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Questions tagged [weak-convergence]

For questions about weak convergence, which can concern sequences in normed/ topological vectors spaces, or sequences of measures. Please use other tags like (tag: functional-analysis) or (tag: probability-theory).

3
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1answer
27 views

Question on Banach-Alaoglu theorem: Bounded subset of a set contained in the dual space

So the Banach-Alaoglu theorem states: Let $X$ be the dual space to some Banach separable space $Z$, i.e $X=Z^*$. Take $M$ a bounded subset of $X$. Then any sequence in $M$ has a weak-* ...
2
votes
2answers
25 views

A weakly convergent sequence in a compact set, is strongly convegnet

Let $E$ be a Banach space, and $K \subset E$, compact set for the strong topology. And let $(x_n)_n$ converges for the weak topology $\sigma(E,E^*)$ to $x$. Why $(x_n)_n$ converges for the strong ...
1
vote
1answer
26 views

$X_n$ ~ Poisson(n), $Y_n$ ~ Geometric($e^{-\frac{1}{n}})$

Let $X_n$ ~ Poisson(n), $Y_n$ ~ Geometric($e^{-\frac{1}{n}})$, everything independent. I want to find the convergence in law of: $$ Z_n = \frac{1}{n}X_n + \beta Y_n $$ With $\beta \in \mathbb{R}$. ...
1
vote
1answer
21 views

Does this conclude that $ \int_E f_jg_j dx \to \int_E fgdx$?

Suppose $f_j \to f$ strongly in $L^2(E)$ and $g_j \to g \ $ weakly in $L^2(E)$, where $E \subset \mathbb{R}^d$ is measurable. Show that $ \ \int_Ef_jg_jdx \to \int_E fgdx$. Answer: I am quoting the ...
0
votes
1answer
19 views

Showing that the projection of a measure on path space onto marginals is continuous

Let $C([0,T], \mathbb{R}^d)$ be the metric space of continuous functions from $[0,T]$ to $\mathbb{R}^d$, endowed with the supremum metric. For any metric space $E$, let $\mathcal{P}(E)$ be the space ...
1
vote
1answer
42 views

Weak convergence of measures and of total variations

I have a sequence of signed, bounded measures $\{\mu_n\}_{n\in \mathbb N}$ and an open, bounded set $\Omega \subset \mathbb R^N$. I know that $\vert \mu_n \vert(\Omega) \to \vert \mu \vert (\Omega)$ (...
1
vote
1answer
19 views

Convergence of sequence of uniforms

Let $X\sim\mathrm{ Uniform}(0,1)$. Consider the sequence $X_n = X^n$. I want to study the convergence in law of this sequence. I did it using the distribution function, I have: $$F_X(x) = x \mathbb{...
1
vote
1answer
47 views

Prove that the weak convergence in $L^p \left(\mathbb{R}^N\right)$ does not imply the weak convergence of the modulus

Let be $p\in [1,\infty)$ and $\{u_n\}\subset L^p\left(\mathbb{R}^N\right)$ a sequence. How do I prove that $u_n \rightharpoonup u$ in $L^p \left(\mathbb{R}^N\right) \not\Longrightarrow|u_n|\...
0
votes
1answer
15 views

Let $X_n$ be a uniform distribution on $(-1,1)$. Let$ Y_n$ ~ Cauchy(0,1). Everything independent.

Let $X_n$ be a uniform distribution on $(-1,1)$. Let$ Y_n$ ~ Cauchy(0,1). Everything independent. Let $Z_n$ = $X_n$ + $Y_n$ I want to study the law convergence of the sample mean of $Z_n$. That is: ...
0
votes
2answers
23 views

Weak Star Convergence of $u_n=\sin(\frac{nx}{n+1})(1+\exp{(-n|y|)})$

Consider the sequence \begin{equation}u_n=\sin(\frac{nx}{n+1})(1+\exp{(-n|y|)}) \text{ where } (x,y)\in I=[-1,1]\times[-1,1], n\in\mathbb{N}.\end{equation} (a) Study the equicontinuity of $(u_n)$ ...
2
votes
1answer
51 views

If $T_n\to T$ strongly and $x_n\to x$ weakly, must $T_nx_n\to Tx$ weakly?

