\}$ is a sequence in $\mathcal{L}^1_{\mathbb{R}}$ such that $$ \sup_{n}{\int_{E}{...
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2answers
59 views

Proving uniformly integrablility of $\{X_n\}$, if $\sup_{n\geq 1} E[f(|X_n|)]<\infty.$

Suppose $\{X_n\}$ is a sequence for which there exists an increasing function $f:[0,\infty)\mapsto [0,\infty)$ such that $\lim\limits_{x\rightarrow\infty} f(x)/x= \infty$ and $$\sup_{n\geq 1} E[f(|...
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1answer
39 views

From $(1)$ and $(2)$, t $\{f_n\}$ has a subsequence equivalent to a uniformly integrable sequence.

Let $(E,\mathcal{A},\mu)$ be a finite measure space and $$ \mathcal{L}^1=\left\{f:E\to \mathbb{R}: \int_{E}{|f(t)|d\mu(t)}<\infty\right\} $$ Let $\{f_n\}\subset \mathcal{L}^1$, such that: $$ \sup_{...
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1answer
95 views

A question about proof of Martingale Convergence Theorem. Why does the Uniform integrability imply the following fact?

Relying on the below definition of uniform integrability: Definition: A subset $\mathcal{U}$ of $\mathcal{L}^{1}$ is said to be a uniformly integrable collection of random variables if \begin{...
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1answer
32 views

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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1answer
31 views

Uniform integrability: other definition

Let $(X_n)_n$ be a sequence of $L^1.$ Prove that $(X_n)_n$ is uniformly integrable if and only if $$\lim_{p \to +\infty}\limsup_n\int_{|X_n|>p}|X_n|dP=0.$$ the implication is easy since $$\...
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1answer
38 views

using DCT to prove Uniform Integrable

If X$_n$ is dominated by some Y in L$^1$ or if it is identically distributed with finite mean, then prove that X$_n$ is uniformly integrable. Approach: Start with dominated convergence theorem but ...
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2answers
54 views

Almost sure convergence implies uniform integrability (Submartingales)

Question: Assume $M_n \rightarrow M_{\infty}$ almost surely where the extended sequence $\{M_0,M_1,...,M_{\infty}\}$ is a submartingale. Show that $(M_n)_{n \ge 0}$ is uniformly integrable. That is $\...
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1answer
49 views

Prove that each sequence of a r.v. is uniformly integrable if $\sup\{ \mathbb{E}(|X_n|^{1+\delta})\}$ is finite

Show that each sequence $(X_n)$ is UI $\forall n \in \mathbb{N}$, if $\sup_{n}\{ \mathbb{E}(|X_n|^{1+\epsilon})\}$ is finite with respect to some probability measure and for positive $\epsilon$, ...
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1answer
44 views

Appling a density function to a sequence of a.s. converging UI random variables

A sequence of random variables $\{X_n\}$ in $[0,1]$ converges to a constant $c\in (0,1)$ almost surely. The sequence is also uniformly integrable. Let $p$ be a probability density function on $[0,1]$, ...
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1answer
14 views

How are these conditions in the definition of uniform integrability different from eachother

My lecture notes define uniform inegrability as follows A family $(X_i)_{i\in I}$ of real random variables is called uniformly integrable, if $\sup_{i\in I} E(|X_i|)<\infty$ $\sup_{i\in I}E\left(...
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1answer
31 views

$\sup L^1$ space with uniform integrability

I have a following question. For $t\in[0,T]$, let $f_t:\mathbb{R}\rightarrow\mathbb{R}$ be uniformly integrable family i.e. $\{f_t,\,t\in[0,T]\}$- uniformly integrable. We consider the space of such ...
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1answer
40 views

Uniform integrability of product of random variables

Let $\mathcal{Y}$ be a collection of random variables on a probability space $(\Omega,\mathcal{F},P)$ that is bounded in $L^p$-norm, $p \in (1,\infty)$. Fix $X \in L^q$ where $L^q$ is the dual space ...
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1answer
64 views

Is $Y_n := \prod_1^n \xi_i$ for $\xi_i$ i.i.d. $\text{Unif}(0,2)$ a sequence of uniformly integrable random variables?

Is $Y_n := \prod_1^n \xi_i$ for $\xi_i$ i.i.d. $\text{Unif}(0,2)$ a sequence of uniformly integrable random variables? I know that $\mathbb E[Y_n]=1$ for all $n$, and looking at the martingale $\{Y_n,...
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1answer
53 views

If $\mathbb E[\sup_{t\in [0,T]}|X_t|]<\infty$, is $(X_t)_{t\in [0,T]}$ uniformly integrable?

