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

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Showing Uniform Integrability of Random variables

Let $X_1,...,X_n$ i.i.d random variables, square integrable, and with $E[X_1]=0$. Let $Y_n = \frac{|X_1 +...+X_n|}{\sqrt{n}}$ I am trying to show that $(Y_n)$ is uniformly integrable, i.e $\...
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21 views

What is the intuition behind Uniform Integrability?

A definition of Uniform Integrability I am currently working with is that: A sequence $X_1, X_2, \ldots$ of random variables is Uniformly Integrable if: $$ \sup_n \mathbb{E}\left(|X_n|\cdot \mathbb{...
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1answer
18 views

Uniform Integrability - different characterisation - prove (ii)

Probability with Martingales: For the 'only if' part assuming the hint is true, then I guess we have $\forall \varepsilon_1 > 0, \exists K \ge 0$ s.t. $$E[|X|1_{|X| > K}] < \...
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1answer
13 views

Uniform Integrability - different characterisation - prove hint

Probability with Martingales: For the 'only if' part how to prove the hint? i'm guessing it's something to do with $$E[X 1_F] \le E[X1_{\Omega}]$$ $$= E[X 1_{|X| > K}] + E[X 1_{|X| \le K}]...
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1answer
16 views

Uniform Integrability - sufficient condition and bounded convergence theorem with weaker hypothesis

Probability with Martingales: How does the result follow? Do we choose $K = (\frac{\varepsilon}{A})^{\frac{1}{1-p}}1_{A \ne 0}$ Why do we have that inequality?
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0answers
17 views

$\lim X_n = 0$ iff $b > 0$

Probability with Martingales: approach 1: Assuming $$\lim E[\exp\{aS_n - bn\}] = E[\lim \exp\{aS_n - bn\}]$$ I can't seem to be able to prove $$\lim E[\exp\{aS_n - bn\}] = 0$$ with just $b &...
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0answers
20 views

Why is ${|f_n-f|^p}$ uniformly integrable and tight iff {$|f_n|^p$} is uniformly integrable and tight ($f_n \rightarrow f$ pointwise)?

Why is ${|f_n-f|^p}$ uniformly integrable and tight iff {$|f_n|^p$} is uniformly integrable and tight ($f_n \rightarrow f$ pointwise)? This is from the last sentence in the proof in the following ...
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1answer
52 views

Is this family of functions uniformly integrable over $[0,1]$

Let $\mathcal F$ be a family of functions on $[0,1]$ each of which is integrable over $[0,1]$ and has $\int_a^b|f|\le b-a$ for all $[a,b] \subseteq [0,1]$. Is $\mathcal F$ uniformly integrable over $[...
3
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1answer
32 views

Doubt on proof of equivalance of conditions for uniform integrability in $L^1$

In a measure space with finite total measure, a family $A$ of r.v's is called U.I if $$ \lim_{N\to\infty}\sup_{X\in A} \int_{|X|>N} |X|\,d\mu=0 $$ I have some doubts on equivalance of this ...
3
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83 views

stochastic exponential uniformly integrable martingale

$N$ is a continuous local martingale and $T_c:=\inf\left\{t>0:\left[N\right]_t>c\right\}$, $c>0$ . I need to show that the stochastic exponential $\mathcal{E}(-N)$ is a uniformly integrable ...
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1answer
31 views

Uniform integrability of a sequence of random variables defined by a recursive relation

I have an i.i.d sequence $(u_j)_{j\in \mathbb{Z}_+}$ with zero mean and finite variance, say $\sigma^2$. Furthermore, I have another random variable $X_0$ (defined on the same probability space) which ...
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1answer
47 views

Uniform Integrability of Random Variables

$\{X_n\}$ is uniformly integrable if $\lim_{M \rightarrow \infty} (\sup_n \mathbb{E}(|X_n| \chi_{|X_n| > M}) = 0$ I would like to know if $\{X_i\}$ uniformly integrable $\implies \sup_n \mathbb{...
3
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1answer
44 views

