# 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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### Pointwise limit of continuous functions, but not Riemann integrable.

I am trying to find a simple example of a function $f:[0,1]\rightarrow\mathbb{R}$ which is a pointwise limit of continuous functions, but is not Riemann integrable. I know the classical example where ...
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### Finding the Pointwise Limit of a Function

If I have a sequence of functions $f_n[0,2] \rightarrow \mathbb{R}$ where $f_n(x) = \frac{x^n}{2^n+n}$. If I attempt to find the pointwise limit, I work out that by taking $x \in [0,2]$: We can ...
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### Prove the compactness and second countability of a subset of $\Bbb{R}^{L^1(G)\times D}$

I'm trying to fulfill the details in a proof whose aim is to guarantee the existence of a compact space with certain properties, but in this case I had no success. Let $X$ be a metric space (actually,...
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### Pointwise convergence vector valued continuous functions

Let $(X,\|\cdot\|)$ be a (infinite dimensional) Banach space and $f_{n}:[0,1]\longrightarrow (X,\|\cdot\|)$ continuous, for each integer $n\geq 1$, a sequence of mappings. There is some "pointwise ...
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### Show convergence of the series of functions $\sum_{n=1}^\infty \frac{x}{n^\alpha (1+nx^2)}$

Show convergence of the series of functions $\sum_{n=1}^\infty \frac{x}{n^\alpha (1+nx^2)}$ on $\mathbb{R}$, where $\alpha > \frac{1}{2}$ My attempt: It suffices to show that for every fixed ...
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### Where to start? Sequence of functions.

I've been playing around with composing translations with the modulus function. Let $f_0(x) = |x|$, and $f_{n+1}(x) = \left| f_n(x)-\frac{1}{n+1}\right|$, where $n$ is a positive integer. I would ...
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### Sequence of continuous functions over a compact set that converges pointwise and monotonically also converges uniformly

I have the following problem: Let $\{f_n\}_{n = 1}^\infty$ be a sequence of functions such that, for all $n \in \mathbb{N}$, $f_n:[0,1] \to [0,1]$ is continuous, and for all $x\in [0,1]$, we have ...
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### Given a $f \in L^1(\mathbb{R})$ is there a $g : \mathbb{R} \to \mathbb{R}$ such that $g(f) = 0$ with $g \neq 0$?

Given a $f \in L^1(\mathbb{R})$ is there a $g : \mathbb{R} \to \mathbb{R}$ such that $g(f) = 0$? I would like to solve the problem myself but don't know how to start. Point of this question; I have ...
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### Does taking the limit in one argument preserve regularity in another argument?

Preliminaries Let $T>0$ be a positive constant, $d,n \in \mathbb{N}_{\geq 1}$ be natural numbers and $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space. For $m \in \mathbb{N}_{\geq 0}$, ...
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### Almost everywhere pointwise convergence vs pointwise convergence

Let $X$ and $Y$ be compact metric spaces. Suppose that $\{f^n\}$ is a sequence of continuous maps $f^n:X\to Y$, and suppose that for every Borel probability measure $\mu$ on $(X,\mathcal B(X))$, there ...
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### Doubts about moving limit under integral

We have to count following integral: $$\lim_{n\to\infty}\int_A\frac{\ln(n+y^3)-\ln n}{\sin(x/n)}d\lambda_2$$ where $A=\{(x,y)\in\mathbb{R}^2_+:1<xy<4,1<y/x<4\}$ Now in order to find ...
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### Why does $h\circ f_{n}\to h\circ f$ when $h$ is continuous and $f_n\to f$ pointwise?

Let $(X,d_{X})$ be a metric space, and for every integer $n\geq 1$, let $f_{n}:X\to\textbf{R}$ be a real-valued function. Suppose that $f_{n}$ converges pointwise to another function $f:X\to\textbf{R}$...
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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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### Mathematical intuition why the iterative Bellman update converges to the optimal solution

I know that the mathematical justification for using the Bellman-equation iteratively to find the optimal policy in Reinforcement Learning is based on convergence results. I wonder however whether ...
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### Beta distribution converges in distribution to a binomial

I have the following problem. Let $X_n \sim \operatorname{Beta}(1/n,1/n)$ be a sequence of beta distributions, and $X \sim \operatorname{Bin}(1,1/2)$ which is equivalent to $\operatorname{bern}(1/2)$....
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### joint convergence in distribution by convergence of conditional expectations

Let $(X_n,Y_n)$ be a sequence of random vectors in $\mathbb R^k\times \mathbb R^l$ such that $X_n\to X$ in distribution. Denote the conditional distributions $Y_n$ given that $X_n=x$ by $P_{n,x}$. I ...
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### Uniform/Pointwise Convergence

So, I have a somewhat arbitrary question about this topic but first I need to state the theorem I am familiar with and from which I will base my question. Theorem: Let $(f_n)_n:[a,b] \to \mathbb{R}$ ...
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### Convergence in distribution of empirical CDF

Without using the fact that the empirical cdf $F_{n}(x)$ converges to $F(x)$ under higher modes of convergence, I want to show that the empirical cdf converges to F(x) in distribution. In other words, ...
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### Continuity and pointwise convergence doesn't imply contiunity

We have a function $f$ and a sequence of functions $f_n$, both on $[a,b] \to \mathbb{R}$. $f_n$ is continuous for each $n \in \mathbb{N}$, and $f_n$ converges pointwise to $f$. I am asked to give an ...
### If a series converges uniformly on $[a+\varepsilon,b)$ for all $\varepsilon>0$ and pointwise on $[a,b)$, is then the convergence uniform on $[a,b)$?
The question arises from the specific series $\sum_{k=0}^\infty \dfrac{k (ex)^k}{(k+1)^2}$, which converges pointwise on $[-1/e,1/e)$ and uniformly on $[-1/e,1/e-\varepsilon]$ for every \$\...