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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Weak Convergence of probability Measures

The definition of weak convergence for a set of borel probability measures $\{\mu_n\}$ is $$\int_{R}f\mu_n(dx)\longrightarrow \int_{R}f\mu(dx)$$ for all bounded continuous functions $f$. I am ...
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43 views

Prove convergence in distribution of sum of non-i.i.d random variables.

$X_1,X_2,X_3 ,\ldots$ be independent random variables with distribution $P(X_i=i)=P(X_i=-i)=1/2$ for all $i$. Define $S_n=X_1+X_2+X_3+\cdots+X_n$. And the question is to show "Does $\{S_n/n^p\}_{n=1}^...
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1answer
34 views

Convergence of a stochastic process

Suppose that $\{X_n(t),t\in\mathbb{R}\}_{n=0,1,...}$ is a collection of stochastic processes, i.e., for any fixed $n$, we have and stochastic process $\{X_n(t),t\in\mathbb{R}\}$. Assume that for any ...
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How to prove weak convergence in homogeneous Sobolev space?

Consider homogeneous Sobolev spaces $$\dot{H}^{s,p}= \{ f: (|\xi|^s\widehat{f})^{\vee} \in L^p\}.$$ Note Soboleve embedding $\dot{H}^{s,p}(\mathbb R^d) \hookrightarrow L^q(\mathbb R^d)$ for $\frac{...
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Real-world application where strong LLN is needed (weak LLN is not enough)

Do you know of any real world (algorithm, physics, ...) application of the law of large numbers where we need the strong LLN and the weak LLN by itself is not enough to prove that the application is ...
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24 views

What exactly is the relationship between weak* convergence and sequential convergence?

I am studying Banach algebras, and the theory frequently makes use of the weak* topology which I am not very familiar with. As I understand it, the weak* topology is defined as follows: let $X$ a ...
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Why weak-$*$ semicontinuity of a functional in $W^{1,\infty}$ implies ONLY qusiconvexity of the lagrangian in the vector valued case?

I am studying theorems regarding the basical equivalence of weakly lower semicontinuity of integral functional with convexity (or quasiconvexity) of the lagrangian. I will restrict to the case $W^{1, \...
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Convergence of moments implies convergence in total variation for Normal distribution?

I would appreciate some proof verification on the following statement concerning convergence in total variation for Normal Distributions, given convergence of the first and second moments. I suppose ...
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Weak convergence result in Levy's Continuity Theorem

I quote a part of Levy's Continuity Theorem and its proof. Theorem Let $\left(\mu_n\right)_{n\geq1}$ be a sequence of probability measures on $\mathbb{R}^d$, and let $\left(\hat{\mu}_n\right)...
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30 views

Weak convergence for $L^2$ on the torus

It is well-known that if we consider, for example, $L^2(\mathbb{R})$, we can pick any function $f\in L^2(\mathbb{R})$ with norm $\Vert f\Vert_{L^2}=1$, and then the sequence $$ f_n:=f(x-n) $$ will ...
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23 views

Weak and strong convergence in the sequence space $\ell^p$

Consider the sequence space $\ell^p=L^p(\mathbb N)$, where $1\leq p \leq \infty$. Let $\mathbf x_n=(x^n_k)_{k=1}^\infty$ be a sequence in $\ell^p$, such that $\mathbf x_n\overset{w}{\to} \mathbf x$. ...
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1answer
35 views

Application of Levy's Continuity Theorem. How is it applied herebelow?

THEOREM (Levy's Continuity Theorem) Let $(\mu_n)_{n\geq1}$ be a sequence of probability measures on $\mathbb{R}^d$, and let $(\hat{\mu}_n)_{n\geq1}$ denote their characteristic functions (or Fourier ...
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Questions about weak star convergence [closed]

Let $X$ be a normed space and $\{x^*_n\} \subseteq X^*$,$\{y_n\}\subseteq X$ a) if $\{x^*_n\} \rightharpoonup x ^*$ then $\{x^*_n\}$ is strongly bounded and $\|x^*\| \le \liminf ||x^*_n||$? b) if $\{...
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boundedness and convergence under integral implies weak convergence

