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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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26 views

Weak convergence and weak* convergence

Let $\Omega$ be some measurable space. Suppose $\{x_n\}$ is a sequence bounded in $L^{\infty}((0,T) ; L^2(\Omega;H^1(\mathbb{R}^d)))$ , then we know there exists a subsequence $\{x_n\}$(denoting same) ...
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1answer
54 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 ...
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1answer
34 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-...
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1answer
28 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 ...
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1answer
41 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 ...
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1answer
51 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 $L^p(\Omega)$ ...
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39 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 ...
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1answer
55 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 ...
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1answer
28 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, &...
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1answer
23 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 ...
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2answers
33 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 ...
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2answers
87 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 ...
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1answer
21 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}...
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0answers
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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}...
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24 views

Asymptotic behavior of sample quantiles with perturbed order.

I am struggling with the following problem. Let $X$ be a real valued random variable whose distribution is absolutely continuous with regard to the Lebesgue measure. Denote $F$ and $f$ the cumulative ...
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1answer
27 views

If the weak limit is zero, is it true that the sequence of functions tends to zero almost everywhere?

Assume that a sequence of continuous functions $(f_n)$, where $f:[0,1]\to\mathbb{R}$ has the following property. For any smooth function $\phi:[0,1]\to\mathbb{R}$ one has $$ \lim_{n\to\infty } \int_0^...
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1answer
38 views

If $v_n \rightharpoonup v$ and $\phi \in C_c^{\infty}(\mathbb{R}^N)$, do we have $v_n \phi \rightharpoonup v \phi$?

I'm stuck trying to solve this question. If I have a sequence in the Sobolev space $D^{1,\vec{p}}(\mathbb{R}^N)$ (or $W_0^{1,p}(\mathbb{R}^N)$ for simplicity) which converges weakly to $v$ and $\phi \...
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37 views

Does weak convergence imply existence of an a.e. convergent subsequence? [duplicate]

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\Dpr}{\mathcal{D}^{1,\vec{p}}(\R^N)}$ I'm reading this article by El Hamidi and Rakotoson. On page 745 (page 5 of the PDF), they construct a bounded sequence ...
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1answer
33 views

Convergence of integrals under weak convergence of measure and compact convergence

I'm trying to solve Problem 2.4.12 on page 64 of Karatzas-Shreve's book "Brownian motion and stochastic calculus": My attempt is to use triangle inequality (denoting $\Omega=C[0,\infty)$) $$|\int_\...
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0answers
37 views

Weak* convergence in $W^{1,\infty}(\Omega)$

Let $\Omega$ be a bounded domain in $\Bbb R^m$ with smooth enough boundary so that $W^{1,\infty}(\Omega)=\text{Lip}(\Omega)$. Let $(u_n)$ be a sequence in $W^{1,\infty}(\Omega)$. What does it mean ...
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22 views

Double Limit of operators converges weakly, does single limit converge?

I am working on a problem from Reed and Simon, which states: Suppose $\{A_\alpha\}$ and $\{B_\alpha\}$, $\alpha \in I$, are nets. Let $A_\alpha^* \to A^*$ and $B_\alpha \to B$ in the Strong Operator ...
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1answer
36 views

Weak operator topology convergence of sequence of bounded operators to a bounded operator

I need to prove the following theorem. Let $X, Y$ be Banach spaces, with $Y$ weakly sequentially complete. Let $\{T_n\} \subset \mathscr{L}(X,Y)$ with $\Lambda(T_nx)$ converges for all $x \in X$, $\...
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0answers
24 views

Relatively compactness of uniformly limited measures in variation w.r.t. weak convergence

Let $f_n(x).dx$ be measures with density wrt Lebesgue measure, $f_n$ derivable, $f_n \rightarrow f$ in uniform norm; $\ n \rightarrow \infty$. Assume that they are uniformly limited in variation, i....
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27 views

Understanding the proof of the Concentration-Compactness principle

$\newcommand{\R}{\mathbb{R}}$ I'm reading parts of the paper The Concentration-Compactness Principle in the Calculus of Variations. The Limit Case, Part 1 by P.L. Lions. I'm trying to understand the ...
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1answer
66 views

Is it possible to extend the notion of $H$-convergence to the case of distributions?

