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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).

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0answers
15 views

Weak convergence improved by Morrey embedding

Let $u_n: [0,T]\times \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be a sequence with \begin{equation} u_n \rightharpoonup u \ \ \ \text{weakly star in } L^2(0,T;W^{1,\infty}(\mathbb{R}^3)) \end{equation} ...
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2answers
33 views

Weak* Convergence exercise

I'm dealing with this exercise about weak* convergence and I'm literally getting lost with indexes. I have this: Let $X := c_0(\mathbb{N}), \hspace{3mm}x_0 \in X^*=\ell^1(\mathbb{N}), \hspace{3mm}\{...
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0answers
12 views

Criterion of weak convergence of probability measures [on hold]

How to prove that $P_n \to P$(weak convergence)$\quad$ if and only if $\quad\forall A- $open set, $P(\partial A)=0 : P_n(A)\to P(A) ?$
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0answers
15 views

Nature of Distribution of a Random Vector whose finite dimensional distributions and distribution of inner products are known

I apologize if this question is vague or trivial. I have a random vector $\mathbf u$ in $\mathbb R^n$ and the following facts about it are true when $n\to \infty$: $u_i \to \mathcal N(0,1)$ in ...
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0answers
20 views

How to prove this result somewhat similar to Du Bois-Reymond's Lemma?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded connected set with smooth boundary. Suppose also that for each index $i, j = 1, \dots, n$, $f_{ij}$ is a smooth function. If for every $v\in C^...
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0answers
11 views

Clarification on the wording of a theorem.

I'm currently reading through van der Vaart and Wellner's Weak Convergence and Empirical Processes, and I am having a bit trouble with the statement of one of the theorems. In particular, after: 1.5....
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1answer
27 views

Does $u'_k \to v$ weakly imply $u_k \to u$ weakly with $u' = v$?

Let's assume $u'_k \to v$ weakly where $u_k \in L^2(\Omega)$ and $\Omega \subseteq \mathbb{R}^n$. Here $'$ denotes the weak derivative. Then, for an arbitrary $\phi \in C^\infty_c(\Omega)$, $\langle ...
1
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1answer
29 views

Weak convergence of $\frac{1}{n} \sum_{i = 1}^{n} \delta_{2 \cos(2 \pi k / n)}$ to $\frac{1}{\pi \sqrt{4 - x^2}} {\large\chi}_{|x|\leq 2} dx,$

My questions come from the pdf https://www.math.univ-toulouse.fr/~bordenave/coursSRG.pdf that I am reading. More specifically, on page 6, the author computes the spectral measure for finite graphs, ...
1
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1answer
14 views

Under what condition on the space X, any Continuous operator will be Completely continuous.

Categorise the spaces $X$ for which $B_{00}(X,X)=B(X,X)$, where $B(X,X)$ is the set of bounded linear operators and $ B_{00}(X,X)$ the set of completely continuous operators, i.e. operators which take ...
1
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1answer
19 views

Spectrum and Convergence of an operator

can you help me to solve this exercice? The first point is ok but I have problems with the others. Let $S_\varepsilon$ an continuous operator on $L^2(\mathbb{R})$ define as $$ S_\varepsilon[f](x)= \...
3
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0answers
35 views

Weak convergence on uniformly convex Banach spaces

I want to prove the following: Let X be a uniformly convex Banach space, $\{x_n\}_{n\in\mathbb{N}} \subset X, x \in X, \|x_n\| \to \|x\|$, $x_n$ converges weakly to $x$. Then $x_n$ also converges ...
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0answers
16 views

convex functions and weak convergence

Let$ f_n$ and $f$ be convex $2\pi-$periodic real functions such that : We have the uniforme convergence of $f_n$ to $f$ by Alexandrov's theorem $f_n$ and $f$ are twice differentiabl so we have ...
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0answers
27 views

Question on convergence in the weak topology of measures

Let $f_n \in W^{2,p}(\Gamma)$ ($\Gamma$ here is the $C^2-$regular boundary of a bounded, open and connected subset of $\mathbb R^3$) and suppose ${\vert \vert f_n \vert \vert}_{L^1} \le M \;\forall n$ ...
1
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1answer
21 views

Convergence in probability of ratio to 1 and convergence of a sequence to a random variable

Suppose $X_n\xrightarrow{d}X$ where is a positive random variable. Now if $X_n/Y_n\xrightarrow{p}1$ do we have that $Y_n\xrightarrow{d}X$? Are the following first steps valid: $P(Y_n<z)\leq P(|X_n-...
2
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1answer
27 views

Characterization of weak convergence in Hilbert $C^*$-modules?

