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

learn more… | top users | synonyms

2
votes
2answers
60 views

Uniform $L^p$ bound on finite measure implies uniform integrability

Suppose that $X$ have finite measure, let $1<p<\infty$, and suppose that $f_n : X\rightarrow \mathbb{R}$ is a sequence of measurable functions such that $\sup_n \int_X |f_n|^p d\mu < \infty$. ...
0
votes
1answer
56 views

Sample uniform direction within cone

My question is pretty much the same as this question below, however I came up with a potential solution to this problem that I didn't see an answer to in the other question and I was wondering if it ...
2
votes
0answers
31 views

How is this the definition of equi-integrable?

Let $Q=(0,T)\times\Omega.$ I am completely lost with this: No definition of equi-integrability I have seen looks anything like this. Can someone help me please? Presumably it is a fact that a ...
-1
votes
1answer
78 views

Convergence of expectations under almost everywhere convergence and uniform integrability

Prove that if a sequence of random variables $X_n$ converges in distribution to $X$, and if the $X_n$ are uniformly integrable (UI), then $$ \lim_{n\rightarrow\infty} E[X_n] = E[X]. $$ Can you ...
1
vote
1answer
34 views

Equivalent condition for equi-integrability

I am looking for a Lemma that gives an equivalent formulation for a family of functions to be equi-integrable: is it true that if $\{f_j\}_j\in L^1$, then we can write $f_j=f^1_j+f_j^2\in L^1+L^p$, ...
0
votes
1answer
37 views

Optional Sampling a.s. finite stopping time

Given a uniformly integrable discrete martingale $M_n$ on prob. space $(\Omega, \mathcal{F}, \mathbb{P})$, and a.s. finite stopping times $T$ and $S$ with $T\geq S$. Show that $E[M_T|\mathcal{F}_S] = ...
1
vote
1answer
38 views

Show the equi-integrability of a finite set of $\mathcal{L}_{\mu}^1$-functions

Let $(\Omega,\mathcal{A},\mu$ be a measure space. A set $\mathcal{F}$ of measurable, numerical functions is called equi-integrable if for any $\varepsilon > 0$ it exists a nonnegative, ...
2
votes
1answer
39 views

Is the product of two independent uniform integrable random variable is uniform integrable?

Is the product of two independent uniform integrable random variable is uniform integrable? What is the role independence plays here?
3
votes
1answer
47 views

Is a compensated Poisson process uniformly integrable

Let $(N_t)_t$ be a Poisson process with intensity $\lambda$. Define $$ \bar{N}_t = N_t - \lambda t $$ which is clearly a martingale. My question is: is $\bar{N}$ uniformly integrable? I strongly ...
2
votes
0answers
103 views

Local martingale is locally uniformly integrable martingale?

Is a local martingale locally uniformly integrable martingale ? Here I define a local martingale to be the process with a localizing sequence $\tau_n$ such that the stopped process is martingale. ...
2
votes
2answers
118 views

Does uniform integrability plus convergence in measure imply convergence in $L^1$?

Does uniform integrability plus convergence in measure imply convergence in $L^1$? I know this holds on a probability space. Does it hold on a general measure space? I have tried googling. It ...
5
votes
1answer
104 views

Uniform integrability (show an equivalence)

Let $(\Omega,\mathcal{A},\mu)$ be a measurable space and $\mathcal{F}$ a set of measurable functions. Show: If $\mu(\Omega)<\infty$, $\mathcal{F}$ is uniformly integrable exactly then, ...
2
votes
1answer
63 views

If $P(X=k)=1/2^k$ for every $k\ge1$ and $h_k(x)$ converges to identity function $h(x)=x$, does $E(X|h_k(X))$ converge almost surely?

$P(X=k)=1/2^k$,k=1,2,3... $h_k(x)$ converges to identity function,i.e $h(x)=x$. The question is that whether $E(X|h_k(X))$ converges almost surely. If true, how to prove it? If not, please give an ...
2
votes
1answer
83 views

Uniform integrability of a set of measurable functions (show an equivalence)

We call a set $\mathcal{F}$ of measurable functions uniformly integrable if for any $\varepsilon >0$ it exists a non-negative, integrable function $h$ so that $$ \sup_{f\in\mathcal{F}}\int ...
1
vote
0answers
29 views

