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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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2answers
37 views

Weak convergence of function to $0$

Suppose $\mu = \mathcal{L} |_{[0,1]}$ where $\mathcal{L}$ is the Lebesgue measure on the real line. Define $$f(x):= \begin{cases} 1 & \text{if } 0 \le x < \frac{1}{2} \\ -1 & \text{if } \...
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1answer
17 views

$\mathcal{F}$ convex and lower continous $\Rightarrow$ $\mathcal{F}$ weakly lower continous

I'm having troubles with one part of a problem consisting out of several subquestions and hope some of you can help me! Let $X$ be a Banach-space and let $\mathcal{F} : X \rightarrow (-\infty,\...
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2answers
28 views

convergence in distribution implies a.s and probability convergence

Let $(X_n)_n$ be a sequence of independent random variable. Let $W_n=\sum_{k=1}^nX_k.$ We suppose that $(W_n)_n$ converges in distribution. Prove that $(W_n)_n$ also converges almost surely and in ...
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0answers
17 views

Convergence of Feller processes implies convergence of the resolvent operators of their generators

Let $E$ be a locally compact separable metric space $(T_n(t))_{t\ge0}$ and $(T(t))_{t\ge0}$ be strongly continuous contraction semigroups on $C_0(E)$ with generator $(\mathcal D(A_n),A_n)$ and $(\...
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0answers
23 views

Find a counterexample of weak convergence and weak* convergence

Given a normed space $X$ over $\mathbb{R}$ (or $\mathbb{C}$), define its dual space $X^*$ as the set of all continuous & linear functionals. $\forall g \in X^*, \|g\|_{X^*}\triangleq \sup_\limits{\...
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1answer
16 views

Sampling with replacement. $P[\upsilon_m>n]=\prod_{i=2}^{n}(1-\frac{i-1}{m})$.

Let $\{X_n,n \leq 1\}$ be i.i.d. and uniformly distributed on the set $\{1,...,m\}$. In repeated sampling, let $v_m$ be the time of the first coincidence; that is, the when we first get a repeated ...
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3answers
62 views

A sufficient and necessary condition for a special linear operator to be compact

Suppose $X$ is Banach and $T\in B(X)$ (i.e. $T$ is a linear and continuous map and $T:X \to X$). Also, suppose $\exists c > 0$, s.t. $\|Tx\| \ge c\|x\|, \forall x\in X$. Prove $T$ is a compact ...
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0answers
31 views

A question regarding Radon-Riesz property

I want to show that if a normed linear space has Radon-Riesz property, then it has the semi Radon-Riesz property. We know that a normed linear space is said to have Radon-Riesz property if for every ...
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0answers
31 views

Quotient space of a a normed linear space with schur's property need have have that property.

I want to show that if a normed linear space has Schur's property (every weakly convergent sequence converges), then it is not guaranteed that every quotient space $X/Y$ will have that property, where ...
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1answer
16 views

Weak convergence of measures and boundedness

Let $(X,d)$ be a Polish space. Suppose that $(\mu_n)$ is a sequence of probability measures on $X$ such that $\mu_n\to \mu$ weakly, where $\mu$ is a probability measure on $X$. Is it true that there ...
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0answers
11 views

Showing that $C=\{h \in L^1(\Omega) : u_1(z) \leq h(z) \leq u_2(z)\}$ is weakly compact.

Exercise : Let $\Omega \subseteq \mathbb R^n$ be open and bounded, $u_1, u_2 \in L^1(\Omega)$ with $u_1(z) \leq u_2(z)$ almost everywhere in $\Omega$. We let $C$ be the set $C=\{h \in L^1(\Omega) : ...
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0answers
39 views

Showing that $A(\overline{B_1^X}) \subseteq Y$ is closed when $A \in \mathcal{L}(X,Y)$.

