# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $C$ ...
### 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,\...