# 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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### How to show $\lim_{n \to \infty}f(x_n)=f(x)?$

consider the Banach space $\ell^1$, and let $e_i$ be the sequence $(0,\dots,1,0\dots)$, with $1$ in the $i$-th position. show that $\{e_i\}$ converges weakly to $0$ in $\ell^1$ but not strongly. My ...
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### Limiting distribution of $Y_n=\frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n X_i^3}$ where $X_i$'s are i.i.d $N(0,1)$

Suppose $X_1,X_2,\ldots,X_n$ are i.i.d standard normal variables. I am looking for the limiting distribution of $$Y_n=\frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n X_i^3}$$ Since $E(X_1^3)=0$, I don't think I ...
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### Closed convex hull of the tail sequence is exactly the weak limit [Haim Brezis Exercise 3.13]

This question is from Haim Brezis' functional analysis exercise 3.13, but generalizes it a little bit. Two related questions are posted here: closed convex hull of weak convergent sequence and here: ...
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### Weak convergence (in the context of Hilbert spaces) of sqaure integrable random variables.

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space and consider the Hilbert space $$\mathcal{X}:=\{X:\Omega\to \mathbb{R}~\mbox{measurable}: \mathbb{E}[X^2]<\infty\}$$ with scalar product ...
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### Compactness in $W^{1,p}_0(B_R)$

I have a sequence of functions $\{u_k\}_k \in u + W^{1,p}_0(B_R, R^N)$, where $B_R \Subset \Omega$ is the ball of radius R and $\Omega \subset R^n$ open and bounded, $u \in W^{1,p}(B_R, R^N)$. I want ...
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### Strong convergence vs Weak convergence _ compactness of integral varifolds

I am reading the proof of the compactness of Integral varifolds on L.Simon's book " Lecture on geometric measure theory", there is a part of the proof concerning the conclusion that the ...
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### Convex Functionals Weak Inferior Semicontinuity

I don't understand the step $\underline{\lim}f(x_n)=\lim f(x_m)$ in the prove. the subsequence in the prove is from Banach-Sacks below, which is the subsequence of a weakly convergent sequence, whose ...
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I have a double sequence of probability density functions $\rho_{n,m}\in C^\infty(\mathbb R^d$). Suppose that for every test function $\varphi\in C_c^{\infty}(\mathbb R^d)$ $\lim_{m\to\infty}\lim_{n\... 1answer 53 views ### Show that$(X_1 + \cdots + X_n )/n \to 0$in distribution by characteristic function There's a random variable$X$whose characteristic function is$\phi(t) = E(e^ {itX})$is given by $$\phi(t) = e^{-|t|^{3/2}}$$ Let$X_1, X_2, \ldots$the independent random variable with the same ... 1answer 26 views ### Are the pointwise and the weak* limit in$L^1$the same? Suppose$\Omega\in \mathbb{R}^n$is open and bounded and$f_n\in L^1(\Omega)$converge weakly* (as signed measures) to some$f\in L^1$and pointwise almost everywhere to some$g\in L^1$. Is$f=g$... 2answers 47 views ### Does weak convergence imply$L^2$-norm convergence? Let$\{f_n\}_{n \geq 1}$be a sequence of functions in$L^2[0,1]$converging to$f$weakly. Does it imply that$f_n \to f$in$L^2$-norm? This question appeared in an exam paper on functional ... 1answer 37 views ### Unitary conserving Schrödinger equation If we work in the spaces of$N\times N$matrices we can write the propagator Schrödinger equation as: $$\dot{U}(t)=-iH(t)U(t)$$ For some time-dependent Hermitian Hamiltonian$H(t)$. If we assume ... 3answers 267 views ### Conditions for weak convergence and generalized distribution functions I am having some trouble proving Corollary 6.3.2 in Borovkov's Probability Theory (for reference, this material is on pages 147 to 149 in the book). For convenience, I provide some definitions and ... 1answer 38 views ### Hilbert And Banach dual spaces Are, and in general, all Hilbert spaces identified to their dual ?, i.e$H'\equiv H$What about Banach spaces? And can someone please state the weak convergence in each? 1answer 32 views ### Weak convergence in$L^2$of powers of a sequence which is weakly convergent in$W^{1,2}$Let$\Omega\subset \mathbb{R}^3$be a bounded domain. Assume that$(f_n)_{n\in \mathbb{N}}\subset W^{1,2}(\Omega)$is weakly convergent to$f\in W^{1,2}(\Omega)$. Consider now$(f_n^3)_{n\in \mathbb{N}...
Can I say that if $\xi_n \xrightarrow {d} \xi$ and $\xi_n, \xi$ are non-negative random variables with finite expected value then $$\mathbb{E}\xi \le \lim{\inf{}_{n\to\infty}\mathbb{E}\xi_n}?$$ I have ...
Suppose $\left\{\mu_n\right\}_{n=1}^\infty$ is a sequence of measures in $\mathbb R$ that converges weakly to $\mu$. The function $x\mapsto h(y,x)$ is bounded and continuous for every $y\in \mathbb R$....