Assume $M$ is a Hilbert $C^*$-module and $(x_n)^{\infty }_{n=1}$ a bounded sequence in $M$. Are these equivalence? $\langle x_n,y\rangle \to 0 $, for all $y\in M$. $(x_n)$ is convergent to $0$ in ...
2
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1answer
55 views

Convexity and strong lower semicontinuity imply weak lower semicontinuity

I have seen that if a set $K$ on an Hilbert space $H$ is convex and strongly sequentially-closed, it is weakly closed. The teacher said that if you take a convex and weakly lower semicontinuous ...
0
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1answer
17 views

Convergence of normal distributions with convergent mean and variance

Let $X_n$ be a sequence of random variables with $X_n \sim \mathscr{N}(m_k,\sigma^2_k)$. a) Assume that $m_k \rightarrow m$ and $\sigma^2_k \rightarrow \sigma^2 $. Show that $X_n \rightarrow^d \...
1
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1answer
46 views

Proving that a sequence converges weak-star in $L^\infty(\mathbb R)$

Let $f \in L^\infty(\mathbb R)$ be a periodic function with period $1$. How can one prove that the sequence $\{f_n\}_{n \in \mathbb N}$ of functions given by $$f_n(x)=f(nx)$$ converges weak-star in $L^...
3
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2answers
72 views

Question about a weak*-norm continuity of a linear operator

Let $X$ and $Y$ be infinite dimensional normed linear spaces and let $S:Y^*\to X^*$ be a one-one linear operator. I want to show that $S$ can not be weak*-norm continuous. My idea is to choose a ...
2
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1answer
33 views

Convergence weakly in Hilbert space.

Why $\lim_k \langle f_{n_k},e_j\rangle =\langle f,e_j\rangle $ with $f=\sum_{k=1}^{\infty} {a_k}^{k} e_k$? I dont understand the diagonal argument...
2
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0answers
23 views

About weakly convergence in Majda-Bertozzi.

I am studying the proof of the existence of local solutions of the Navier-Stokes equation, in Majda.Bertozzi book, page 109. Fixed $m$, I have a sequence $\{v^{\varepsilon}\}$ such that $v^{\...
1
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1answer
102 views

Can weak* convergence be upgraded to strong convergence in this situation?

$\newcommand{\mr}{\mathscr}$ Definitions. Let $X$ be a compact metric space and $\mr P(X)$ be the set of all the probability measures on $X$. We say that a sequence $(\nu_n)$ in $\mr P(X)$ converges ...
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1answer
18 views

Convergence in distribution of the sum of two dependent random variables

I have the following question about the limiting distribution of the sum of two random variables say $Z_n = X_n+Y_n.$ I know the following: Conditioned on $X_n,$ $Y_n$ has a CLT i.e., $$\mathbb P (...
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0answers
71 views

How is the weak law of large numbers applied here?

Let $d\in\mathbb N$ with $d>1$ $\lambda^d$ denote the Lebesuge measure on $\mathcal B\left(\mathbb R^d\right)$ $f\in C^2(\mathbb R)$ be positive and $$\pi(x):=\prod_{i=1}^df(x_i)\;\;\;\text{for }x\...
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0answers
15 views

the convergence of the first and second derivative of a sequance of bounded functions

Let $f_n$ be a sequence of continuos function defined as $f_n:\mathbb{S}^{n-1}\rightarrow \mathbb{R} $ such that $||f_n-f ||_{\infty} \rightarrow 0 $ as $n \rightarrow +\infty$ I want to prove ...
0
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1answer
21 views

Inner product of weakly convergent sequences in Hilbert space

Let $(H, \langle \cdot, \cdot \rangle)$ be a Hilbert space, and let $(u_n), (v_n)$ be two weakly convergent sequences in $H$ with limits $u$ and $v$ respectively, i.e. $\langle u_n, y \rangle \to \...
3
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0answers
36 views

