# Tagged Questions

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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### Missing crucial step in the derivation of the Stirling's Formula via the Poisson Distribution and CLT

I believe that this is a particular neat example that we've done in class. Unfortunately there is one step I do not quite understand and my Professor had to skip due to the lack of time. I think this ...
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### Showing weak law of large numbers holds

My question: $\{X_n\}$ is a sequence of random variables. Var$(X_n)\le C\ \ \forall \ n$ and $\rho_{ij}=$Cov$(X_i,X_j)\to 0$ as $|i-j|\to \infty$ . Show WLLN holds. In my book there are 3 ...
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### Equivalence of norm and weak topologies in finite dimensional space.

I haven't solved a lot of problems on functional analysis, so I don't have intuition whether my idea of proof is correct or not. The weakness of weak-topology is obvious from the definition. To show ...
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### Every bounded sequence of dual space contains a subsequence which is weak* convergent

I were doing this problem in Functional Analysis of Erwin Kreyszig(part 4.9, problem 10, page 269), but got stuck in the last point to come to the conclusion. Can anyone give me some hint to move on? ...
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### Prove that a set which satisfies every nonempty subset contains a weak Cauchy sequence must be bounded

I got stuck on this problem about weak convergence in normed space. This problem is exercise 9, page 263 in Functional Analysis of Erwin Kreyszig. So the question is basically on the title, let me ...
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### Student test statistic and self normalizing sum.

I study the asymptotic distribution of self normalizing sums which are defined as $S_n/V_n$ where $S_n=\sum_{i=1}^n X_i$ and $V_n^2 = \sum_{i=1}^n X_i^2$ for some i.i.d RV's $X_i$. Motivation ...
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### A question concerning Mazur's Lemma

I have a problem with application of Mazur's Lemma. Just consider $B(H)$ when $H=\ell_2$. Then, $B(H)$ is a normed vector space. Then, take operators $$X_n:={\rm diag}(0,0,\cdots,0,1,1,1,1,\cdots)$$...
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### Showing a sequence converges weakly.

Let $f \in L_2(\mathbb{R})$. How can I show that the sequence ${g_n}$ converges weakly to $0$ in $L_2(\mathbb{R})$, where $g_n(x) = f(x − n)$? If this is not true could someone provide a counter ...
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### A weaker characterization of convergence in distribution

$X_n$ and $X$ are real-valued random variables. If $X_n \to X$ in distribution, then we know that $$P\{X_n \leq a\} \to P\{X \leq a\}$$ for every $a$ at which $x\mapsto P\{X \leq x\}$ is continuous. ...
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### Problem. Convergence. Banach space. Weak topology

Let E be a Banach space and let $(X_n)$ be a sequence such that $X_n \rightharpoonup x$ in the weak topology σ(E,E'). Set: $S_n=\frac{1}{n}\sum_{k=1}^n(-1)^kX_k$ Does $Sn \rightharpoonup x$ in the ...
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### Weakly convergent series

Consider sequence $\{x_n\}_{n = 0}^\infty$ in complex Banach space X. Suppose that for all $z \in \mathbb{C}$ such that $|z| < 1$ the series $\sum_{n = 0}^\infty z^nx_n$ is weakly convergent. How ...
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### Problem following proof of Šmulian theorem for separable space

I tried to solve Problem 10 on p. 464 of Brezis to get a proof of part of the Eberlein-Šmulian theorem, precisely the equivalence between compactness and sequential compactness in the weak topology of ...
Let E,F be any two Banach space and let $\mathcal{B}(E,F)$ be the space of all bounded linear operators from E to F. I can show that this space is a complete space with respect to the norm and strong ...