Tagged Questions

For questions about families of uniformly integrable random variables. Use the tags [tag: measure-theory] or [tag: probablity-theory].

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-4
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0answers
29 views

integrability of limits and integrals?

Suppose that $h_n$ is a sequence of non-negative integrable functions on $[a,b]$, such that $\lim_{n \to +\infty} \int_a^b h_n(x)dx = 0$. Show that if f is integrable on [a,b], then the $\lim_{n \to ...
0
votes
1answer
41 views

uniform integrability characterization

How to show the following: When a family of random variables $ \{X_n\}_{n \geq 1}$ is $L^p$ bounded for some $p > 1$ then $ \{X_n\}_{n \geq 1}$ is uniformly integrable. Also why does the above ...
1
vote
1answer
48 views

Uniformly Integrable of sets in $L_{1}(\mu)$ is equivalent to almost order boundedness

A bounded set $F\in L_{1}(\mu)$ is said to be uniformly integrable if : $\forall \epsilon$ there is a $\delta>0 $, such that $\forall$ measurable set $A$, and $\forall f\in F$ , if $\mu(A)< ...
2
votes
1answer
17 views

Equivalent condition of uniform integrability of a sequence of random variables

Here's the definition I have for a sequence of random variables to be uniformly integrable: $(1)$ A sequence of random variables $X_1, X_2, \ldots$ is uniformly integrable (U.I.) if for every ...
3
votes
2answers
63 views

Extracting a subsequence from a sequence of $\mathcal{L}^1$ functions

Any help with the following problem is appreciated. Given: a sequence of nonnegative functions $(g_n)$ which are U.I. (uniformly integrable) in $\mathcal{L}^1(0,1)$ with $\sup_n \Vert g_n \Vert_1 ...
3
votes
1answer
111 views

Martingale not uniformly integrable

I've come across a statement that implies that non-negative martingales for which $\{M_{\tau}\mid \tau \ \rm{stopping} \ \rm{time}\}$ is not uniformly integrable exist. I personally can't think of an ...
1
vote
1answer
54 views

Example for a sequence of functions in $\mathcal{L}^1[0,1]$

I am looking for an example of a sequence of functions $(g_n)$ that is in $\mathcal{L}^1[0,1]$ and U.I. so that the following three conditions are satisfied: $\forall \, n\, \, \vert g_n \vert ...
1
vote
1answer
55 views

In this Cumulative Distribution Function, am I finding the wrong term?

Question I was given: Let V be a uniform random variable distributed over the interval (0,1). Let $\ X = \frac{1}{\sqrt(U)}$. What is the cumulative distribution function and probability density ...
1
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1answer
62 views

Question about integration (related to uniform integrability)

Consider a probability space $( \Omega, \Sigma, \mu) $ (we could also consider a general measure space). Suppose $f: \Omega -> \mathbb{R}$ is integrable. Does this mean that $ \int |f| \chi(|f| ...
7
votes
2answers
148 views

Measure theory questions

i. If $1 < p < \infty$ and $E = \{f_a, a \in A\}$ set of measurable functions of $\mathbb{R}$ and $\sup_{a \in A} ||f_a||_p < \infty$, I want to show that for $ 0 < q < p$, $\lim ...
3
votes
1answer
77 views

A Question About Uniform Integrability

Proposition: Assume $E$ has finite measure. Let the sequence of functions $\{f_n\}$ be uniformly integrable over $E$. If $\{f_n\} \rightarrow f$ pointwise a.e. on $E$, then $f$ is integrable ...
1
vote
1answer
170 views

Show that if $f_n \leq g$ for all $n$ and $g$ is integrable, then $\{f_n\}$ is uniformly integrable

A sequence {$f_n$} of measurable functions is called uniformly integrable if $$\lim_{M \to \infty} \sup_{n} \int_{[|f| >= M]} |f_n|\ \mathrm{d}\mu = 0$$ Show that if $|f_n| \leq g$ for all $n$ ...
2
votes
0answers
39 views

Uniform integrability, book

I search about this theme, in the books is as exercise. But I want some more theory. What book recommend?
1
vote
1answer
52 views

Uniform integrability for a single random variable

Let $X$ be a random variable. Are the following three equivalent? $X \in L^1$, i.e. $E |X| < \infty$. $X$ is uniformly integrable. That is, if given $\epsilon>0$, there exists $K\in[0,\infty)$ ...
5
votes
1answer
95 views

Equi-integrability of a single function: is it the same as summability?

Let $(\Omega, \mathcal{M}, \mu)$ be a measure space and let $f\ge 0$ be a measurable function on $\Omega$. Suppose that $f$ satisfies the following properties: For all $\varepsilon > 0$ there ...
1
vote
1answer
99 views

Convergence in the absence of Dominated Convergence Theorem, and uniform integrability

This question is extended from Resnick's exercise 5.13 in his book A Probability Path. Let the probability space be the Lebesgue interval, that is, $(\Omega=[0,1],\mathcal{B}([0,1]),\lambda)$ and ...
1
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0answers
41 views

Question (3) on Uniform Integrability (simpler)

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^m$, so that $m(W)=1$. Consider $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$, $X \subseteq \mathbb{R}^n$ such that $\forall w \in W$ $\ ...
2
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1answer
73 views

Uniform Integrability after composition

Let $\mu$ be a finite measure on $X \subseteq \mathbb{R}^n$. Consider the Uniformly Integrable family $\{ f_n(\cdot) \}_{n \in \mathbb{N}}$ of functions $f_n : X \rightarrow \mathbb{R}_{\geq 0}$. ...
0
votes
2answers
154 views

Non uniformly integrable stochastic process if we consider a larger range

This is probably a stupid question, but am I mislead if I think that as soon as a stochastic process indexed by $t$ (continuous time) is not uniformly integrable (UI) for a certain range of $t$, say ...
4
votes
0answers
489 views

Dunford-Pettis Theorem

The Dunford-Pettis Theorem (see Uniform Integrability Wiki) states that: A class of random variables $X_n \in L^1(\mu)$ is Uniformly Integrable if and only if it is relatively weakly compact. Now ...
2
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1answer
183 views

An equivalent definition of uniform integrability

Let $(X,\mathcal{M},\mu)$ be a measure space and $\{f\}$ be a sequence of functions on $X$, each of which is integrable over $X$. Show that $\{f_n\}$ is uniformly integrable if and only if for each ...
5
votes
1answer
233 views

Uniform Integrability

Let $\mu$ be a probability measure on $X$, so that $\int_X \mu(dx) = 1$. I have a family $\{f_i\}_{i=1}^{\infty}$ of functions $f_i: X \rightarrow \mathbb{R}_{\geq 0}$ such that $$ \displaystyle ...
1
vote
0answers
261 views

showing that a sequence is uniformly integrable

I am currently reading the new edition of Royden and I've gotten to a part where the book made some comments without justification and I'm trying to verify these facts on my own. I want your help in ...
8
votes
1answer
444 views

I want to understand uniform integrability in terms of Lebesgue integration

According to my Real Analysis textbook, a family $\scr{F}$ of measurable functions on $E$ is said to be uniformly integrable over $E$ provided for each $\epsilon$ $>$ $0$, there is a $\delta$ ...
2
votes
4answers
1k views

Do convergence in distribution along with uniform integrability imply convergence in mean?

There are at least 2 places in Wikipedia saying that $X_n$ converges to $X$ in mean in and only if $X_n$ converges to $X$ in probability and $X_n$ is uniformly integrable. See the following link for ...