# 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 showing that if $\mu_n$ converges to $\mu\in M_1(X)$ in weak* topology, then $\lim_n\int_X fd\mu_n=\int_X fd\mu$ for every Lipschitz function $f$

Let $(X,d)$ be a metric space and $M_1(X)$ the set of Borel probability measures over $X$. Suppose that $\mu_n$ is a sequence in $M_1(X)$ such that $\mu_n$s converge to some $\mu\in M_1(X)$ in weak* ...
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### Weak convergence of functionals $g_n^*(f) = n\int_0^1 x^nf(x)dx$ [closed]

Show that sequnce of functionals $g_n^*(f) = n\displaystyle{\int_0^1 x^nf(x)dx}, f \in C[0,1]$ converges weakly and find its limit functional. Does it converge in the norm of space $C^*[0,1]$? I don'...
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### Proving that convergence of a series implies weak convergence in an inner product space

I'd like to prove the following: If $\sum_{n=1}^\infty u_n = u$, then $\sum_{n=1}^\infty \langle u_n, x \rangle = \langle u, x \rangle$ for any $x$ in an inner product space. Is the following proof ...
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### Proof of Helly Bray theorem: $X_n\overset{d}{\rightarrow}X\Leftrightarrow Eg(X_n)\rightarrow Eg(X)$ for bounded continuous $g$

Helly Bray Theorem $X_n\overset{d}{\rightarrow}X\Leftrightarrow Eg(X_n)\rightarrow Eg(X)$ for all bounded, continuous functions, $g$. For the only if part of the proof, I am a little stuck. I have ...
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### Folland: Convergence in weak topology iff convergence in dual

I am self-learning real analysis from Folland and got stuck on weak topology convergence. He defines weak topology as follows: If $X$ is a normed vector space, the weak topology on $X$ is the weak ...
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### Metrization of topologies of weak convergence of measures and other notions of convergence

Let $(X,d)$ be a metric space. Denote $\mathcal P(X)$ the space of probability measures on $X$. Let, $(\mu_n) \subset \mathcal P(X)$, the usual notion of weak convergence of probability measures reads:...
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### Carleman's theorem on a compact interval

Let $\mu_n$ be Borel probability measures on $[0, 1]$. I'm trying to prove a special case of Carleman's theorem, i.e., Theorem If the sequence $(\int_0^1 x^k \mathrm d \mu_n (x), n\in \mathbb N)$ ...
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### Some problems in the application of Arzelà–Ascoli theorem

Let $\Omega$ be a bounded domain in $\mathbb{R}^d$ with a smooth boundary. Consider the sequence $\{u_n(\cdot,s)\} \subset L^2(0,T;L^2(\Omega))$ such that $\lVert u_n(\cdot,s) \rVert_{ L^2(\Omega)}$ ...
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### Weak Convergence in Hilbert Spaces & Inner Product

I have a question about weak convergence in Hilbert spaces. Suppose we have a Hilbert space $H$ equipped with a scalar product $(\cdot / \cdot)$, and that in this space we have $x_n \rightharpoonup x$ ...
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### Clarifying when a weak limit can be assumed to be a strong limit

In the second answer to this post this it is stated: Since you've already proved that there is a strongly convergent subsequence, let's say $Tu_{n_k} \to u^*$ for $k \to \infty$. Then by the weak ...
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I tried to solve the following problem and want to know if my approach is correct and some help in order to finish. Thanks. Let $V = L^\infty(\mathbb{R})$ equipped with the norm $\left\lVert \cdot\... 1 vote 1 answer 68 views ###$ f_n(x) := n^{\frac{1}{p}} \chi_{[0,1]}(n x) $weakly converges to$0$for$p>1$, but not for$p=1$As the title says, I am trying to show the weak convergence of the function sequence$ n^{\frac{1}{p}} \chi_{[0,1]}(n x) $, which is supposed to converge to$0$for$p>1$but not for$p = 1$. I ... • 67 1 vote 0 answers 41 views ### showing weak convergence for scaling limit of basic Markov jump process Let$X_t$be a continuous-time, finite state, real-valued Markov jump process with generator matrix$G$. We can just assume the process takes values in$\{0,1,2,\ldots,m\}$. I think this works more ... • 5,890 2 votes 0 answers 30 views ### Notion of convergence of random variables of growing dimensionality When$Y_1, Y_2, \dots Y_n$be a sequence of$d$-variate random variables we have the notion of convergence in distribution where we say$Y_n \implies Y_{\infty}$if the corresponding distribution ... • 21 2 votes 2 answers 70 views ### Failure of Banach Alaoglu in$L^1$We know that dual of$L^{\infty}(\mathbb{R})$is not$L^{1}(\mathbb{R})$and hence Banach Alaoglu theorem is not applicable for the$(L^1,L^{\infty})$pairing. In other words if$\{f_n\}$is a ... • 364 0 votes 0 answers 43 views ### Is there a version of Levy's Continuity Theorem for stochastic processes? We know that a stochastic process$X = (X_{t})_{t \in \mathbb Z}$defined in$(\Omega, \mathcal F, \mathbb P)$with real values can be expressed as a measurable function with values in some space of ... • 616 0 votes 0 answers 40 views ### Application of Continuous Mapping theorem to logarithm of random variables In the Wikipedia page of Continuous Mapping Theorem, it reads if$g$is any function between two metric spaces and$X_n$converges in law to$X$which almost surely will not take value in the ... • 1,011 1 vote 1 answer 40 views ### Why is the operatornorm not a weak* continuous map? Let$X$be a normed space and consider $$||\cdot ||:X^*\rightarrow \Bbb{R};~~f\mapsto ||f||$$ Then in a lecture side note I read that$||\cdot||$is not weak* continuous. As I understood it, we say$|...
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$\textbf{Question}$ Let $\left(\mathcal{H}, \| \cdot \| \right)$ be a real Hilbert space and let $C$ be a subset of $\mathcal{H}$ such that $( \forall n \in \mathbb{N} ) ~ C \cap B(0;n)$ is weakly ...
Consider a sequence of probability measure $\mu_1, \cdots, \mu_N$ with support on non-negative real numbers, $\mathbb{R}_{\geq 0}$. Now, let for each $1 \leq i \leq N$, $\hat{\mu}_i$ be the ...