Weak convergence and compact embeddings

Assume that $X,Y$ are Banach spaces, such that $X$ is compactly embedded into $Y$. Further, let $f_n: [0,T]\rightarrow X$ be a sequence of functions such that \begin{equation} \sup_{t \in [0,T]}|\...
2
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0answers
29 views

Joint convergence in distribution of independent random variables

Question: My approach: Fix any $x$ and $y$ in the codomains of $X$ and $Y$ respectively. $\mathbb{P}(X_n \leq x, Y_n \leq y) = \mathbb{P}(X_n \leq x)\mathbb{P}(Y_n \leq y) \rightarrow \mathbb{P}(...
2
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0answers
23 views

Exchangeable sequence and convergence

Let $(X_1,...,X_n)$ an exchangeable family of real random variables such that its empirical measure $\mu_n:=\frac{1}{n} \sum_{k=1}^n \delta_{X_k}$ converges to some probability measure $\mu$. The ...
1
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1answer
19 views

Weak convergence of discrete random variables, interchanging order of limits

The question says $\{X_n\}^{\infty}_{n=1}$ and $X$ are real valued discrete random variables supported on the integers. Given: $P(X_n = j) \Rightarrow P(X=j)$ for all integers $j$. We are required ...
1
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1answer
32 views

In what way is convergence in operator norm 'better' than other forms of convergence?

If we show a sequence of operators $T_n$ converges uniformly/in operator norm to some operator $T$ this implies the sequence converges strongly/pointwise, which in term means it converges weakly. I ...
3
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1answer
50 views

Convergence in measure and boundness implies weak convergence in $L^p$

Let $(f_n)_{n=1}^\infty$ be a sequence in $L^p ([0,1]), 1\leq p<\infty$. Suppose that $f_n\rightarrow f$ in measure and that $\sup\limits_{n\in\mathbb{N}} \|f_n\|<\infty$. Show that $f_n\...
4
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1answer
47 views

Convergence of integrals of all smooth functions implies weak convergence of measures

Let $\mu$ be a Borel probability measure on $\mathbb{R}^d$ and and let $(\mu_n : n \in \mathbb{N})$ be a sequence of such measures. Suppose that $\mu_n(f) \to \mu(f)$ for all smooth $f$ of compact ...
0
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0answers
25 views

$X_n$ is a.s. convergent to zero? Given $\sqrt{n}X_n\implies N(0,1)$ [duplicate]

Suppose that a random sequence $\sqrt{n}X_n$ weak converges to a standard normal distribution $N(0,1)$. Then $X_n$ is converges to 0 in L2 and convergent with probability.. I am wondering $X_n$ ...
1
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0answers
26 views

weak convergence and trace operator (Neumann boundary condition)

We have $f_k \rightharpoonup f$ in $L^2(0,T;W^{1,2}(\Omega))$, where the $f_k$ are said to satisfy the boundary condition $\nabla f_k \cdot\text{n}|_{\partial \Omega} = 0$ in the sense of traces. ...
3
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1answer
37 views

Weak convergence of measures implying almost sure convergence of random variables

Suppose $\mu,\mu_n$ are Borel probability measures on $\mathbb{R}$ with $\mu_n$ converging weakly to $\mu$. I am asked to find some probability space $(\Omega,\mathcal{F},\mathbb{P})$ and random ...
2
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2answers
17 views

A (sort of) converse to the monotone class theorem

In my notes we have the monotone class theorem which says that if a vector space of functions contains all the indicator functions of measurable sets and is closed under bounded monotone limits then ...
1
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1answer
57 views

Sequence of functionals in $ L^{\infty}(0, + \infty)$

For each $ n \in \mathbb{N} $ let $ T_n : L^{\infty}( (0, + \infty)) \to \mathbb{R} $ be defined as $$ T_n(f) = n \Biggl ( \int_0^1 x^n f(x) dx + \int_1^{+ \infty}e^{-nx}f(x)dx \Biggr )$$ Does there ...
2
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1answer
36 views

Approximating states of the enveloping von Neumann Algebra

Let $A$ be a $C^*$-Algebra and $A''$ its enveloping von Neumann Algebra. Is the state space $S(A)$ of $A$ weak*-dense in the state space of $S(A'')$? I Know that every state on $A$ extends as a vector-...
2
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1answer
44 views

Compactness / sequentially compact / reflexive space / separable spaces / weak topology

I'm a bit confuse with all theses notions. Let $E$ a normed vector space of infinite dimension (also Banach, but it's probably not important). The theorem of Eberlin Smulian theorem says that : all ...
0
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1answer
48 views

Approximating integrals of continuous functions.