Convergence in mean with logarithm

The problem is to prove that $E[\log|X_1-\bar{X}_n|]$ converges towards $E[\log|X_1|]$ for i.i.d. continuous random variables $X_1,\ldots,X_n$ with $E[X_i]=0$ and $Var[X_i]=1$, for example for Laplace ...
1
vote
1answer
73 views

Convergence in probability of the means of a uniformly integrable sequence

Suppose $\{X_n\}$ is a uniformly integrable sequence of independent random variables with zero mean. Prove that $1/n \sum\limits_{i=1}^n X_i \rightarrow0 $ in probability. I tried to ...
2
votes
1answer
176 views

Sequence of independent random variables: Convergence, martingales, uniform integrability

I am having some problems with the following exercise: Let $(Y_n ,n ≥ 1)$ be a sequence of independent random variables such that: $P(Y_n = e^n − 1) = e^{−n}$, $P(Y_n = −1) = 1 − e^{−n}$, $∀n ...
5
votes
3answers
74 views

The Probabilistic Method, Section 2.5. Unbalancing Lights

I am working through Alon's and Spencer's Probabilistic Method book. In Section 2.5, Unbalancing light, in the proof of Theorem 2.5.1, it is mentioned that $R_i$ has distribution $S_n$, the ...
1
vote
2answers
92 views

$\int_0^1|f_n|^3\leq 1\Rightarrow \int_E |f_n|<\varepsilon$ when $|E|$ is small

Let $f_n\colon [0,1]\to\mathbb{R}$ be Lebesgue measurable with $$\int_0^1|f_n|^3\leq 1 \mbox{ for all } n.$$ Show that for all $\varepsilon>0$ there exists $\delta>0$ so that if $E\subset[0,1]$ ...
2
votes
2answers
231 views

When does a Riemann sum converge uniformly?

Consider the series $$f_n=\sum_{k=0}^{n-1}\frac{\sin^2(x+\frac{k}{n})}{n}.$$ Since $\lim_{n\rightarrow\infty}f_n=\int_x^{x+1}\sin^2(y)dy$, we know that $f_n$ converges. And my question is: will the ...
1
vote
1answer
203 views

How to prove convergence in mean implies uniform integrability?

My class notes and wikipedia both say that $X_n \xrightarrow{L^1} X$ $\Leftrightarrow \; X_n \xrightarrow{P} X$ and $X_n$ are uniformly integrable. I am trying to work through the proof. I am able ...
2
votes
1answer
187 views

exercise on uniform integrability

I cannot figure out the following exercise: Let $F$ be the family of functions $f$ on $[0,1]$, each of which is (Lebesgue) integrable over $[0,1]$ and has $\int_a^b|f|\le b-a$ for all ...
1
vote
1answer
171 views

Uniform integrability of a backward submartingale

Let $\{\mathcal{F}_n\}_n$ be a decreasing sequence of sub-$\sigma$-fields of $\mathcal{F}$($\mathcal{F}_{n+1}\subset\mathcal{F}_n$) and let $\{X_n\}_n$ be a backward ...
3
votes
2answers
131 views

uniform integrability of a squared sum of iid variables

I'm trying to prove that if $X_i$ are independent, identically distributed random variables such that $E X_i = 0$ and $E X_i^2 < \infty$ then the sequence $\frac{(\sum_{i=1}^{n} X_i)^2}{n}$ is ...
1
vote
1answer
80 views

Uniform integrability of RV's

$\mathbf{Theorem}$: Let $Y \in \mathbb{L}_1$, then the RV $(\mathbb{E}[Y \mid \mathcal{F}], \mathcal{F} \subset \mathcal{A} \space \sigma\text{-algebra})$ are uniformly integrable. $\mathbf{Proof}$: ...
0
votes
1answer
61 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
150 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
118 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
0answers
107 views

Convergence In $L^{1}$ in the Strong Law of Large Numbers

I'm trying to prove that if $(X_n)_{n\geq 1}$ is uniformly integrable, then $X_n$ almost surely converging to $X$ implies $X_n$ converges to $X$ in $L^{1}$. How is this done? Generally speaking: ...
3
votes
2answers
148 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 ...
4
votes
1answer
276 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
60 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
175 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
vote
1answer
88 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
189 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
213 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
199 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
1answer
63 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
130 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
190 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
191 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
vote
0answers
55 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
votes
1answer
114 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
171 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
1k 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
votes
1answer
433 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 ...
4
votes
1answer
355 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 ...
8
votes
1answer
767 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
2k 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 ...