The usual $H$-convergence is defined for operators of the following form (for the sake of simplicity, restrict ourselves with the one-dimensional case): $$ \frac{d}{d x}\left[A_\varepsilon(x) \frac{d ...
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35 views

Weak$^*$ convergence, uniformly on compacta

Consider the following result: Let $E$ be a Banach space, and let $X\subseteq E^*$ be a subspace. For any $\mu$ in the weak$^*$-closure of $X$ we can find a net $(\mu_\alpha)$ in $X$ which ...
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30 views

Equivalence over convergence in distribution

Let $X$ be a metric space and $\mu_1,\mu_2,...,\mu$ be Borel probability measures on $X$. The Portmanteau theorem says the following are equivalent: (a) $\int_X gd\mu_n \to \int_X gd\mu$ for each $...
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0answers
51 views

Is the topology of weak convergence of probability measures first-countable?

Let $S$ be a separable metric space, and $\mu,\mu_1,\mu_2,\ldots$ be Borel probability measures on $S$. We know that $\mu_n \to \mu$ weakly if and only if $\pi(\mu_n,\mu) \to 0$ where $\pi$ is the ...
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1answer
37 views

Do operators from $L(X', Y')$ preserve weak*-convergence?

I am wondering whether the following is true: Let $X, Y$ be normed spaces and $T \in L(X', Y')$. If $x_n' \overset{*}{\rightharpoonup} x'$ in $X'$, then $Tx_n' \overset{*}{\rightharpoonup} Tx'$ ...
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1answer
25 views

Weak convergence implies stochastic boundedness

I'm trying to prove the following implication: Every sequence of random variables which converges weakly is also stochastically bounded. I know that this is an implication of Prohorov's Theorem but ...
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2answers
37 views

Why $\frac{\varepsilon}{x^2+\varepsilon^2}$ converges in the sense of distributions to a constant times the Dirac delta

The integral of $f_\varepsilon(x)=\frac\varepsilon{x^2+\varepsilon^2}$ is the tan inverse, which is well behaved anywhere on $\mathbb{R}$, and so $f_\varepsilon$ is in $L^1_\text{loc}(\mathbb{R})$. ...
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0answers
25 views

weak convergence of composition

Given $v_k \rightarrow v \ \ \ \text{weakly star in } L^2(0,T;W^{1,\infty}(\mathbb{R}^n)) $ $\eta _k(t,\cdot) \rightarrow \eta(t,\cdot) \ \ \ \text{in } C_\text{loc}(\mathbb{R}^n)\ $ uniformly in $t,...
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0answers
95 views

weak convergence, relative compactness

I am interested in weak convergence of stochastic processes with sample paths in $D_{\mathbb{H}}[0,1]$. Let $\mathbb{H}$ denote a separable Hilbert space and $D_{\mathbb{H}}[0,1]$ the space of ...
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3answers
38 views

Can we prove the weak convergence of sequence bounded by another weakly convergence sequence?

A series $a_n$ in some Hilbert space $H$ weakly converges to zero (meaning $f(a_n) \to 0$ $\forall f \in H^{'}$, the dual space of $H$). It is also given that for another sequence $b_n$, norm of each ...
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2answers
51 views

Rate of weak convergence of sin(nx)

Since $\sin(n\cdot)$ converges weakly to zero, we know that $$ \lim_{n\rightarrow\infty} \int_a^b g(x)\sin(nx)\mathrm{d}x = \int_a^b g(x)\cdot 0\,\mathrm{d}x = 0 $$ holds for all $g\in L^2([a,b])$. ...
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1answer
50 views

Bounded sequence has Cauchy subsequence w.r.t. $ (x|y)_0:=\sum^\infty_{n=1} 2^{-n}\phi_n(x)\phi_n(y).$

Let $X$ be a separable reflexive real Banach space and let $(\phi_n)$ be a dense sequence in $$\{ \phi\in X'\,|\,\|\phi\|\leq 1 \}.$$ Consider in $X$ the scalar product $( \cdot| \cdot)_0 $ defined by ...
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1answer
25 views

If $\phi: [0,\, 1] \to X $ be a sequentially weakly continuous, then $f(t) = \|φ(t)\| $ is bounded

Let $X$ be a Banach space. Let $\phi: [0,\, 1] \to X $ be a sequentially weakly continuous function, that is, $$\forall \,(t_n) \subset [0, 1],\,\,\, t_n \to t \Rightarrow \phi(t_n) \rightharpoonup \...
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2answers
80 views