Let $X$ and $Y$ be Banach spaces, and let $\{T_n\}\subset L(X,Y)$, where $L(X,Y)$ denotes the space of bounded linear operators from X to Y.If $T_n\to T$ strongly and $x_n\to x$ weakly, must $T_nx_n\...
2
votes
1answer
96 views

Compact operators and weak convergence

Let $X$ and $Y$ be Banach spaces. (a) Let $T \in \mathcal{L}(X, Y )$. For each sequence $(x_n)_{n \geq 1}$ in $X$ and each $x \in X$, show that $x_n →x$ weakly, as $n \rightarrow \infty$ ,implies ...
1
vote
1answer
13 views

A clarification on Stable Convergence of Triangular Arrays of Random Variables

Consider the following important theorem, due to Jacod (1997), on stable convergence of triangular arrays of random variables. In what follows $\mathcal{F}_t$ indicates a filtration on $[0,1]$ and $t_{...
2
votes
1answer
60 views

Compactness of a specific set in weak topology

I have the following question: Let $E$ be a polish space (that is, a topological space, which is separable and metrizable, such that $E$ would be complete if equipped with this metric). Consider the ...
0
votes
1answer
26 views

Boundedness and Strong convergence

$f_n\rightarrow f$ in $L^2(0,1)$, $\{ f,f_1,f_2,\ldots \}\subset H^1(0,1)$, $||f_n||_{H^1(0,1)}\leq M,\ \forall n\geq 1 $, Is is true that $f_n\rightharpoonup f$ in $H^1(0,1)$? If not, then what is a ...
2
votes
1answer
48 views

Weak convergence of bounded sequence $(x_n)$ in Hilbert space where $\langle{x_n,y\rangle}\rightarrow \langle{x_n,y\rangle}$ for all $y\in D\subset H$

Let $H$ be a Hilbert Space endowed with the inner product $\langle{.,.\rangle}$ and $D$ a subset of $H$ such that span$(D)$ is dense in $H$. Show that, given a bounded sequence $(x_n)$ in $H$, such ...
3
votes
0answers
52 views

Show convergence of inner product of an operator and a weak convergent sequence

Let $(u_n)_{n\in\mathbb{N}}\subseteq (W^{1,p}_0(\Omega))$, $u_n \rightharpoonup u \in (W^{1,p}_0(\Omega))$ and $B$ be an operator on $(W^{1,p}_0(\Omega))\times(W^{1,p}_0(\Omega))$. B is defined ...
1
vote
2answers
16 views

A Question about Weak Convergence of a Ratio

I have a question regarding convergence in distribution, which arose during one of my research works. First, let me describe the setup a bit: $\textbf{Description of the Problem}$: Suppose that $X_n$...
2
votes
2answers
53 views

Bounded sequence in $L^\infty$ which converges in $L^1$

I have sequence in $L^\infty(\mathbb{R}^n)\cap L^1(\mathbb{R}^n)$ such that 1. $(u_n)_n$ is bounded in $L^\infty$ : there exists $a>0$ such that $\|u_n\|_{L^\infty}\leq a$. 2.$(u_n)_n$ converges (...
2
votes
0answers
26 views

Why are the finite-dimensional sets of $\mathbb{R}^{\infty}$ convergence-deterimining sets?

I'm examining proofs made on Page 12 of this document and in Billingsley's book Convergence of Probability Measures (first edition, page 12). I'm trying to extend this to an infinite product space ...
2
votes
1answer
30 views

Does Weak Convergence in $W^{1,2}$ imply weak convergence in $W^{1,4}$

Say I have a sequence $u_{n} \in W^{1,4}(\mathbb{T}^2)$, i.e $u^{2}_{n} \in W^{1,2}(\mathbb{T}^2)$. If $u^{2}_{n}$ converges weakly to $v$ in $W^{1,2}(\mathbb{T}^2)$, does $u_{n}$ converge weakly ...
0
votes
1answer
23 views

Compensated Compactness And Conservation laws

I am trying to understand Compensated compactness. I am new to this area. I have the following doubts to start with 1) I have been reading many books where its been written in differnet ways. So What ...
1
vote
1answer
23 views

Problem about weak convergence

I am reading a lecture note about Radon Riesz theorem, the resources is here: https://faculty.etsu.edu/gardnerr/5210/notes/Radon-Riesz.pdf At page 4, fourth line from the bottom, it says convergence ...
3
votes
1answer
39 views

About weak convergence in $L^{\infty}$

doing my homework I'm dealing with this: Let, for all $n\in \mathbb{N} \quad f_n(t) := e^{-nt^2}, \quad t \in [-1,1]$ Show that 1)$f_n \overset{\ast}{\rightharpoonup} 0$ in $L^\infty(-1,1)$ 2)$f_n$ ...
1
vote
0answers
35 views

Proving the weak closure of Sobolev Spaces,

I am trying to show the space $W^{1,4}[\mathbb{T}^{2}]$ is weakly closed in $W^{1,4}[(0,1)^{2}]$. $\bf{That ~~ is :} $ If $u_{n}\rightharpoonup u$. Where $u_{n}\in W^{1,4}[\mathbb{T}^{2}]$ and $u\...
0
votes
1answer
63 views

