# 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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-...
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### 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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### The limit distribution of Wilcoxon signed rank statistic?

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