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

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Uniform integrability of specific sequence of RV

I am investigating the following limit $$ \lim_{n \to \infty} E \left[ n \ln^-\left(1 - 2 \frac{\sigma}{n} [{\cal N}]_1 + \frac{\sigma^2}{n^2} \underbrace{ \| {\cal N} \|^2}_{\chi^2 \mbox{ ...
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1answer
45 views

Uniformly integrable martingales and stopping time

I want to prove the statement below: Theorem: Let $(Y_n,\mathfrak{F})$ be a uniformly integrable martingale. Show that $(Y_{T\wedge n},\mathfrak{F})$ is a uniformly integrable martingale for any ...
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2answers
72 views

A convergence question concerning the uniformly integrability

If $\{\xi_n\}$ is an uniformly integrable r.v.s., then $$\lim_{n\rightarrow\infty}E\left(\frac{1}{n}\sup_{1\leqslant k\leqslant n}|\xi_k|\right)=0.$$ Why?
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1answer
154 views

Convergence in $L^1$ and uniform integrability

Suppose that $X_n\leq Y_n\leq Z_n$ where $X_n\to X$, $Y_n\to Y$, $Z_n\to Z$ in probability. If $E(X_n)\to E(X)$ and $E(Z_n)\to E(Z)$, show that $E(Y_n)\to E(Y)$. I want to solve the task above. ...
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1answer
35 views

Stopped process of Brownian motion

I am baffled about the following problem: Let $(B_t)$ be a standard Brownian motion. Let $$ \tau:= \inf\{ t \geq 0 :B_t = x \} \wedge \inf\{ t \geq 0 :B_t = -y \}$$ be a stopping time, where $x,y ...
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1answer
47 views

Uniformly integrable martingale in a finite time horizon

Let $\{ M (t) \mid t \in [0,T] \}$ be a martingale and $\{ \tau_n \mid n = 1, 2, \ldots\}$ be an increasing sequence of stopping times such that $\tau_n \rightarrow \infty$ as $n \rightarrow \infty$. ...
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1answer
48 views

Proving a certain limit for uniformly integrable random variables

There is an interesting problem that has been resisting my efforts for a while. Assume that $\{X_n: n = 1, 2, \ldots\} $ is a sequence of uniformly integrable random variables. I would like to show ...
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1answer
77 views

If a family $\{f_n\}$, is uniformly integrable, then also $\{|f_n|\}$ is.

I want to show that if a set of functions $\{f_n\}_n$ is uniformly integrable, then also $\{|f_n|\}_n$ is also uniformly integrable. How can I show this? My guess is to use separate $f_n$ into real ...
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1answer
25 views

Compacting in Uniform Integration

If I have that $f$ is integrable, how show that for all $\varepsilon>0$, there is some $h>0$ for which we have that $$\int_{\{x\in X \colon |f(x)|<h\}} |f(x)|d m<\varepsilon$$ in a ...
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1answer
90 views

$\sup_{n \geq 0} E[|X_n|\ln(|X_n|)] < \infty$ implies that random variables $X_n$ are uniformly integrable

$X_n$ are uniformly integrable if $\lim_{R \rightarrow \infty} \sup_n E[|X_n|,|X_n| \geq R] = 0$. Show that if $\sup_{n \geq 0} E[|X_n|\ln(|X_n|)] < \infty$, then $X_n$ are uniformly integrable. ...
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1answer
100 views

Product of sequence of uniform integrable random variables are uniformly integrable?

If $\{X_i\}$ for $i \in I $ and $\{Y_j\}$ for $j \in J$ are uniformly integrable.Then prove that, $\{X_i+Y_j\}$ for $(i,j) \in I \times J$ is uniformly integrable.What about $ \{X_iY_j\} $ for $(i,j) ...
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1answer
101 views

Uniform integrability of a sequence of functions.

I am considering the following sequence of functions. I think it converges pointwise to $0$ because the intervals in the domain in which the nth function is greater than zero eventually shrink and ...
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1answer
166 views

How to understand uniform integrability?

From the definition to uniform integrability, I could not understand why "uniform" is used as qualifier. Can someone please enlighten me?
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2answers
189 views

Uniform integrability of a function in $L^1$

A collection of functions $(\phi_i)_{i\in I}\in L^1(\mu)$ is called uniformly integrable if given $\epsilon>0$ there exists $\delta>0$ such that : $$\int_E|\phi_i|d\mu<\epsilon~~~~\forall ...
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1answer
159 views

Conditions for convergence of moments

Let ${X_n}$ be a sequence of r.v. such that $X_n\xrightarrow [d]{}X$, with $E(X)$ finite, and with $E(|X_n|^{1+\delta})\leq K<\infty$ for all $n$. We know that: a) For $\delta>0$, we have ...
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110 views

Different definitions of uniform integrability

In different books and resources, I saw different definitions of uniformly integrable. For example, in some books the definition is like: Definition 1: $\{f_k\}\in L^1(E)$ is uniformly integrable ...
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1answer
75 views

If $u_n$ are equi-integrable and $f:\mathbb{R} \to \mathbb{R}$ is continuous, is $f(u_n)$ equi-integrable?

is it true that if $u_n$ are equi-integrable and $f:\mathbb{R} \to \mathbb{R}$ is continuous, is $f(u_n)$ equi-integrable? Here $u_n \in L^1(\Omega)$ where $\Omega$ is a finite measure space with ...
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1answer
48 views

Find a sequence of r.v's satisfying the following conditions

I think part a) can be solved by using $X_n=\frac{1}{n}\chi_{[0,n^2]}$ Not sure about part b).
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1answer
133 views

question about uniform integrability

Am I correct with usage of this generalised Dominated Convergence lemma: a sequence $(f_n)$ in $L^1$ on a bounded domain is strongly convergent if and only if $(f_n)$ is uniformly integrable and ...
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2answers
97 views

Uniformly integrable martingale

I have the following martingale. $M_n=\exp\left(aB_n-\frac{1}{2}a^2n\right)$ for $n\geq0$ and $a\neq0$, $B_n$ is a BM. I have to show that for $a>0$, $M_n\rightarrow0$ in probability. Is $M_n$ ...
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88 views

Uniform integrability of the maximum of a random walk with negative drift

Given $S_k^{(n)} = X_1^{(n)} + ... + X_k^{(n)}$ for all $k,n\in\mathbb{N}$, where the $X_i^{(n)}$'s are iid with mean $-\gamma$ for some $\gamma > 0$ and unit variance. Let ...
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2answers
203 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 $\sup_n \int_X |f_n|^p d\mu < \infty$. ...
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1answer
91 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 ...
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81 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 ...
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163 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 ...
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1answer
54 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 $f_j=f^1_j+f_j^2\in L^1+L^p$, ...
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1answer
60 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] = ...
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1answer
66 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, ...
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1answer
117 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?
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104 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 ...
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1answer
228 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. ...
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2answers
196 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 ...
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165 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, ...
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1answer
76 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 ...
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1answer
138 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 ...
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49 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 ...
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1answer
102 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 ...
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1answer
216 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 ...
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3answers
87 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 ...
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$\int_0^1|f_n|^3\leq 1\Rightarrow \int_E |f_n|<\varepsilon$ 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 $E\subset[0,1]$ ...
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457 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 ...
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1answer
418 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 ...
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1answer
248 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 ...
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1answer
252 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 ...
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225 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 ...
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1answer
107 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. $\mathbf{Proof}$: ...
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1answer
84 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 ...
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1answer
197 views

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

A bounded set $F\subseteq L_{1}(\mu)$ 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 ...
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1answer
261 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 ...
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141 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: ...