A simple proof of the Portmanteau Theorem

A premise. Let $X_n$ be a sequence of random variables and let $X$ be a random variable. Call $$ F_n(x)=\mathbb{P}[X_n\leq x] $$ and $$ F(x)=\mathbb{P}[X\leq x]. $$ I say that $X_n$ converges in ...
2
votes
0answers
29 views

weak* convergence of convolution between mollifiers and Radon measure

I've got a question concerning mollifiers. If $\Omega \subset \mathbb{R}^N$ is open and $\mu = (\mu_1,..., \mu_m)$ is a Radon measure in $\Omega$. Let $(\rho_{\epsilon})_{\epsilon > 0}$ be a family ...
0
votes
1answer
33 views

Point-wise convergence but not weakly

Let $X$ be a reflexive Banach space and $x_n\in C(0,\tau;X)$ be a bounded sequence. We know that a subsequence of $x_n$, denote it by the same symbol, converges weakly to $x$ in $L^2(0,\tau;X)$. Can ...
1
vote
0answers
25 views

Prove convergence in distribution.

We have real-valued random variables $\{X_n\}_{n=1}^\infty$, $\{Y_n\}_{n=1}^\infty$, $X$ and $Y$. $X_n \rightarrow X$ in distribution and $Y_n \rightarrow Y$ in distribution, respectively. Also, $X$ ...
2
votes
1answer
40 views

Prove weak-weak-continouity.

Let $p,q\in ]1,\infty[,$ $f:\mathbb{R}\to\mathbb{R}$ continuous, $|f(s)| \leq C|s|^{\frac{p}{q}}$, where $C>0$ is a constant. Let $A:\ell^p \to \ell^q, (x_n)_{n\in\mathbb{N}}\mapsto (a_n+f(x_n)...
3
votes
0answers
30 views

Characterization of Wasserstein convergence

Let $(X,d)$ be a complete metric space and define $$\mathcal{P}_2(X) := \{ \mu \text{ Borel probability measure} \mid \int_X d^2(x,x_0) d\mu(x) < \infty \text{ for some } x_0 \in X \}$$ endowed ...
2
votes
1answer
20 views

Weak cauchy operator sequence over hilbert spaces

Supose that $X$ and $Y$ are two Hilbert spaces and $(T_n)$ a weakly Cauchy sequence. That means for all $x \in X$ and $y \in Y$, $(\langle x, T_ny \rangle )_{n \in \mathbb{N}}$ is a Cauchy sequence in ...
1
vote
2answers
82 views

$A_1,A_2$ fulfill property, but their sum $A_1+A_2$ does not

Let $V$ be a real, reflexive, separable Banach space. Are there operators $A_1,A_2: V \to V^*$ that fulfill the property \begin{cases} u_n \rightharpoonup u \\ A_iu_n \rightharpoonup b \\ ...
1
vote
1answer
23 views

Proof verification: Tightness of a family of random variables converging in distribution.

I'm trying to solve the following exercise: Suppose that $X_n \to X$ in distribution. Show that $(X_n)_{n \geq 1}$ is a tight family. First, just to recall the definition for those possibly ...
0
votes
2answers
46 views

Does strong convergence in $H$ (or $L^2$) imply convergence in $V$ (or $W_0^{1,2}$)?

Suppose: $\mathbf{V}$ and $\mathbf{H}$ are Hilbert spaces. $\mathbf{V} \hookrightarrow \mathbf{H}$ is compact embedding. $\mathbf{V}$ is dense in $\mathbf{H}$. For example $\mathbf{V} = W_0^{1,2}(\...
1
vote
1answer
21 views

Showing weak convergence in $\sigma(L^p,L^{p'})$

I am solving the following exercise. I have followed the below solution for the first part, and it seems okay. Except for why (S2) and (S3) implies that $g_n\to f$ almost everywhere. We should not ...
1
vote
0answers
41 views

Convergence in distribution of $\frac{1}{n}max(X_n,Y_n)$, where $X_n\text{ }U(-n-5,4n-5)$, $Y_n\text{ }Poiss(10n)$

Convergence in distribution of $\frac{1}{n}max(X_n,Y_n)$, where $X_n\text{ is }U(-n-5,4n-5)$, $Y_n\text{ is }Poiss(10n)$. My idea is to look how $X_n, Y_n$ behave as n tends to infinity, so my first ...
2
votes
1answer
24 views

If $E[h(M_n)]\to h(x)$ for all bounded and continuous functions $h$ then $M_n\to x$ in probability

Let $M_n$ be a sequence of random variables and let $x\in\mathbb{R}$ such that $$\mathbb{E}[h(M_n)]\to h(x)$$ as $n\to\infty$ for all bounded and continuous functions $h$. How to show that for all ...
1
vote
0answers
38 views

Convergence in distribution of sum of random, independent variables.

