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).
2,752 questions
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$f_n(x) := 1+\sin(n \pi x)$, $f_n^k$ converges weakly to a constant $c_k$. [closed]
I have the following exercise:
Let $p \in (1, +\infty)$. Consider the sequence $\left\{ f_n \right\}_{n \in \mathbb N}$ in $L^p ([0, 1])$ where $f_n (x) := 1 + \sin(n \pi x)$. Let $k \in \mathbb N$
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Conditions under which the potential energy is weakly continuous
This question is concerned with the following result in Lieb & Loss' Analysis:
More precisely, condition 11.3(14) is stated as follows:
My question is concerned with the case $n \geq 3$. The ...
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Does uniform convergence on compact subsets of $\mathbb{R}$ imply weak convergence in $L_p(\mathbb{R})$?
Let $f_n$ be a sequence in $L^p(\mathbb{R})$ with $1 < p < \infty$ so that $f_n$ converges uniformly to $f$ on every compact subset of the real line. Find whether or not $f_n$ converges weakly ...
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Set of uniformly bounded integer-valued measures is closed in weak star topology
I am trying to prove the following statement:
Let $(X,d)$ be a compact metric space. The set of all integer-valued Borel measures on $X$, that are uniformly bounded by an integer $\alpha$ (that is, $a(...
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When is the weak limit of indicator/characteristic functions an indicator/characteristic function?
I've been reading the following posts (Link1, Link2) about weak limits of indicator/characteristic functions. It is clear to me that in general the weak limit of indicator functions may not be an ...
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Distribution of sum of product [closed]
Let $x_n\overset{p}{\to}c$ and $x_n\overset{d}{\to}N(0,\sigma^2)$ denote convergence in probability to a constant $c$ and convergence in distribution to a random normal variable (with some abuse of ...
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Convergence of the product of weak $L^1$ convergent functions in kinetic contexts
"…and that we can extract a subsequence, again denoted by $f^n(t)$, which converges weakly in $L^1(\Re^3)$ to $f(t)$, for all $t \in [0,T]$. Therefore:
$$ \int d\xi \, d\xi_* \, \phi(\xi,\xi_*) \,...
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Finding sequences of Gaussian measures for different modes of convergence (norm, narrow, weak*)
I am working on a problem about modes of convergence for measures and would like to find sequences of Gaussian measures that satisfy specific criteria.
Let $\mu_{m,s}$ be the Gaussian probability ...
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Is conditional mutual information $I(X;Y\mid Z)$ lower semicontinuous for general $X,Y,Z$?
Assume $(X_n,Y_n,Z_n)\Rightarrow (X,Y,Z)$ weakly on standard Borel spaces. Is it always true that
$$I(X;Y\mid Z)\ \le\ \liminf_{n\to\infty} I(X_n;Y_n\mid Z_n)?$$
It is classical that relative entropy $...
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Integral operator on the space of Radon measure
Let $X \subset \mathbb{R}^d$ be compact and let $ K : X \times X \to \mathbb{R}$ be a continuous function. Call $\mathcal{M}(X)$ the space of Radon measure on $X$.
We define the integral operator $T:\...
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A question related to a convergence of Levy measures
I am working on the preservation of symmetry for Lévy measures under a specific type of convergence. First, recall that a Lévy measure $\mu$ on $\mathbb{R}^k$ is symmetric if $\mu(E) = \mu(-E)$ for ...
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Does a kind of subsequential weak lower semicontinuity imply weak lower semicontinuity?
Let $X$ be Hilbert and let $f\colon X \to \mathbb{R}$ satisfy the following: for every sequence $x_n$ such that $x_n \rightharpoonup x$ (weakly) in $X$, there exists a subsequence $n_j$ such that
$$f(...
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Does the adjoint of a compact operator map a weak* convergent net to a norm convergent net?
Continuing from the following discussion:
Does the adjoint of a compact operator maps a weak* convergence sequence to norm convergence?
Let $X, Y$ be two Banach spaces and $T:X\to Y$ be a compact ...
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Does Weak Convergence and Variance Convergence Imply Uniform Integrability of the Lévy Measures' Second Moment?
$\DeclareMathOperator{\varr}{\operatorname{var}}$
$\DeclareMathOperator{\tr}{\operatorname{tr}}$
Let $\{X_n\}_{n \ge 1}$ be a sequence of zero-mean, $p$-dimensional infinitely divisible (ID) random ...
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Does Weak Convergence plus a Uniform Moment Bound Imply Uniform Integrability for Lévy Measures?
Let $\{X_n\}_{n \ge 1}$ be a sequence of $p$-dimensional infinitely divisible (ID) random vectors, and let $\{\mu_n\}_{n \ge 1}$ be their corresponding Lévy measures.
Suppose we have two conditions:
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