Martingale convergence for UI martingales

I started reading this paper (Lamb, Charles W.. “Shorter Notes: A Short Proof of the Martingale Convergence Theorem”. Proceedings of the American Mathematical Society 38.1 (1973): 215–217) today. In ...
4
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1answer
53 views

Uniform integrability and stopping times

I want to know whether there is any example where $X_n$ is uniformly integrable, $N$ is a stopping time and $E[X_N] =\infty$? Or uniform integrability of $X_n$ implies that $E[X_N]< \infty$?
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33 views

Stopped process not uniformly integrable

I need to construct a counter example such that the process $\{X_n\}_{n \ge 1}$ is uniformly integrable; however, the stopped process $X_{\tau \wedge n}$ where $\tau$ is a stopping time, is NOT ...
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1answer
65 views

Counterexample for uniform integrability of a stopped process

I want to find an example where $X_n$ is uniformly integrable, $N$ is a stopping time, but $X_n^N = X_{\min\{n,N\}}$ in not uniformly integrable. There is a theorem saying that if $M_n$ is a uniform ...
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0answers
19 views

convergence of scores of log-concave densities

Given a sequence of log-concave densities $f_n = \exp\phi_n(x)$ converges uniformly in exponential weighting norm to some log-concave density $f_0 = \exp\phi_0(x)$, i.e. $\sup_x\exp a(x)|f_n(x)-f_0(...
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1answer
64 views

Counterexamples for expressions regarding uniform integrability

I have two questions regarding uniform integrability. 1- Is there any example such that $E[\sup_{n \geq 1} \vert X_n \vert] < \infty$ but $\sup_{n \geq1} E [\vert X_n \vert^p] =\infty$ for any $p ...
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1answer
48 views

Example for uniformly integrable $\mathbb{L}^2$-bounded sequences

How can I construct an example to show that the sequence $\{X_n\}_{n \ge 1}$ can be $\mathbb{L}^2$-bounded, however it has $\mathbb{E}\left[\sup\limits_{n\ge 1} |X_n|\right]=\infty$. Basically I need ...
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1answer
43 views

A uniformly integrable sequence that is not uniformly integrable in $\mathbb{L}^p$ for $p>1$

I need to construct an example to show that if for the sequence $\{X_n\}_{n \ge 1}$ we have that $\mathbb{E}\left[\sup\limits_{n \ge 1} |X_n|\right] < \infty$, the sequence in not necessarily ...
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1answer
39 views

Counterexample for uniform integrability of an $\mathbb{L}^1$-bounded sequence

I need to find an example such that $\sup\limits_{n \ge 1} \mathbb{E}[|X_n|] < \infty$ but $\{X_n\}_{n \ge 1}$ are not uniformly integrable. I can show that if $\{X_n\}_{n \ge 1}$ are uniformly ...
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1answer
66 views

Application of the theorem stating that “almost sure convergence plus uniform integrability implies $L^1$ convergence”

I'm confused on an application of the theorem stating that "almost sure convergence plus uniform integrability implies $L^1$ convergence". The example that follows is from van der Vaart "Asymptotic ...
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46 views

Stochastic processes closed under truncation

I'm currently studying some properties of general stochastic processes, and am having some issue understanding how to prove this (probably simple) example. First, let me introduce the notation & ...
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1answer
57 views

Uniform integrability of “exponent martingale”

Suppose that $X_n$ is an iid sequence of random variables with $P[X_i=1]=p$ and $P[X_i=-1]=q:=(1-p)$. Then, $S_n=X_1+\cdots+X_n$ (with $S_0=0$) is a simple random walk. We can easily check that $T_n=(...
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0answers
69 views

Uniform integrability of process with bounded conditional expectation

Let $[0, T]$ be a finite time horizon, i.e., $T < \infty$. Consider a complete filtered probability space $(\Omega, {\cal F}, {\mathbb F}, P)$, where ${\mathbb F} = \{ {\cal F}_t \}_{t \in [0, T]}$ ...
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1answer
55 views

