My intended answer to a weak LLN problem on Math.SE.

Problem: Suppose $(X_n)$ is a sequence of r.v’s satisfying $P(X_n=\pm\ln (n))=\frac{1}{2}$ for each $n=1,2\dots$. I am trying to show that $(X_n)$ satisfies the weak LLN.

The idea is to show that $P(\overline{X_n}>\varepsilon)$ tends to 0, but I am unsure how to do so.

My solution: As in the accepted answer in OP’s previous question https://math.stackexchange.com/q/3021650/290189, I’ll assume the independence of $(X_n)$. By Chebylshev’s inequality,

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The old icon for Staticman Lab was made by GIMP from Staticman’s icon in PNG in the GitHub repo. Recently, I’ve found the SVG version of this icon. To serve customers better, I’ve recreated the logo from this SVG file so that the edges in the logo become sharper.

I intended to answer Maddle’s $p$-test question, but T. Bongers has beaten me by two minutes, so I posted my answer here to save my work.

The problem statement

This is the sum: $$\sum\limits_{n=3}^\infty\frac{1}{n\cdot\ln(n)\cdot\ln(\ln(n))^p}$$ How do I tell which values of $p$ allow this to converge? The ratio test isn’t working out for me at all.

Unpublished solution

The integral test will do.

When $p \ge 1$, the improper integral diverges. When $p < 1$, it converges.

Aim

To merge a pull request.

How?

Let’s take Staticman PR 231 as an example. I would like to test it before commiting this merget to Heroku.

$ cd ~/staticman
$ git branch -a
* deploy
  dev
  master
  ...
$ git remote -v
eduardoboucas https://github.com/eduardoboucas/staticman.git (fetch)
eduardoboucas https://github.com/eduardoboucas/staticman.git (push)
heroku  https://git.heroku.com/staticman3.git (fetch)
heroku  https://git.heroku.com/staticman3.git (push)
...
$ git pull eduardoboucas pull/231/head:deploy
remote: Enumerating objects: 10, done.
remote: Counting objects: 100% (10/10), done.
remote: Total 18 (delta 10), reused 10 (delta 10), pack-reused 8
Unpacking objects: 100% (18/18), done.
From https://github.com/eduardoboucas/staticman
 ! [rejected]        refs/pull/231/head -> deploy  (non-fast-forward)

I executed the last command on branch dev. I didn’t have time to figure out the reason for this error. The following commands should work.

Quick links

1. A minimal demo site on GitLab (Source)
2. Beautiful Hugo pull request 222
3. Pre-release notes for this pull request

Motivation

For the mathematical ones, please see my previous post.

As a math student, it’s inefficient to reinvent the wheel like engineering students. Thanks to three existing examples, I had convinced myself that I could bring this to the theme Beautiful Hugo.

• Zongren’s Hexo theme (worked best)

• Made Mistakes Jekyll theme

• Network Hobo’s customization of Beautiful Hugo (inspired by the second one, but contains a logic error)

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Vision

• gain autonomy: freedom is the basis of moral actions. No freedom, no morality.
• transcend ourselves: change/improve our lives through free thoughts

Goal

Convert our free thoughts into free code.

Free code allows users around the world to run and/or improve them. This would bring real enhancement to our tools.

For example, beautiful math writing used to be a complicated process. A decade ago, this required the installation of a typesetting engine called $\LaTeX$. Thanks to freely available scripts like MathJax and $\KaTeX$, it’s now possible to write math viewable by any modern web browser by writing the content in the middle.

Oriented affine $k$-simplex $\sigma = [{\bf p}_0,{\bf p}_1,\dots,{\bf p}_k]$

A $k$-surface given by the affine function

$$ \sigma\left(\sum_{i=1}^k a_i {\bf e}_i \right) := {\bf p}_0 + \sum_{i=1}^k a_i ({\bf p}_i - {\bf p}_0) \tag{1}, $$

where ${\bf p}_i \in \R^n$ for all $i \in \{1,\dots,k\}$.
In particular, $\sigma({\bf 0})={\bf p}_0$ and for each $i\in\{1,\dots,k\}$, $\sigma({\bf e}_i)={\bf p}_i$.

Standard simplex $Q^k := [{\bf 0}, {\bf e}_1, \dots, {\bf e}_k]$

A particular type of oriented affine $k$-simplex with the standard basis $\{{\bf e}_1, \dots, {\bf e}_k\}$ of $\R^k$.

$$ Q^k := \left\{ \sum_{i=1}^k a_i {\bf e}_i \Biggm| \forall i \in \{1,\dots,k\}, a_i \ge 0, \sum_{i=1}^k a_i = 1 \right\} $$

Note that an oriented affine $k$-simplex $\sigma$ has parameter domain $Q^k$.

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Dans l’article de Robert Fortet et Edith Mourier en 1953, une distance entre deux mesures de probabilité sur un espace métrique est définie.

De nos jours, je trouve la façon dont ils l’ont écrit assez difficile à comprendre. Je suis plus à l’aise avec $\sup$ que “b.s.” que désigne “borne supérieure”. Ils se sont servi de $M[f]$ pour $\lVert f \rVert_{\rm Lip}$, où

$$ \lVert f \rVert_{\rm Lip} = \sup_{x \ne y} \frac{|f(x) - f(y)|}{d(x, y)}. $$

Notations

Supposons toutes les notations dans Espace de trajectoires.

Problématique

La mesurabilité de l’application dans le sous-titre est basée sur l’égalité suivante.

J’ai passé quatres heures pour comprendre

• pourquoi ça entraîne la mesurabilité ?
• pourquoi l’égalité elle-même est vraie ?

Réponses

Mesurabilité de la trace sur $\CO$ de $\Bor{\R}{\OXT}$

A la première lecture, je ne connaisais même pas la définition de la trace d’une tribu sur un emsemble. En effet, c’est une définition universaire sur des ensembles, selon une question sur la trace sur Math.SE.

Motivation

The books that I read in the past didn’t explain what a dataframe meant.

Definition

Dataframe

A table of data in which the values of each observed variable is contained in the same column.

Counterexample

I’ve difficulty in reading long lines of text like the above definition, so let’s illustrate this definition with a counterexample.

We have carried out repeated experiments with four types of things and obtaine some data. (Say, poured some liquid into an empty cup and take the temperature.)