Let $(X_t)_{t\in [0,T]}$ a stochastic process. If $$\mathbb E\left[\sup_{t\in [0,T]}|X_t|\right]<\infty \,,$$ is $(X_t)_{t\in [0,T]}$ uniformly integrable ? For me it's uniformly integrable if ...
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1answer
109 views

Doob's Decomposition Theorem and Uniform Integrability

Let {$X_n$} be a uniformly integrable $(F_n)$-submartingale and $ \tau$ be the collection of all $F_n$-stopping times. Prove that {${X_T: T \in \tau}$} is uniformly integrable. I want to use Doob's ...
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1answer
97 views

Does there exist $\{X_n\}$ for which $\liminf_{n\to \infty}X_n$ doesn't exist but the negative parts $\{X^-_n\}$ are uniformly integrable?

Are there random variables $\{X_n\}_{n\ge 1}$ for which the expected value of $ \liminf\limits_{n\to \infty} X_n $ doesn't exist but the negative parts $\{X^-_n\}_{n\ge 1}$ are uniformly integrable? ...
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1answer
57 views

Uniform integrability for a Martingale (defined via Poisson process)

I'm trying to solve the following exercise: Let us consider the martingale $M_t = (N_t - \theta t)^2 - \theta t$. where $(N_t)_t$ is a homogeneous Poisson process with parameter $\theta$. Decide ...
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1answer
34 views

Almost surely finite stopping time and the limit of a martingale

I am working on this exercise: Let $(X_{n},\mathcal{F}_{n})$ be a martingale and $\tau$ a $\mathcal{F}_{n}$ stopping time that is almost surely finite. Further assume that $\mathbb{E}|X_{\tau}|<\...
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1answer
46 views

Equivalent Definition of Uniformly Integrable

This question is about the equivalent definition of uniformly integrable, which A possible uncountable collection of random variables $\{X_{\alpha}, \alpha\in I\}$ is said to be uniformly integrable ...
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61 views

Product of uniformly integrable functions

Let $S$ be a set of measurable functions $E \to [- \infty, \infty]$. We say that $S$ is uniformly integrable iff for every $\epsilon>0$ there exists a $\delta >0$ such that, for any $f \in S$ ...
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2answers
65 views

Karatzas and Shreve Problem 1.3.16. Proving nonnegative right continuous supermartingale convergence

This is Problem 3.16 from Chapter 1 of Karatzas and Shreve. Let $\{X_t, \mathscr{F}_t: 0 \le t < \infty\}$ be a right-continuous, nonnegative supermartingale; then $X_\infty(\omega) = \lim_{t ...
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1answer
50 views

Vallée Poussin's Theorem on Uniform Integrablity

I've started to read Rao's Theory of Orlicz Spaces book. There are two points I could not get well at proof of Vallée Poussin's uniform integrablity theorem. Please find the theorem above. The two ...
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1answer
40 views

Karatzas and Shreve solution to uniform integrability of backward martingale

Let $\{\mathscr{F}_n\}_{n=1}^\infty$ be a decreasing sequence of sub-$\sigma$-fields of $\mathscr{F}$, i.e. $\mathscr{F}_{n+1} \subset \mathscr{F}_n \subset \mathscr{F}$, and let $\{X_n, \mathscr{F}_n\...
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1answer
32 views

Show that this definition of uniform integrability implies the other one

Assume $\mathcal{F}$ is set of integrable functions which satifies $$\forall \varepsilon >0 \exists g_\varepsilon \forall f\in\mathcal{F}: ~ \int\limits_{[\vert f\vert>g_\varepsilon]}\vert f \...
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1answer
26 views

Show that : $\mathcal{X} +\mathcal {Y} =\{X+Y : X\in \mathcal{X} , Y\in\mathcal {Y} \}$ is uniformly integrable [closed]

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $\mathcal{X} $, $\mathcal {Y} $ be two uniformly-integrable families on the same probability space $\Omega $. Show that the ...
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1answer
164 views

Convergence in measure implies $L^1$ norm-(Vitali Convergence Theorem) [closed]

Let $(X, \nu)$ be a finite measure space and let $f_n$ be a sequence of integrable functions on $X$ converging in measure to a function $f$. Suppose that $\int |f_n|^p d\nu\leq A$ for some constant $A$...
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1answer
111 views

l1-bounded local martingale is uniformly integrable martingale

Let $M$ be a cadlag local martingale, i.e. there exists a sequence of stopping times $\{ \tau_k\} $ such that $\tau_k \to \infty$ and $M^{\tau_k}$ is a u.i. martingale for all $k$. Let $M_t$ be also $...
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68 views

Showing $E[X_n]\to E[X]$

Given $\{X_n\}_n$ are uniformly integrable non-negative random variables with $X_n\rightarrow X$ almost surely then show that $E[X_n]\to E[X]$ Uniform integrability implies $\sup_nE|X_n|<\infty$ ...
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1answer
37 views

Sufficient condition for uniform (Lebesgue) integrability

This exercise is from Bass's book on graduate real analysis. I am stuck in one question. Let $\mu$ be a finite measure and $\sup_n \int |f_n|^{1+\gamma} d\mu<\infty$ for any $\gamma > 0$. Then, ...
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1answer
69 views