I am trying to proof the following: Let $(\Omega,\mu)$ be a measure space, $1<p<\infty$. Then weak convergence of $(f_n)$ to $f$ in $L^p(\Omega,\mu)$ is equivalent to $(f_n)$ being bounded ...
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1answer
56 views

Weak convergence in $L^p=L^p(\Omega,\mu)$

Let $\{f_n\} \subseteq L^p,\{g_n\} \subseteq L^q, f \in L^p,g \in L^q,$ and ${1} \over {p}$ + ${1} \over {q}$$=1$. Suppose $\|f_n-f\|_p \rightarrow 0$ and $\int_{\Omega} g_n \varphi d\mu \rightarrow \...
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Exercise 5.22 in Brezis, “Functional Analysis Sobolev Spaces and Partial Differential Equations”.

Let $H$ be a Hilbert space, $C\subseteq H$ a nonempty closed convex set and $T:C\to C$ a nonlinear contraction, that is $$ (*)\qquad|Tu - Tv| \leq |u-v|. $$ Let $(u_n)$ be a sequence in $C$ such that $...
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Reference for the Dunford-Pettis Theorem.

Does anyone have a book reference for the Dunford-Pettis theorem as stated in the Wikipedia article https://en.wikipedia.org/wiki/Uniform_integrability.
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1answer
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why $x_m$ converges weakly to $x_\infty$?

Let $(X,\|.\|)$ be reflexive Banach space and $Y$ be a closed separable subspace of $X$ $\big((Y ,\|.\|)$is clearly a separable reflexive Banach space$\big)$, then the dual space $Y^*$ of $Y$ is ...
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Application of a Central Limit Theorem (proof strategy)

Let $\{X_{n,i}:1\leq i \leq d_n\}$ be a triangular array of mean zero random variables where $d_n$ is a positive increasing sequence ($d_n\leq n$). Under some conditions, a Central Limit Theorem ...
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Weak* cluster points

Let $X$ be a banach space. I know that a convergent sequence $\{f_n\}\subset X^*$ in the weak* topology on $X^*$ is a sequence such that for all $x\in X$, $\{f_n(x)\}_{n=1}^{\infty}$ is convergent in $...
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$n^{1/p}\chi_{[0,1/n]}$ DOES NOT converge weakly to $0$

I'm working on this problem "Let $X=[0,1]$ with the usual measure. For $n=1,2,...$, let $f_n=n^{1/p}\chi_{[0,1/n]}$. Prove that $f_n$ does not converge weakly to $0$ in $L^1([0,1])$." Yes, this ...
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Detail on mean, distribution and probability convergence

The following three exercises are extracted from the following Book. Possible solutions are also given. In exercise 3.81, I agree that $(\xi_n)_n$ and $(\nu_n)_n$ should be independent, the same ...
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1answer
29 views

If densities converge then the corresponding RV converge in distribution

I tried to prove the following Theorem: Given $(X_n)_{n\in\mathbb{N}}$ iid. random variables with $\mathbb{E}[X_i^2]<\infty$. If the rv's have respective densities $(f_n)_{n\in\mathbb{N}}$ and $...
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56 views

Convergence of sequences in D' is not metrizable

I'm struggled with the problem. I need to proove that the convergence of the sequence of generalized functions in D' can't be defined by any metric (can't be metrizable). I've tried to use the fact ...
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1answer
61 views

Weak Convergence in the Space $L_1$, Why So Special?

For $1< p< \infty$ and $U\subset \mathbb{R}^d$, and let $p'$ be the conjugate of $p$. We say that a sequence $\{f_n\}\in L^p(U)$ converges weakly to $f\in L^p(U)$ if $$ \lim_{n\to \infty}\...
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1answer
21 views

Convergence of polynomial to $0$ in uniform norm in relation with convergence to $0$ of its coefficients

The problem is the following : Let $f(x) = x^{k} + \epsilon_{k,n}(x)$,$x \in I := [0,1]$, where $\epsilon_{k,n}(x)$ is a polynomial of degree at most $k$; Suppose we know that $f \overset{||.||_{\...
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Show that $\sqrt{n}(g(Y_n)-g(m))$ converges in distribution to $N(0,(g'(m))^2σ^2)$

I want to solve the following exercise, but I'm having problems handling all the information given. Let $Y_n $ be a sequence of real random variables and $m, σ > 0$ be real numbers such that as $n ...
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How can I conclude that the weak topology on $~\mathcal l^2~$ is a proper subset of the norm topology from what I've done?