Let $(E,d)$ be a complete and separable metric space. Let $\mu$ be a probability measure on $E$. Let $f \colon E \rightarrow \mathbb{R}$ be continuous with $f \geq 0$. I want to find continuous and ...
0
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1answer
55 views

product of weakly convergent sequences [closed]

Let $\Omega$ be a bounded subset of $\mathbb{R}^n$. Let $f_\epsilon, g_\epsilon, f, f^*$ and $g$ be real-valued functions. Suppose that $f_\epsilon\left(x\right) \rightharpoonup f(x)$ in $L^p(\Omega)$ ...
4
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1answer
62 views

Portmanteau Theorem in Weak Convergence ( Lipschitz functions and lim sup / lim inf )

I have recently embarked on an endeavor to understand weak convergence and, consequently, have stumbled across the Portmanteau theorem. As I am not a mathematician, I presume that my questions are ...
4
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1answer
63 views

Proving convergence in probability.

Let $Y_k$ be independently Bernoulli distributed rvs holding $P(Y_k = 1) = \frac{1}{k}$ for every $k = 1,2,...,n$. I need to prove that the sum $\frac{1}{\log n} \sum_{k=1}^n Y_k$ converges in ...
2
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1answer
32 views

Weak convergence(i.e convergence in distribution) of first order statistics # problem 1.1(ch 6) of “Intermediate course in Probability” by Allan Gut

For each $n = 1, 2, ....$, suppose that $X_n$ is a continuous random variable with density $$\hspace{10mm}\mathrm{f}(x) = \begin{cases} \frac{1}{2}(1+x)e^{-x}, & \text{if $x \ge 0$ } \\[2ex] 0, &...
0
votes
1answer
26 views

Why a set is weakly sequentially precompact set?

I'm struggling to understand the following assertion: Since $\lim_{t\rightarrow \infty}\|x(t)-y_0\|$ exists, $\{x(t):t\geq 0\}$ is weakly sequentially precompact. Please give thorough explanations ...
1
vote
2answers
34 views

Why is there no subsequence of $\frac{1}{n}\boldsymbol 1_{[0,n]}(x)$ has no weak subsequence that converge weakly in $L^1$?

Let $f_n(x)=\frac{1}{n}\boldsymbol 1_{[0,n]}(x)$. This sequence is bounded in $L^1(\mathbb R)$ since $\|f_n\|_{L^1}=1$. But why is there no subsequence that convergent weakly ? I know that if such ...
3
votes
2answers
98 views

Weak convergence implies strong convergence in $L^1$ for Fourier series?

We say $\{f_n\}$ weakly converge to $f$ in $L^1[-π,π]$ if for each $g \in L^\infty[-π,π]$, $$\lim_{n\to\infty}\int_{-π}^{π}f_n(x)g(x)dx=\int_{-π}^{π}f(x)g(x)dx.$$ There is a question in my homework ...
0
votes
1answer
23 views

Convergence in measure of a bounded sequence in $L^{2}[0, 1]$ implies weak convergence

Suppose a sequence $\{f_{n} \}$ of functions in $L^{2}[0, 1]$ converges in measure to $f$, and furthermore, assume there exists constant $K$ such that $||f_{n}|| \leq K$ for all $n$. Show that $\{f_{n}...
1
vote
0answers
10 views

Asymptotic behavior of the empirical mean of weakly convergent processes.

Let $M,n \in \mathbb{N}$. For $i \in \{0,\dots,M\}$, let $(X_{n,i})_{n\geq 1}$ be $M$ independent sequences of real valued random variables, each weakly converging to a standard gaussian, i.e. $X_{n,i}...