# 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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### 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 ...
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Let us say that a normed vector space has the a) RR (Radon-Riesz) property if for any sequence, norm convergence is equivalent to weak convergence + convergence of norms. b) KK (Kadec-Klee) property ...
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### Is there a version of Slutsky theorem for stochastic process?

To be more specific, if a stochastic process $X_n(t)$ converges weakly to a tight Gaussian process $G(t)$, and another stochastic process $Y_n(t)$ converges uniformly to a deterministic function $H(t)$...
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### A lemma in the application of Concentration compactness principle in Hardy-Littlewood-Sobolev inequality

I'm encountering some problems when reading Lions' paper "the concentration-compactness principle in the calculus of variations. The limit case, Part 2". The Hardy-Littlewood-Sobolev (HLS) ...
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### 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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### Relation between weak convergence of probability measures and weak-* convergence

I am trying to nail down the relation between probability and functional analysis. In particular, how the notion of weak convergence used in probability theory is related to the weak-* convergence of ...
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### Convergence of Hilbert transform of a converging sequence

Fix $n$, and consider random variables $x_1, \dots, x_n$ whose joint p.d.f. is $p_n$. Assume that the empirical distribution of $x_1, \dots,x_n$ converges weakly almost surely to the probability ...
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