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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Convergent subsequences in $L^p(\mathbb R^n)$

This is a particular fact that I ended up proving in the process of attempting one of my recent homeworks, but I don't think I've seen this particular fact online even though it feels like a fairly ...
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If $\sum_{n=1}^\infty a_n^2$ converges, then $\lim_{n\to \infty} \frac 1 {\sqrt n} \sum_{k=1}^n a_k =0$

Problem. Prove that if $\sum_{n=1}^\infty a_n^2$ converges, then $\lim_{n\to \infty} \frac 1 {\sqrt n} \sum_{k=1}^n a_k =0$. The problem arises from the following question: Let $(e_i)_{i=1}^\infty$ ...
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Convergence of scalar product in $L^2$ space. [closed]

Let $\Omega\subset\mathbb{R}^3$. If $u^\epsilon\to u ,\text{in}\ (L^2(\Omega))^3$, as $\epsilon\to 0$, and $v^\epsilon\to v ,\text{in}\ (L^2(\Omega))^3$, as $\epsilon\to 0$, then can we ...
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Weak convergence in $L^2$ equivalence

Problem statement: Denoting by $B_r$ the open ball of $\mathbb{R}^N$ centered at the origin with radius $r$, consider a sequence $f_n \in L^2(B_1)$ which is bounded in the $L^2$ norm. Prove that $f_n$ ...
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Extracting a subsequence which converges in the $t$-Wasserstein distance?

Posted this to MathOverflow as well. Assume that $\mu_n$ are probability measures on $\mathbb R ^d$ with finite moments of order $t$, and $\mu_n\to\mu$ weakly. Clearly, $\int |x|^t d\mu_n(x)$ is a ...
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Conditions on the weak * convergence to be strong.

I know, thanks to Brezis, that on uniformly convex Banach spaces we have that the strong convergence is equivalent to weak convergence and the convergence of norms. I cannot find anything comparable ...
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Interchanging infinite sum and limit in distribution

I'm trying to do a proof for a project and I've run into the following problem. For each $j$ consider a sequence $(Y_{j,n})_{n \in \mathbb{N}}$ of random variables such that the different sequences ...
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Show that $(X_n, Y_n) \stackrel d \to (X,Y).$

Let $\{X_n \}_{n \geq 1}$ and $\{Y_n \}_{n \geq 1}$ be independent random variables such that $X_n \stackrel d \to X$ and $Y_n \stackrel d \to Y.$ Suppose that $X$ and $Y$ are also independent. Then ...
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Sufficient conditions for finitely supported measures being dense

Let $(X,\mathcal{B})$ be a Hausdorff topological space with its Borel $\sigma$-algebra. What are some general conditions we could impose on $X$ so that finitely supported measures (i.e. finite affine ...
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Show that there exists $x \in \mathbb R$ such that $\mathbb P (Y = x ) = 1.$

Let $\{x_n \}_{n \geq 1}$ be a sequence of real numbers and $\{X_n \}_{n \geq 1}$ be a sequence of random variables such that $\mathbb P (X_n = x_n) = 1,\ n \geq 1.$ Let $Y$ be a random variable such ...
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Does the weak limit of a sequence in $L^2([0,1])$ vanish on the limit set of vanishing sets?

Suppose $h_n$ is a sequence of non-negative functions in $L^2([0,1])$ converging weakly to $h$ (i.e., for every $g\in L^2([0,1])$ it holds $\int g\cdot h_n \,d\lambda \to \int g\cdot h\, d\lambda$). ...
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Is it true that any sequence of random variables that converge in distribution is tight?

Let the sequence of real random variables $\{X_n\} \to X$ in distribution (but not necessarily in probability). Is it true that $\{X_n\}$ form a tight sequence, and if yes, how do we prove it? So we ...
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Definition of a weakly continuous map

I am familiar with the weak topology and a linear functional on a Banach space being weakly continuous, but in PDE I often see a statement along the lines of "$f$ is a weakly continuous solution ...
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How does the weak topology imply the definition of weak convergence?

Let $X$ be a Banach space. Then the weak topology on $X$ is defined as the coarsest topology such that the linear functionals in $X^*$ are continuous. A sequence $x_n \in X$ weakly converges to $x$ if ...
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Let $Q \in P(P(\Sigma))$ be given, whereby $\Sigma$ is a Polish space (complete and separable metric space) and $P(\Sigma)$ denotes the space of probability measures on $\Sigma$ and $P(P(\Sigma))$ the ...