There is a sequence $(X_n)_{n≥1}$ of independent random variables, where for n ≥ 1 the distribution for $X_n$ is given $$P (X_n = 0) = \frac{1}{n}$$ $$P(X_n = 2n) = 1 −\frac{1}{n}$$ Examine if the sum ...
1
vote
2answers
78 views

Why weak convergence and a.e. convergence imply the convergence of this integral?

In the proof of the Brezis-Nirenberg theorem on the book "Nonlinear Analysis and Semilinear Elliptic Problems" by Ambrosetti and Malchiodi, it is used lemma 11.11 at pag. 183, in whose proof we have a ...
1
vote
1answer
68 views

Central limit theorem for sequence of Gamma-distributed random variables.

Suppose that $X_ n \sim \text {Gamma}\ (n\alpha , \lambda)$ for all $n \ge 1$, for fixed $\alpha,\lambda >0.$ Show that $$\frac {1} {\sqrt n} \left (X_n - \frac {n \alpha} {\lambda} \right ) \...
1
vote
0answers
20 views

Push forward of measures and weak* converence

I have to prove the following: Proposition: Let $$\mu := \mathcal{L}^1 \big|_{[0,1]} $$ and $1<p< \infty$. Consider the sequence of functions $ \{f_h \}_{h>0} \subset L^p(\mathbb{R}, \...
1
vote
1answer
37 views

Central Limit Theorem - Different Forms

Given the following function: $W_n = \frac{1}{\sqrt{n}}\Pi_{k=1}^{\infty}\log(U_k)$ where $U_k$ is uniformly distributed from $1$ to $e$. Does $\{W_n\}_{n\geq 1}$ converge in distribution? I found ...
2
votes
1answer
24 views

Convergence of third moments when second moments are uniformly bounded.

Suppose $\lambda_n, n\geq 1 $ are $\sigma $-finite measures on $\mathbb{R}$ such that $$ \sup_{n\geq 1} \int_{[-1,1]}x^2 \lambda_n(dx) < \infty $$ and $$\forall \delta > 0, \exists n_0 \geq 1, \...
0
votes
1answer
31 views

Weak convergence in closed subspace of a Hilbert Space

I get stuck in following problems Let $(x_n)_n \subset Y$ be such that $x_n \xrightarrow{w} x$ (converges weakly), and Y a closed subspace of a Hilbert Space. Show that $x \in Y$ My try I didn't ...
1
vote
1answer
54 views

Weak convergence and pointwise convergence of norm

Assume that $f_n\rightharpoonup f$ weakly in some $L^p$ space, and $|f_n|\rightarrow|f|$ pointwise. Does this imply that $f_n\rightarrow f$ pointwise? (or a subsequence?)
1
vote
1answer
26 views

Prove $\{f\in L^\infty : \|f\|_\infty \leq 1- \epsilon\}$ for $\epsilon\in (0, 1)$ is $w^*$- closed

I want to prove the following statement Prove $\{f\in L^\infty : \|f\|_\infty \leq 1- \epsilon\}$ for $\epsilon\in (0, 1)$ is $w^*$- closed I've started: suppose $f_n \xrightarrow{w^*}f$ so tht ...
1
vote
0answers
36 views

Rate of convergence in distribution

Imagine a sequence of random variables $\{X_n\}_{n=1}^\infty$ in [0,1] converging pointwise (i.e, sure convergence) to the random variable $X$, with convergence factor $\lambda$: $$ |X_n(\omega)-X(\...
0
votes
0answers
25 views

$\sqrt{n}(x_n-x_0) \to N(0,\sigma_0^2)$ in distribution implies $x_n \to x_0$ almost surely

Is this true? I figured that if not, there will some positive probability $\sigma$ that $\sqrt{n}(x_n-x_0)$ takes $\sqrt{M} \cdot \epsilon$ for infinitely many large $M$. Even though this "blowing up" ...
1
vote
0answers
21 views

An example of weakly convergent probability measures that do not form a family of relatively compact probability measures.

We have space {$C_{[0,1]},\mathfrak B(C_{[0,1]}) $}. What is an example of weakly convergent probability measures, that don't form a family of relatively compact probability measures ?$$$$ $C_{[0,1]}$-...