Exercise : Let $X,Y$ be Banach spaces and $X$ be reflexive, $A \in \mathcal{L}(X,Y)$. If $\overline{B_1^X} = \{ u \in X : \|u\|_X \leq 1\}$, show that $A(\overline{B_1^X}) \subseteq Y$ is closed. ...
3
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1answer
50 views

prove a series weakly converges to its mean value

Let $u ∈ L^∞ (\Bbb R)$ such that $u(x + 1) = u(x)$ almost for every $x ∈ \Bbb R$. Let $ū = \int_0^1 u(y) dy$ and let's define $(u_n )_{n∈\Bbb N}$ by $u_n (x) = u(nx)$ $\forall n ∈ \Bbb N$ and for ...
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1answer
17 views

Weak convergence of functionals in dual space given by a sum of functionals

Suppose that $X_1$, $X_2$ are Hilbert spaces and $Y=X_1\times X_2$. Let $y\in Y$ be $y=(x_1, x_2)$ and define $F\in Y^*$ by $$\langle F,y\rangle_{Y^*\times Y}=\langle f,x_1\rangle_{X^*_1\times X_1}+\...
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1answer
33 views

Weak star convergence

Let $\{f_n\}$ be a sequence converging weakly in $C( [0,T] ; L^2(\mathbb{R}^n))$ and also converges in weak star sense in $L^\infty([0,T]; L^2(\mathbb{R}^n))$ to $f$. Can it be concluded that $f_n(t)...
1
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1answer
28 views

Proving a sequence of functions converges weakly*

I am reading a book about functional analysis and I found the following example proving that some sequence converges weakly* but not weakly. Let $X=C[-1,1]$ be the space of continuous functions ...
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2answers
47 views

weak convergence: Show that $\mathbf{P}(X_n = q) \to 0$ as $n \to \infty$ when the cdf $F$ of the limit $X$ is continuous at $q$

The question might seem easy to some of you, but I'm still pretty new to probabilty theory (and also not very deep in mathematics) and appreciate any help. Here is the full statement I'm trying to ...
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1answer
35 views

Finding the weak limit of a sequence of random variables.

Suppose that $X_1,X_2,\ldots,X_n$ are independent and identically distributed random variables having characteristic function $\chi(t)=e^{-|t|^{1.9}}$.Then what is the weak limit of $n^{-5/9}S_n$ as n ...
2
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1answer
49 views

$A \in \mathcal{L}(X,Y) \implies A^* \in \mathcal{L}(Y^*_{w^*}, X^*_{w^*})$

Exercise : Let $X,Y$ be Banach spaces and $A \in \mathcal{L}(X,Y)$. Show that $ A^* \in \mathcal{L}(Y^*_{w^*}, X^*_{w^*})$. Attempt : The linearity is trivial. Τo show that $A^*$ is $w^*$ to $w^*$...
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2answers
61 views

closed convex hull of weak convergent sequence

Le $H$ be a Hilbert space and Suppose $x_n$ converges weakly to $x$ in $H$. Let $K_n$ be the closed convex hull $\bar{co}\{x_k:k\geq n\}$. I would like to show $\bigcap K_n=\{x\}$. What I know so far ...
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0answers
51 views

Example for Lévy's continuity theorem

I am searching a sequence of RV $(X_n)$ for which we prove a convergence in distribution to a random variable $X$, using the fact that the characteristic functions $(\varphi_n)_n$ converges pointwise ...
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0answers
11 views

Convergence of Frechet Derivative Functional related to Palais-Smale Sequence

Let $\phi : H_{0}^{1}(\Omega)\to\mathbb{R}$ be an energy functional for bounded domain $\Omega$ such that $\phi'$ be its Frechet derivative, and there exists a bounded sequence $\{u_{n}\}_{n\in\mathbb{...
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1answer
24 views

Orthogonal Projection in Hilbert Space to prove weak convergence

Let $H$ be a hilbert space and $(h_{n})_{n\in\mathbb{N}}$ be a bounded sequence in $H$. Define $H_{0}:= \text{cl}(\text{span}(h_{1},h_{2},...))$. Then, $H_{0}$ is a separable space since the set of ...
3
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1answer
97 views

How can we show that $\left\{x\mapsto\prod_{i=1}^kf_i(x_i):f_i\in C_0(E_i)\right\}$ is convergence determining on $\times_{i=1}^ kE_i$?

Let $E$ be a complete locally compact separable metric space and $\mu,\mu_n$ be probability measures on $\mathcal B(E)$. We say $\mu_n\to\mu$ weakly if $$\int f\:{\rm d}\mu_n\to\int f\:{\rm d}\mu\tag1$...
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1answer
15 views

Weak convergence for linear operator in Hilbert Space

Assume $T : X \to Y$ be a linear operator for $X,Y$ Hilbert spaces. Moreover, let $<\,\cdot\, , \,\cdot\,>_{X}$ and $<\,\cdot\, , \,\cdot\,>_{Y}$ be inner dot product of $X$ and $Y$ ...
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1answer
32 views

If $F$ is a family of probability measures on $C(K,E)$ and $π_t$ is the evaluation map at $t∈K$, can we show that $(μ∘π_t^{-1})_{μ∈F}$ is tight?

