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).

# Questions tagged [weak-convergence]

2251 questions

**33**

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### Strong and weak convergence in $\ell^1$

Let $\ell^1$ be the space of absolutely summable real or complex sequences. Let us say that a sequence $(x_1, x_2, \ldots)$ of vectors in $\ell^1$ converges weakly to $x \in \ell^1$ if for every bounded linear functional $\varphi \in (\ell^1)^*$,…

Zhen Lin

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### Weak convergence in probability implies uniform convergence in distribution functions

Exercise 1: Let $\mu_n$, $\mu$ be probability measures on $\left(\mathbb{R}, \mathcal{B}\left(\mathbb{R}\right)\right)$ with distribution functions $F_n$, $F$. Show: If $\left(\mu_n\right)$ converges weakly to $\mu$ and $F$ is continuous, then…

Keith

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### A closed subspace of a reflexive Banach space is reflexive

Let $X$ be a reflexive Banach Space. Let $Y$ be a closed subspace of it.I need to show that $Y$ is reflexive as well. So as usual I consider the inclusion map $$J: Y \to Y'', J(y)=j_{y}, j_{y}(y')=y'(y)$$, where $Y''$ denotes the bidual space of…

tattwamasi amrutam

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### Weak convergence, together with convergence of norms, implies strong convergence in a Hilbert space.

Let $(x_n)$ be a weakly convergent sequence in a Hilbert space $H$. If $\| x_n \| \to \| x \|$, show that $x_n$ converges strongly to $x$.
Context
This problem comes from a question in my exam paper; the original problem was incorrect.

Ricardo Gomes

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### Characterization of weak convergence in $\ell_\infty$

Is there some simple characterization of weak convergence of sequences in the space $\ell_\infty$? If yes, is there some similar claim for nets?
I was only able to come up with a characterization of sequential weak convergence using limit along…

Martin Sleziak

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### Example that in a normed space, weak convergence does not implies strong convergence.

The book "Introductory Functional Analysis with Applications" (Kreyszig) presents the following definitions.
A sequence $(x_n)$ in a normed space $X$ is said to be strongly convergent if there is an $x\in X$ such that $\lim\|x_n-x\|=0$. (page…

Pedro

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### Uniform convergence and weak convergence

Assume $F_{n},F$ are distribution functions of r.v.$X_{n}$ and $X$, $F_{n}$ weakly converge to $F$. If $F$ is pointwise continuous in the interval $[a,b]\subset\mathbb{R}$, show that
$$\sup_{x\in[a,b]}|F_{n}(x)-F(x)|\rightarrow 0,n\rightarrow…

Jacky Zhang

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### On the weak and strong convergence of an iterative sequence

I have some difficulties in the following problem.
I would like to thank for all kind help and construction.
Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a mapping. Suppose that there exists $\gamma>0$ such…

blindman

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### Every bounded sequence has a weakly convergent subsequence in a Hilbert space

I tried to prove the following theorem and was wondering if someone could please tell me if my proof can be fixed somehow...
Theorem: Let $H$ be a Hilbert space and $x_n\in H$ a bounded sequence. Then $x_n$ has a weakly convergent subsequence.
My…

user167889

**13**

votes

**2**answers

### Intuitive explanation of Lyapunov condition for CLT

I found the Lyapunov condition for applying the central limit theorem, which is useful in settings where one has to deal with non-identically distributed random variables:
Lyapunov CLT. Let $s_n^2 = \sum_{k=1}^n \text{Var}[Y_i]$ and let $Y=\sum_i…

Mathrobot

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### A characterization of Weak Convergence in $L^p$ spaces

I'm working on the following problem, I'm having trouble with the reverse direction. My question is bolded below. Also could someone check my forward direction?:
Let $(X, \mathcal{M}, \mu)$ be a $\sigma$ finite measure space and $\{f_n\},f \in…

yoshi

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**1**answer

### Confusion with the narrow and weak* convergence of measures

Think of a LCH space $X.$ Consider the spaces $C_{0}(X)$ of continuous functions "vanishing at infinity" and the space $BC(X)$ of bounded continuous functions. Consider as well the space of Radon (Borel regular) measures $M(X).$
What follows is an…

Qwertuy

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### $\ell_\infty$ is a Grothendieck space

The problem I am considering stated formally is this:
Show that if a sequence in $\ell_\infty^*$ is weak*-convergent, then it is also weakly convergent.
We may reduce this to the case where the sequence is weak$^*$-null, and show that it is weakly…

Keaton

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votes

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### uniform convergence of characteristic functions

Assume that a sequence of probability measures $\mu_n$ converges weakly to $\mu$. Let $\phi_n$ and $\phi$ denote respetively the characteristic function of $\mu_n$ and $\mu$. Prove that $\phi_n$ converges uniformly to $\phi$ on any bounded…

Shine

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### Convergence in probability implies convergence in distribution

A sequence of random variables $\{X_n\}$ converges to $X$ in probability if for any $\varepsilon > 0$,
$$P(|X_n-X| \geq \varepsilon) \rightarrow 0$$
They converge in distribution if
$$F_{X_n} \rightarrow F_X$$
at points where $F_X$ is…

Hawii

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