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

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1answer
21 views

Uniform integrability of the stopped compensated Poisson process

Let $N(t)$ be a Poisson process of rate $\lambda$ and consider the compensated Poisson process $$\bar{N}(t):= N(t) - \lambda t.$$ It was already shown in another post (Is a compensated Poisson ...
2
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3answers
47 views

Family of functions that are bounded in $L^1$ but *NOT* Uniformly Integrable

I'm having a difficult time constructing a counter example to this. My intuition (sloppily) is to construct a family of functions {$X_n$} that have Dirac pulses at $n$ and $-n$. Such that $\sup_n \Bbb ...
2
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1answer
66 views

Exponential of a uniform integrable martingale is a submartingale

For reference I want to prove this Lemma: Let $M$ be a uniformly integrable martingale with the additional property that $\mathbb{E}[ \exp(M_\infty)] < 1$. Then $\exp(M)$ is a uniformly ...
2
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1answer
135 views

Continuous time Uniform Integrability and convergence in L1 for (super)martingale

I need to prove the following theorem by reasoning as in discrete-time, since we have proven this theorem in discrete time. Theorem Let $M$ be a right-continuous supermartingale that is bounded in ...
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1answer
32 views

Uniform integrability of $\cos ^n\left(\frac{x}{\sqrt{n}}\right) 1_{|x| \leq c \sqrt{n}}$

I am trying to show uniform integrability of $\cos^n \left(\frac{x}{\sqrt{n}}\right) 1_{|x| \leq c \sqrt{n}}$, where $c$ is some positive constant. I was able to show that $\cos^n ...
4
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1answer
95 views

Uniform Integrbility

I need to show that if $|f_n|\leq g$, and $g$ is integrable, then $\{f_n\}$ is uniformly integrable, i.e., $\underset{a\rightarrow\infty}\lim\sup_n\int_{[|f_n|\geq a]} |f_n| d\mu=0$. Here is how I ...
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0answers
63 views

Continuity of $t\mapsto \mathbb E X_t$

Is the following statement and its proof correct? Do you know this or related results and where I could read more about such things? Lemma: If $X$ is a right-continuous process, and the collection ...
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1answer
89 views

Tightness and Uniform Integrability

I'm trying to develop some intuition for the concepts of tightness and uniform integrability, in a probabilistic context. (i) Does uniform integrability imply tightness? (ii) If not, is ...
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1answer
75 views

Prove that process is uniformly integrable

Let $(X_t)_{t\ge 0}$ be a stochastic process, and let $Y$ be an integrable random variable, such that $|X_t|\le Y$ for $t\ge0$. Prove that $(X_t)_{t\ge 0}$ is uniformly integrable. From ...
2
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1answer
55 views

Product of two uniformly square integrable random variables

Take two classes of uniformly square integrable random variables $\lbrace X_t:t\in T\rbrace$ and $\lbrace Y_t:t\in T\rbrace$. Is the class $\lbrace X_tY_t :t\in T\rbrace$ uniformly integrable? My ...
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0answers
61 views

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
151 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 ...
4
votes
1answer
275 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. ...
4
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1answer
74 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 ...
4
votes
1answer
87 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 $n \rightarrow \infty$. ...
2
votes
1answer
57 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 ...
2
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1answer
91 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
34 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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vote
1answer
115 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. ...
2
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1answer
178 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) ...
2
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1answer
116 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
189 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
220 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
188 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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0answers
122 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
85 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 ...
3
votes
1answer
49 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).
3
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1answer
159 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
115 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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0answers
98 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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votes
2answers
238 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
108 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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0answers
93 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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1answer
192 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
56 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
71 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
73 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
167 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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1answer
122 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 ...
3
votes
1answer
296 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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votes
2answers
217 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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votes
1answer
179 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
77 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
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1answer
155 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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0answers
54 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
118 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
248 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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votes
3answers
94 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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2answers
99 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
558 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 ...