Let $K$ be a compact metric space, $E$ be a separable metric space, $C(K,E)$ denote the space of continuous functions from $K$ to $E$ equipped with the supremum metric and $$\pi_t:C(K,E)\to E\;,\;\;\;...
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0answers
14 views

Are normed spaces equipped with the weak topology sequential [duplicate]

Let $X$ be a normed space equipped with the weak topology. Is $X$ a sequential space (using this definition)? That is can we test closedness in the weak topology using weakly convergent sequences? I ...
3
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1answer
47 views

Weak Convergence in Lp space

I'm trying to solve the following problem. Let $f_0 \in L^p(\mathbb{R})$ and let $f_n(x)=f_0(x+n)$. Show that $f_n$ converges weakly to zero in $L^p(\mathbb{R})$. I know that if $(x_n)$ is a ...
0
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1answer
23 views

convergence for the weak-* topology

Let E be a Banach space. Let $(x^∗_n )$ be a sequence in $E^∗$ verifying $(<x^∗_n , x>)$ converges for any $x ∈ E$. Prove that $\exists x^∗ ∈ E^∗: (x^∗_n )$ converges vers $x^*$ for the weak-∗...
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0answers
58 views

I'm trying to understand a weak convergence result for Feller processes in Ethier and Kurtz

Let $E$ be a locally compact separable metric space, $D([0,\infty),E)$ denote the Skorohod space, $C_0(E)$ denote the space of continuous functions $E\to\mathbb R$ vanishing at infinity and $E^\ast:=E\...
1
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1answer
68 views

Every operator $T : X \rightarrow L^1$ is weakly compact.

I am trying the following problem. Let $X$ be a Banach space that does not contain a copy of $l^1$. Show that every operator $T : X \rightarrow L^1$ is weakly compact. I am not sure how to start. ...
2
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0answers
94 views

If $X^n$ is a càdlàg process in a locally compact space $E$, show that $\{X^n\}$ is relatively compact as processes in the compactification of $E$

Let $E$ be a locally compact separable$^1$ metric space, $D([0,\infty),E)$ denote the Skorohod space, $C_0(E)$ denote the space of continuous functions $E\to\mathbb R$ vanishing at infinity and $E^\...
3
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1answer
32 views

A Problem on Tightness of Measures

Can someone provide an example of probability measures $\{\mu_n\}$ and $\{\nu_n\}$ such that although $\int_{\mathbb{R}}f d\mu_n - \int_{\mathbb{R}}f d\nu_n \rightarrow 0$ for all continuous real-...
2
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2answers
51 views

Weak convergence in $C([0,+\infty))$ and convergence in probability

I'm reading an article and I can't manage to solve something. They say "It's not hard to check that $\sup_{[0,T]}\sqrt{\epsilon}|V_{(t-\epsilon \tau)/\epsilon }-V_{t/\epsilon}|$ converges to 0 in ...
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0answers
17 views

The limit distribution of Wilcoxon signed rank statistic?

An alternative representation of the Wilcoxon signed rank statistic $V$ is $V=\sum_{i\le j}\mathbb{I}_{\{X_i+X_j>0\}}=\sum_i\mathbb{I}_{\{X_i>0\}}+\sum_{i<j}\mathbb{I}_{\{X_i+X_j>0\}}$ ...
2
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1answer
31 views

What is the set of pointwise limits of polynomials?

The set of pointwise limits of continuous functions from from $\mathbb{R}$ to $\mathbb{R}$ is the set of Baire class 1 functions. My question is, my question is, what is the set of pointwise limits ...
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2answers
25 views

Hilbert space and weak convergence

I have some dubts about this exercise: Let $(H, (\cdot, \cdot))$ a Hilbert space e let $(x_{n})_{n}\subset H$ and $x \in H$. Prove that If $x_{n}\xrightarrow{\tau_{w}}x$ if and only if $(x_{n},y) \...
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1answer
24 views

$X$ Banach, $u_n \to u$ and $x^*_n \xrightarrow{w^*} x^*$ implie that $\langle x^*_n, u_n\rangle \to \langle x^*,u\rangle$.