Uniform Integrability and relation to $L^p$ for $p>1$

Let $X_n$ be a martingale. Then we know that for $p> 1$ the conditions $\sup_n E[|X_n|^p] < \infty$ and $E[\sup_n |X_n|^p] < \infty$ are equivalent. For $p=1$ this does not hold, because ...
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1answer
65 views

Conditions implying uniform integrability

We say that a family of random variables $X_n, n \geq 1$ is uniformly integrable if $$\lim_{M \rightarrow \infty} \sup_{n} E[|X_n| 1_{|X_n|>m}]=0.$$ I am struggling with some proofs and could ...
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2answers
52 views

Monotone convergence theorem without monotonicity

I have a question related to the monotone convergence theorem. Consider a situation in which all assumptions of the monotone convergence theorem are satisfied except the monotonicity, i.e. we have a ...
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1answer
73 views

Exchanging limits and Riemann Integral

Consider a function $f:\Theta \subseteq \mathbb{R}\rightarrow \mathbb{R}$ continuous at $\theta=\theta_0$ such that $f(\theta)\geq 0$ $\forall \theta \in \Theta$. Consider a sequence of real numbers $\...
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1answer
32 views

Uniform integrability of reciprocal of random variables

Let $\{X_n, n\geq 0 \}$ be a sequence of positive random variables that are uniformly integrable. Assume that $\frac{1}{X_n}$ is integrable. Then, is it true that $\left\{\frac{1}{X_n}, n \geq 0\...
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1answer
78 views

For a martingale $X$ does uniform integrability imply integrability of $\sup |X_{n}|$?

All is in the title: if $(X_{n})$ is a uniformly integrable martingale is it true that $\sup_{n\in \mathbb{N}} |X_{n}|$ is an integrable variable ? If I had to take a guess I'd say the answer is no, ...
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39 views

Uniform integrability of functional of an ergodic Markov Chain?

Suppose $\{X_n\}$ is an ergodic Markov Chain with general state space $\mathcal{X}$ and stationary distribution $\pi$. Suppose $\forall x\in\mathcal{X}, f(x)\geq 0$ and $\mathbb{E}_{\pi}[f(X)]<\...
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1answer
85 views

A Probabilistic Approach to Stirling's Formula

I am working on the following problem: Suppose $X_1, X_2,\dots$ are i.i.d Poisson$(1)$ random variables, and let $S_n=X_1+\dots+X_n$. a)Compute $E\left[\left( \frac{S_n-n}{\sqrt{n}} \right)^-\...
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23 views

Uniform convergence of 2-norm of a multinomial vector

Let $(X_1,X_2,\ldots,X_k)$ be distributed according to a multinomial distribution with parameters $(n;p_1,p_2,\ldots, p_k),$ i.e. $$P(X_1=n_1,\ldots,X_k=n_k) = {n\choose n_1,n_2,\ldots,n_k} p_1^{n_1}...
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1answer
58 views

Showing $|S_n/\sqrt{n}|^\alpha$ is uniformly integrable given certain conditions

Let $X_1, X_2, \dots$ be i.i.d. with $E(X_1)=0$ and $E(X_1^4)<\infty.$ Let $S_n=X_1+\dots +X_n$. Show that $|S_n/\sqrt{n}|^\alpha$ are uniformly integrable for any $0<\alpha<4$. I am trying ...
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53 views

Characterization of uniform Integrability

How can show that that $\lim_{k \to \infty} \sup_{i \in J} E[(|Y_i|-k)^+]=0$ implies uniform integrability of the set of r.v's $(Y_i)_{i \in J}$ I have spent quite some time ,unsuccessfully trying to ...
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1answer
85 views

$\sup\limits_{X \in \mathcal X, Y\in \mathcal Y}E[|Y| \mathbb 1_{\{ |X| \ge K\}}]\to0$ for $\mathcal X$, $\mathcal Y$ uniformly integrable