I'm doing a problem in topology. In a) I proved that the weak topology is coarser than the norm topology, and in b) I proved that the standard one sequence $~(e_n)~$ in $~\mathcal l^2~$ approaches $~(...
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39 views

Basic sequence weak convergence implies convergence in norm

A sequence $(x_n)_n$ of a Banach Space $X$ is called basic if it is a basis of $\overline{span\{x_n\}}$. Prove that if $x_n\overset{w}{\to}0$ then $\|x_n\|\to 0$. I was trying to define a $T\in X^*$ ...
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1answer
44 views

weak convergence in $c_0(\mathbb{N})$

Let \begin{equation}X := c_0(\mathbb{N}), \hspace{3mm}x_0 \in X, \hspace{3mm}\{x_n\}_{n \in \mathbb{N}} \subset X \quad bounded\end{equation} . Show that \begin{equation} x_n \rightharpoonup x_0 \...
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1answer
29 views

Lebesgue-Stieltjes measure on $\mathbb R$ vs on $[a,b]$ and weak convergence

Let $F : \mathbb R \rightarrow \mathbb R$ be non-decreasing and right-continuous. Then there exists a unique Borel measure $\mu_F$ on $\mathbb R$ such that for any interval $J \subset \mathbb R$ : $$\...
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1answer
46 views

A simple ODE with sign nonlinearity on the RHS

Suppose we have the following ODE: $$ x'(t) = \begin{cases} 1 & \text{if } x(t) < 0 \\ -1 & \text{if } x(t) \ge 0 \end{cases} \qquad x(0) = 0. $$ I am not sure whether the solution exists ...
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1answer
38 views

Weak convergence of $\Bigl(\sum\limits_{k=n}^\infty e_k\Bigr)_n$

Let $e_n:=δ_{kn}$, for $k\in\mathbb N$. Given the sequence $(a_n):=\sum\limits_{k=n}^∞e_k\subset\ell^\infty$, i.e.$$((1,1,\cdots),(0,1,1,\cdots),(0,0,1,1,\cdots),\cdots).$$ I want to know if $(a_n)$ ...
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1answer
29 views

Not bounded function in $W^{1,1}(\mathbb{R})$ [duplicate]

I have seen in page 627 (Exercise 1) of Taheri's book: Function spaces and partial differential equations that if $n\geq2$, then $W^{1,n}(\mathbb{R}^n)$ does not embed into $L^\infty(\mathbb{R}^n)$. ...
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3answers
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Weak convergence of $f_n$ in $L^p$ implies that of $|f_n|$?

I am wondering that if $f_n$ converge weakly to $f$ in $L^p(\mathbb R^d)$, for $1<p<\infty$ then also $|f_n|$ converge weakly to $|f|$ in $L^p(\mathbb R^d)$? I think it is true but I do not ...
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1answer
41 views

A metric for weak* convergence on bounded sets

Let $X$ be a separable Banach space and let $(x_n)_{n \in \mathbb{N}}$ be a fixed sequence such that $x_n \neq 0$ for all $n \in \mathbb{N}$ and $\lbrace x_n: n\in \mathbb{N}\rbrace$ is dense in $X$. ...
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1answer
39 views

Counterexample for weak convergence in $L^p$ if p=1

For $1<p<\infty$, a sequence $(f_n)_{n \in \mathbb{N}}$ in $L^p(\mathbb{R}, \mathcal{B}(\mathbb{R}), \lambda)$ and $f \in L^p(\mathbb{R}, \mathcal{B}(\mathbb{R}), \lambda)$, I have already ...
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0answers
188 views

Minimizing problem (1-dimensional phase transition) [duplicate]

I need some help in this calculus of variation exercise: Let $X$ to be the set of functions $\chi\in H^1_{loc}(\mathbb{R})$ such that $\chi(0)=1/2$ and such that the following limits do exists: \...
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2answers
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The time convergence of stochastic integral and Doob's convergence.