Uniform integrability for a specific martingale

Suppose I have an i.i.d sequence of random variables $X_1,X_2 \ldots $ such that $\mathbb{P}(X_i =+1)=\mathbb{P}(X_i = -1)=0.5$. I need to prove that the random series $\sum_{k \geq 1} \frac{X_k}{k}$ ...
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1answer
45 views

Equivalence condition to uniform integrability

Please help me with this exercise: Let $\mathcal{F}$ be a family of functions, each of which is integrable over $E$. Show that $\mathcal{F}$ is uniformly integrable over $E$ if and only if for each $\...
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2answers
33 views

Does $E(X_{n} 1_{\vert X_{n} \vert \geq 1 })=\frac{1}{n}$ imply uniform integrability

Say $E(X_{n} 1_{\vert X_{n} \vert \geq 1 })=\frac{1}{n}$ for all $n \in \mathbb N$. Can I immediately deduce that $(X_{n})_{n}$ is uniformly integrable? My idea: $E(\vert X_{n}\vert)= E(\vert X_{n}...
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2answers
70 views

Determine if a sequence of functions is uniformly integrable

Let {$f_n$} be a sequence of continuous functions on $[0,1]$, and $|f_n|\le1$ on $[0,1]$. Is {$f_n$} uniformly integrable over $[0,1]$? For the same sequence, if we assume {$f_n$} is integrable and $\...
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1answer
38 views

Uniform integrability of random variables $~e^{(X_1+X_2+…+X_{n-1}+X_n - \frac{n}{2})}$

I have a task: $(X_n)$ are i.i.d. $$P(X_n=1)=P(X_n=-1)=\frac{1}{2}~.$$ Prove that $$Z_n=e^{(X_1+X_2+...+X_{n-1}+X_n - \frac{n}{2})}$$ is uniformly integrable. We have to prove that $$\lim_{b \...
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1answer
42 views

Equivalent definitions of uniform integrability

According to Wikipedia, A class $\mathcal{C}$ of random variables is $\textbf{uniformly integrable}$ if given $\epsilon > 0$, there exists $K \in [0. \infty)$ such that $\textbf{E}(|X|I_{|X| \...
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2answers
35 views

A question about a UI martingale with 2 stopping times

If $L$ and $M$ are stopping times with $L\leq M$ and $Y_{n\wedge M}$ is a uniformly integrable submartingale, then $EY_L\leq EY_M$ and $Y_L\leq E(Y_M|\mathcal{F}_L)$. Since $Y_{n\wedge M}$ is UI, $Y_{...
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2answers
65 views

Uniform convergence of expectation and probability

I have got a statement which seems really trivial to me, but I was not able to prove it, so maybe someone here can help me out. We consider a Family of random variables $(X_n)_{n\in\mathbb{N}}$ such ...
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1answer
45 views

Random probability measures and uniformly integrable functions

Let $\mu_n$, $\mu$ be random probability measures on a Polish space for all $n\geq 1$. Also let $m_n$, $m$ be the mean measures of $\mu_n$ and $\mu$, respectively - so for example $m(\cdot)=E[\mu(\...
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1answer
76 views

How do I proof a characterisation of uniform integrability?

The remark of interest is given in my measure-theory script but without a proof and I don't know how to proof it. I am using the following definition. Let in the following $(\Omega,\mathfrak{A},\mu)$ ...
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1answer
42 views

Why does this expectation tend to 0 for random variables converging in probability?

Suppose $(X_n)_n$ are identically distributed random variables with $\mathbb{E}(|X_1|^2)<\infty$. I have shown that $n^{-1/2}\max_{k\leq{n}}|X_k|\rightarrow0$ in probability as $n\rightarrow\infty$ ...
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1answer
66 views

Difficult example of Uniformly absolute continuous sequence but not $L^1$ bounded

Let $(f_n)_{n \in \mathbb N} \subset L^1( X, \Sigma, \mu )$ be a sequence of integrable functions with $\mu $ a finite measure. Suppose further that the sequence is uniformly absolutely continuous ...
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0answers
41 views

Proof: Every supermartingale is locally of class $(D)$

An adapted RCLL process $(X_t)_{t\leq T}$ is said to be of class $(D)$ is the family $$ \{ X_\tau \mid \tau\leq T \text{ is a stopping time}\}$$ is uniformly integrable. I've heard a lemma that ...
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1answer
39 views

Uniform integrability and convergence in probability [closed]

Let $X$ and $Y$ be two families of random variables, such that $X$ is uniformly-integrable and $Y \rightarrow c$ in probability to $c \in \mathbb{R}$. I have to show that for all $\epsilon >0, $ $$ ...
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1answer
35 views

Uniformly Integrability and Non-Tightness

I want to costruct a measure space $(X,\mathcal{F},\mu)$ and a $\mathcal{C}\subset\mathrm{m}\mathcal{F}$, where $\mathrm{m}\mathcal{F}$ be the set of extended real-valued measurable functions on $X$, ...

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