I want to show that $\lim_{k \to \infty } \sup_{X \in \mathcal{X}, Y\in \mathcal{Y}}E[|Y| \mathbb{1}_{\{ |X| \geq K\}}]=0$ It is given that the set of random variables $\mathcal{X}$ and $\mathcal{Y}$...
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1answer
34 views

$L^2$ convergence from convergence in distribution and uniform integrability

Is it true that if $X_n \to X$ in distribution and $\{X_n^2\}$ are uniformly integrable then $X_n \to X$ in $L^2$
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1answer
38 views

An example of a non trivial weakly compact in $L^1$

Let us consider the space $L^1$ of measurable functions associated to a probability space. I would like to see non trivial examples of weakly-compact sets. Thank you.
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1answer
47 views

Uniform integrability in central limit theorem

Suppose $X_1,X_2,\ldots$ are i.i.d. with $P(X_1=+1) = P(X_1=-1) = \frac 12.$ We know that $n^{-1/2}\sum_{i=1}^n X_i \stackrel{d}{\to} Z$ where $Z\sim\mathcal{N}(0,1).$ How steeply can a continuous ...
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1answer
96 views

Uniformly integrability and convergence

Question 1: $X_n$'s are non-negative, uniformly integrable. Then $E\left[\dfrac{\max_{1\leq k \leq n} X_k}{n}\right]\rightarrow 0$. Question 2: If u.i. is dropped then above the may fail. My thought:...
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1answer
86 views

Characterization of uniform integrability of random variables

Let $\{X_n \}$ be a sequence of random variables on a probability space $(\Omega,\mathcal{F},P)$. Then, $\{X_n\}$ is uniformly integrable if $$\lim_{M \to \infty} \sup_n \int_{|X_n| > M } |X_n| = ...
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2answers
58 views

Uniform integrability of stopped martingale

Let $(M_t,\mathcal{F}_t)_{t\geq 0}$ be a martingale with continuous paths and $(\tau_k)_{k\geq 0}$ stopping times. Hence we know that $M_{t\wedge\tau_k}=\mathbb{E}[M_t|\ \mathcal{F}_{t\wedge\tau_k}]$. ...
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1answer
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$E(f(|X_n|))$ property implies uniform integrability?

This is exercise 6.10 in Resnick's book "A Probability Path". We're given a sequence of random variables $(X_n)$ and an increasing function $f: [0, \infty) \rightarrow [0, \infty)$ such that $$ \...
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55 views

Collection is uniformly integrable, but individual is not integrable

Could you give me an example about: "a collection of functions that is uniformly integrable but each (or some) function in the collection is not integrable." This sounds counterintuitive? However ...
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1answer
76 views

Show that $(W_t)_{t \geq 0}$ and $(W_t^2-t)_{t \geq 0}$ are not uniformly integrable

I'm considering the following martingale $M_t:=W_t^2-t,\ t\geq 1$, where the $W_t$ is a Brownian motion. I want to prove that this martingale and the Brownian motion are not uniformly integrable. I ...
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1answer
66 views

Dominated convergence martingale and uniform integrability

For a fixed $t\in [0,1]$ I have a sequence $(X^t_n)_{n\geq 1}$ of normal distributed random variables which is a martingale and bounded in $L^2$. So by the martingale convergence theorem there exists ...
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1answer
27 views

$MN-\langle M,N \rangle$ is uniformly integrable when M, N are $H^2$?

This proof below really seems to have little to do with uniform integrability, what does the DCT application give exactly? Any alternative proofs would be good too
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39 views

Is this modified Dirichlet function integrable and/or uniformly integrable?

I was considering the following functions $X_n(x)$ , similar to the Dirichlet function: \begin{Bmatrix} q\: \; \textrm{when} \; x\:=\frac{p}{q} \in \mathbb{Q} \cap [0, \frac{1}{n}]& \; p \in \...
2
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1answer
52 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 ...