Consider the process $$X_{t}=\int_{0}^{t}e^{-s}dW_{s},$$ where $e^{-s}$ is deterministic. I am wondering if $\lim_{t\rightarrow\infty}X_{t}$ exists almost surely... I understand that $X_{t}$ in the ...
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1answer
43 views

Weak Limit of $f_n = \sum_{i=1}^{2^n} (-1)^{i-1} 1_{\big[ \frac{i-1}{2^n}, \frac{i}{2^n} \big) }$ in $L^p$

Consider $\Omega=[0,1]$ equipped with Borel $\sigma$-algebra and Lebesgue measure. Consider the sequence $f_n = \sum_{i=1}^{2^n} (-1)^{i-1} 1_{\big[ \frac{i-1}{2^n}, \frac{i}{2^n} \big) }$ in $L^p$ ...
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0answers
35 views

Limit of Poisson random variables with parameters tending to infinity [duplicate]

I have two independent Poisson distributed random variables: $\xi$ with parameter $\lambda$ and $\eta$ with parameter $\mu$. I need to show that $${{(\xi - \lambda) - (\eta - \mu)}\over{\sqrt{\xi + \...
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2answers
52 views

Weak and strong convergence of a sequence of operators in $L^2(\mathbb{R})$

Let $M_\alpha = sin(\frac{x}{\alpha})f(x)$ a sequence of operators in $L^2(\mathbb{R})$. Prove that $M_\alpha$ does not converge strongly to $0$, but converges weakly to $0$, for $\alpha \to 0$. ...
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1answer
43 views

A question in the proof that Lipchitz continuous functions implies $W^{1,\infty}$. [duplicate]

In $\S 5.8.2$ of Evan's PDE book, there is a theorem about characterization of $W^{1,\infty}$. Here it says On the other hand assume now $u$ is Lipschitz continuous; we must prove that $u$ has ...
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2answers
158 views

Pratt's lemma and convergence in distribution

Let $(X_n)_n,$ $(Y_n)_n$ be sequences in $L^1,$ $(X_n)_n$ converges in distribution to $X \in L^1,$ $(Y_n)_n$ converges in distribution to $Y \in L^1,$ $\lim_nE[X_n]=E[X],\lim_n E[Y_n]=E[Y].$ Let $(...
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2answers
38 views

Weak*-convergence to 0 on L^\infty and convergence almost everywhere

I am stuck with something standard... Let $f_n \in L^\infty(\mathbb{R}^d)\cap L^1(\mathbb{R}^d)$, $n\geq1$, be such that $$ \sup_{n\geq1} \|f_n\|_{L^\infty(\mathbb{R}^d)}<\infty $$ and $$ \lim_{n\...
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0answers
26 views

About weak convergence and derivative

The following question had appeared in so many places, but none justify it, I tried a lot but . If someone can give me a hand. Let $X$ be a Hilbert space and $I:X\rightarrow \mathbb{R}$ a ...
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12 views

Is the bound of different quotients the same as that of weak derivatives?

In $\S 5.8.2$ of Evan's PDE, there is a theorem relating to different quotients and weak derivatives. Theorem 3 (ii) Assume $1 < p < \infty$, $u \in L^p(V)$, and there exists a constant $C$ ...
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1answer
17 views

Bounded weakly* convergent sequence in $L^\infty(\Omega)$ with bounded limit …(with the same limits)

Let $\left(a_{i}\right)_{i}$ be a sequence in $L^{\infty}\left(\Omega\right)$ ($\Omega$ is a regular bounded domain of $\mathbb{R}^{n}$,$n=2,3$) and $c_{1},c_{2}$ be real numbers such that $c_{1}\leq ...
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1answer
27 views

Convergence problem with a sequence of operators

Is it true that the sequence $\{T_n:H\rightarrow H\}_n$ with $H=l^2(\mathbb{N};\mathbb{R})$ ($\mathbb{N}$ is just the index set of the series) defined by $T_n(x_0,x_1,x_2,...)=(x_n,.x_{n-1},...,x_0,0,...
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121 views

Why is this $L^1$-sequence relatively weakly sequentially compact?

Let $(E,\mathcal E,m)$ be a probability space, $\theta$ be a measurable map on $(E,\mathcal E)$ with $m\circ\theta^{-1}=m$, $s_n$ be a real-valued nonpositive integrable random variable on $(E,\...

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