Exercise : Let $X$ be a Banach space, $u_n \to u$ and $x^*_n \xrightarrow{w^*} x^*$. Show that $\langle x^*_n, u_n\rangle \to \langle x^*,u\rangle$. Attempt-Discussion : I know that a sequence $...
3
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1answer
51 views

If $\text P\left[|X_n|>n^{-\alpha}\right]\to0$ as $n\to\infty$ for some $\alpha>0$, does $(X_n)_{n\in\mathbb N}$ converge in probability?

Let $(X_n)_{n\in\mathbb N}$ be a sequence of real-valued random variables on a probability space $(\Omega,\mathcal A,\operatorname P)$ and $\alpha>0$. Is there some relation between convergence in ...
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0answers
39 views

Conclude convergence in probability from uniform convergence on a set of limiting probability 1

Let $\kappa_d$ be a Markov kernel on $(\mathbb R^d,\mathcal B(\mathbb R^d))$ for $d\in\mathbb N$ $f_d:\mathbb R^d\times\mathbb R^d\to\mathbb R$ be Borel measurable for $d\in\mathbb N$ $B_d\in\mathcal ...
3
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2answers
54 views

$u_n$ converges weakly to $u$ $\Leftrightarrow$ $\{u_n\}$ is bounded and $\langle x^*, u_n \rangle \to \langle x^*, u \rangle$

Proof Request : I am seeking for a proof of the following Lemma defining Weak Convergence. I am aware of a similar statement regarding Hilbert spaces but it seems to differ. I know that for the $(\...
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0answers
38 views

Which version of the central limit theorem do we need to apply here?

Let $f\in C^3(\mathbb R)$ with $f>0$ and $g:=\ln f$. Assume $g'$ is Lipschitz continuous. Let $d\in\mathbb N$ and $X$ be a $\mathbb R^d$-valued random variable with density $$\mathbb R^d\ni x\...
1
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1answer
29 views

Weakly null sequence in Banach lattices

Let $(x_n)_n$ be a positive, disjoint, weakly null sequence in a Banach lattice $E$. If $(y_n)_n$ is a sequence such that $0\leq y_n\leq x_n$ for every $n\in \mathbb{N}$, we can garantee that $y_n$ is ...
0
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0answers
55 views

Show that the limit inferior of the sum of third moments of mutually independent Gaussians tends to negative infinity

Let $d\in\mathbb N$, $\sigma:=\ell d^{-\alpha}$ for some $\ell>0$ and $\alpha>0$, $Y$ be a Gaussian $\mathbb R^d$-valued random variable with mean $0$ and covariance matrix $\sigma^2I_d$$^1$, $Z$...
1
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1answer
69 views

On weak*-sequentially completeness

I want to prove that every dual space is weak*-sequentially complete. Let $X$ be a normed linear space and let $(f_n)$ be a weak* Cauchy sequence in $X^*$. Thus for all $x\in X$, $(f_n(x))$ is a ...
0
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0answers
26 views

Different topologies in Sobolev space $W^{1,p}$

In paper [1], L.Ambrosio talks about the space $W^{1,p}(\Omega),\ 1\leq p<+\infty$ endowed with four different topologies: The strong topology, denoted by $W^{1,p}(\Omega)$. The weak topology, ...
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0answers
23 views

Weak star convergence of Borel probability measures on a metric space

Let $(X,\rho)$ be a compact metric space and let $P(X)$ be the set of Borel probability measures on the Borel $\sigma$-algebra of $X$. Suppose $\mu_n,\mu \in P(X)$ for $n \in \mathbb{N}$ such that $\...
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0answers
27 views

Weak Convergence from Strong Convergence

Let $\Omega \subset \mathbb{R}$ be a bounded domain, $v \in H_{0}^{1}(\Omega)$ such that $||u_{n}-v||_{H_{0}^{1}(\Omega)}\to 0$ as $n\to\infty$ for a bounded sequence $\{u_{n}\}_{n\in\mathbb{N}} \...
1
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1answer
24 views

Convergence in Probability - Random function evaluated in random argument.

Suppose that $g_n: \mathbb{R}^k \to \mathbb{R}^h$ are random linear mappings such that $$ g_n \stackrel{P}{\to} g, \quad \quad \text{as } n\to\infty, $$ where the non-random limit $g$ is injective. ...
1
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0answers
88 views

Merging with respect to bounded uniformly continuous functions in terms of characteristic functions

I would like to know if there are any results, where merging of probability measures in $R^n$ with respect to bounded uniformly continuous functions is deduced from some conditions on characteristic ...