# 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 against upper invariant measure

Setting I am studying invariant measures and their weak limits. In a book about probability on graphs the following setting is presented in chapter 6.3 (this is a short form of the actual presentation)...
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### Setwise convergence of measures implies weak convergence under special hypothesis

I'm struggling with producing a proof of the following result: Let $X = \overline{\mathbb{C}}$ be the Riemann sphere, and consider $M(X)$ the space of finite Borel measures on $X$ with norm given by ...
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### $\lim_{\delta\to 0} \delta^{-1}\int \phi\, dL^n\vert_{B(0, 1+\delta)\setminus B(0,1)} = \int \phi \, d\sigma^{n-1}$ for every $\phi\in C_0(\Bbb R^n)$

The surface measure $\sigma^{n-1}$ on the sphere $S^{n-1}$ is defined in Folland text, in the following way: Consider the homeomorphism $\Phi:\Bbb R^n\setminus\{0\} \to (0,\infty)\times S^{n-1}$ ...
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### Weak Convergence for a continuous function without compact support

Let $X$ be an open subset of $\mathbb R^n$ and let $\Omega$ be a relatively compact, open subset of $X$. Let $\{ \mu_n\}$ be a sequence of positive measures that converge weakly to the positive ...
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### How to show $\frac{1}{\delta} \mathbb{P}[ \sup_{t\leq s \leq t+\delta} |X(s)-X(t)|\geq \epsilon] \leq \eta$?

If a random element $X$ of $D[0,1]$ has the property that $$\lim_{\delta\to 0} \sup_{0\leq t\leq 1-\delta} \frac{1}{\delta}\mathbb{P}(|X(t+\delta)-X(t)|\geq \epsilon) = 0.$$ for every $\epsilon >0$,...
• 851
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### Question about proof that every probability measure on a metric space is regular

I have a question from the first proof in Billingsley's Convergence of Probability measures. In the following proof, why does the result follow from showing that $\mathcal{G}$ is a $\sigma$-field? ...
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1 vote
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### Is $\{a_n X_n +c\}$ this sequence bounded in probability?

Suppose $\{X_n\}$ converge weakly to standard normal distribution. Now we define a sequence $\{a_nX_n +c\}$ like that where $c>0$ is any constant and $a_n\rightarrow\infty$. Can we conclude that ...
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### Brownian motion has unbounded variation.

I tried to solve this problem in the following way. Suppose $\{B_t | t\in [0,1]\}$ is our Brownian motion. Define, $$f_n(w) = \sum_{k=1}^{2^n} \bigg|B_{\frac{k}{2^n}}(w)-B_{\frac{k-1}{2^n}}(w)\bigg|.$$...
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### Show there exists a subsequence that converges weakly

In PDE by Evans, Chapter 8.4.2. we want to minimize the energy functional $$I[w]: = \int \frac{1}{2} |Dw|^2 - fw dx$$ among all functions in $$A: = \{ w \in H^1_0(U): w \geq h \; \; a.e. \}$$ with ...
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I understand that $T_n\xrightarrow{\mathbb P}t$ does not imply that $\mathbb E[T_n]$ converges (including to $t$). At the same time, we know that convergence in probability, implies convergence in ...