$f_n \rightharpoonup f$, $g_n \to g$ in measure, $\|g_n\|_{L^\infty} \leq M$, then $f_n g_n \rightharpoonup fg$,

Let $f_n$, $f \in L^2(0, 1)$ be such that $f_n$ converges to $f$ weakly in $L^2(0, 1)$. Let $g_n, g : (0, 1) \to \mathbb{R}$ be measurable functions such that $g_n$ converges to $g$ in measure and $\|...
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2answers
31 views

Almost everywhere convergence and $L^1$ convergence

In class we have seen how, given the solution for the heat problem (the one done by convoluting with $\frac{1}{(4 \pi t)^{n/2}} e^{-|x|^2 /4t}$), even if the initial condition was only $L^1$, we still ...
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1answer
34 views

Weak convergence and infinite square mean

$(\Omega , \mathscr{F}, P)$ probability space, $X_n$ are i.i.d., $S_n:=X_1+\cdots X_n$ Suppose $a_n, b_n>0 $ with $\frac{S_n-a_n}{b_n}$ weakly convergent to $N$ standard normal distribution and $E[...
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2answers
40 views

Does the notion “weak convergence” coincide with that using in functional analysis?

Let $\mu_k$ be a sequence of probability (Borel) measures on $\mathbb{R}^n$. We say, $\mu_k$ converges to a probability measure $\mu$ weakly if $\int gd\mu_k \to \int gd\mu$ for every continuous ...
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1answer
18 views

Convergence in coordinates vs convergence in norm

In the proof of the statement that weak convergence is equivalent to strong convergence in finite dimensional normed vector space, we need to use the fact that in finite dimensional space convergence ...
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2answers
26 views

Show that every closed linear subspace $ \ Y \ $ of a Banach space $ \ X \ $ is weakly sequentially closed

Show that every closed linear subspace $ \ Y \ $ of a Banach space $ \ X \ $ is weakly sequentially closed , that is , $ \ Y \ $ contains weak limits of all weakly convergent sequences $ \ \{y_n \} ...
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1answer
80 views

Book of Functional analysis

I'm looking for a book/note or something else of functional analysis which has a lot of exercises and examples. In particular I need convergence (strong, weakly) in C^m spaces, L^p and also in sobolev ...
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1answer
35 views

Sequence of normal random variables converging in distribution

Let $X_{n}$ be a sequence of normal random variables with mean $\mu_{n}$ and variance $\sigma^{2}_{n}$ for $n \geq 1$. Suppose $X_{n} \rightarrow X$ in distribution where $X \neq c$ almost surely for ...
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1answer
39 views

Convergence in distribution under special condition

Let $a\in \mathbb R $ and $v>0$ and $(X_n)_n$ a sequence of real valued random variables with $\sqrt n (X_n-a) \xrightarrow{d} N (0,v) $ for $n \to \infty $. Let $f:\mathbb R \to \mathbb R $ be ...
4
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1answer
37 views

Counterexample $X_n \to X$ in distribution but $\lim_{n \to \infty} E[ \log(1+X_n) ]\neq E[ \log(1+X) ]$

Let $X_n \to X$ in distribution where we only consider non-negative random variables. I am looking for a counterexample that for the followig limit \begin{align} \lim_{n \to \infty} E[ \log(1+X_n) ...
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1answer
26 views

$\limsup P[X_n\leq x]\leq P[X\leq x]$

Let, $X_n$ converges $X$ in distribution. Give a counterexample of this statement: $\limsup P[X_n\leq x]\leq P[X\leq x]$ I know $\limsup P[X_n\leq x]\leq P[\limsup X\leq x]$. Then if I can give an ...
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1answer
26 views

Exercise on Sobolev Spaces and strong/weak convergence

I have to solve this exercise: "Fix $v \in \mathcal{C}^\infty_c(\mathbb{R})$. Discuss the strong and the weak convergence of the sequence $u_n(x) = \frac{v(nx-n^2)}{n}$ in the spaces $W^{k,p}(\mathbb{...
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0answers
33 views

$X_n$ converges to $X$ in distribution", is equivalent to…

The statement "$X_n$ converges to $X$ in distribution", is equivalent to A. $\limsup P[X_n< x]\leq P[X<x]$ for all real $x$. B. $\liminf P[X_n< x]\geq P[X<x]$ and $\liminf P[X_n> x]\...