# Questions tagged [pointwise-convergence]

For questions about pointwise convergence, a common mode of convergence in which a sequence of functions converges to a particular function. This tag should be used with the tag [convergence].

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### Given $U\subseteq\mathbb{C}$ an open set and $f : X\rightarrow \mathbb{C}$ measurable then $f^{-1}(U)$ is measurable

Given the measurable space (X,$B$) where $B$ is a $\sigma$-algebra, I want to show that if $f : X\rightarrow \mathbb{C}$ is measurable (in the sense that, there exists a sequence of simple functions ...
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### Almost sure convergence for GARCH(1,1)-process

I'm proving the conditions for strict stationarity of GARCH(1,1)-process: $$X_t=\sigma_t Z_t\qquad \sigma_t^2 = \alpha_0 + \alpha_1 X_{t-1}^2 + \beta_1\sigma_{t-1}^2.$$ We can rewrite the process to a ...
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### Evaluate point convergence and uniform convergence of given funcion sequence

I'm given a function sequence as such: $$f_n(x) = \frac{x^n}{n^n}$$ over $$x \in [0,1]$$ I have to find what is the point convergence and uniform convergence of this sequence. What is the proper ...
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### Weak and Pointwise Convergence question

I found an interesting property in some lecture notes on weak convergence: lemma 3.2 (2) on page 10: https://www.uio.no/studier/emner/matnat/math/MAT4380/v06/Weakconvergence.pdf . The idea is as ...
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### If a function does not converge point-wise, then it doesn't converge uniformly as well, right?

Here is the function: $f_n(x) = \sqrt{\sin^2 x + n^{-4}}, \ \ x \in R$. If $x \neq \frac{\pi}{2}+2\pi k \text{ or } x \neq 2\pi k, \ \ k \in Z$ then the function doesn't converge at all even point ...
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### Convergence of a series with bounds depending on the iteration

I have a positive sequence $f_k$. Every $T$ iterations, an event $A$ happens and I can prove that, for this iteration, I have the bound \begin{align} f_{k+1} - f_k \leq - (\alpha_k)^2 ~~~~~(1) \...
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### Almost sure convergence of an algorithm

You designed an algorithm and you proved that this algorithm converges almost surely. Let's note $X_n$ the value of your algorithm at step $n$, which is a random variable, and $X$ the true converged ...
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### What does $\sum_{m=1}^n \ln (1 -\frac{\theta^2}{n}+ \frac{\theta^2}{2n m(\log m)^2} )$ converge to?

What does the following sum converge to as $n\rightarrow\infty$? $$\sum_{m=2}^n \ln (1 -\frac{\theta^2}{n}+ \frac{\theta^2}{2n m(\log m)^2} )$$ I think it converges to $-\theta^2/2$, but I am not ...
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### Prove point-wise and uniform convergence of function (Carothers 10.9.f)

I am trying to complete exercise 10.9.f in Carothers Real Analysis. The ask is to provide a formal proof that $nxe^{-nx}$ converges pointwise and determine if it uniformly converges (if not, find a ...
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### Does pointwise converge imply uniform convergence in some interval?

I was wondering if $f_n$ converges pointwise to $f$ on a closed interval, $E$, must there be closed interval $E' \subseteq E$ such that $f_n|_{E'}$ converges uniformly to $f|_{E'}$. Currently, with ...
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Let $M_n$ be a square integrable martingale with $M_0 = 0$ and let $M_n^2 = M_0^2 + X_n + A_n$ be the Doob decomposition of $M_n^2$ i.e. $A_n = \sum_{i=1}^nE(M_i^2 - M_{i-1}^2|\mathcal{F}_{i-1}) = \... 2answers 44 views ### Will we have uniform convergence in this case? Assume that$f: [a,b] \rightarrow \mathbb{R}$is continuously differentiable. Then we know that for each$t \in [a,b]$we have that $$\frac{f(t+\Delta t)-f(t)}{\Delta t}-f'(t)$$ will converge to ... 1answer 46 views ### Limit of sequence of functions$g_n(x) = n - n^2x $I found the following sequence in an answer of a different question, see https://math.stackexchange.com/a/1919759/579544 Consider the sequence of functions$g_n : [0,1] \to \mathbb{R}$defined by ... 3answers 47 views ### An increasing sequence of non-negative functions in$\mathcal{L}_{1}(X, \mu, \mathbb{R})$is$\mathcal L_1$-Cauchy To avoid any ambiguity, I first present the related definitions: Let$(X, \mathcal{A}, \mu)$be a complete,$\sigma$-finite measure space and$(E,|\cdot|)$a Banach space. We say$f \in E^{X}$is$\...
I need some help with this exercise: Let $n \in N$ and $f_n:[0,1]\Rightarrow R$ with $f_n(x):=\frac{n^\alpha \cdot \ x}{1+n^2x^2}$, and $\alpha\ge0$ a) For which $\alpha$ is $f_n$ pointwise ...