First moment

Population: $ \Omega = \{ x_1, \dots, x_N \} $
Collection of $n$-samples: $\mathcal{S} = \{ s \in \Omega^n \mid \forall i,j \in s, i \ne j \} $
Collection of $n$-samples containing $x$: $ \mathcal{S}_x = \{ s \in \mathcal{S} \mid x \in s \} $
Observe that $ |\mathcal{S}_x| = \binom{N-1}{n-1} $.
Let population mean be zero. $\mu = 0$, i.e. $ \sum_{i = 1}^N x_i = 0 $
Fix an order for $\mathcal{S}$: $ \mathcal{S} = \{ s_1, s_2, \dots, s_{|\mathcal{S}|} \} $.
$j$-th $n$-sample mean $ m_j = \frac1n \sum_{x \in \mathcal{S}_j} x $
Remark: I don’t use $ \sum s_j $ as in $ \cup \mathcal{T} $ in topology to avoid misreading the $n$-sample $ s_j $ as an element.
mean of $n$-sample mean $ m = \frac{1}{|\mathcal{S}|} \sum_{s_j \in \mathcal{S}} m_j $

I intend to post this for a Borel-Cantelli lemma exercise on Math.SE.

The target event is ${\exists i_0 \in \Bbb{N} : \forall i \ge i_0, X_i = 1}$, whose complement is

$$ {\forall i_0 \in \Bbb{N} : \exists i \ge i_0, X_i > = 0} = \limsup_i {X_i = 0}. $$

To apply Borel-Cantelli, one has to determine whether $\sum_i P(X_i = 0)<+\infty$.

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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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.

I jotted down only a few keywords that might be reusable. I didn’t understand any of the talks.

Functional Data Analysis

• Goal: predict equipment temperature
• Tools: Fourier coefficients (trigo ones), followed by discretisation, min-error estimation, cross-validation 10-folds, $R^2$ adjusted ?, MAE, MSPE
• Comparison with non-functional data

Tolérancement

• Thème : Traiter les incertitudes sur les dimensions des pièces de l’avion
• Objectif :
  • établir une modélisation mathématiques
  • construire un virtual twin de l’avion
• Outils :
  • Modèle de variabilité
  • Modèle d’assemblage $\text{airbus}: Y = \sum_{i = 1}^n a_iX_i?$
  • Notion de risque … calculs des coefficients de convolution

SVM

• Multiclass vs structual, hidden Markov model
• Plan for this year:
  • apply structual SVM for real SVM
  • apply structual SVM for deep neural network

Auxiliary information

• auxiliary function given in one partition
• auxiliary function given in mutiple partitions
• bootstrap
• law of iterated logarithms
• Kullback–Leibler distance
• convergence: Donsker class, var, covar
• ranking ration method: convergence to Gaussian process, entropy conditions, Telegrandś inequality
  • weak convergence: KMT, Berthet-Maison
  • strong convergence: ?
    • consequences: Berry-Essen bound, bias & variance estimation of ranking ration method

Euler scheme SDE

I could only write “Toeplitz tape operator”.

Problem

To show that a measure $\mu$ defined on a metric space $(S,d)$ is regular.

1. outer regularity: approximation by inner closed sets
2. inner regularity: approximation by outer open sets

Discussion

Since this problem involves all borel sets $A \in \mathcal{B}(S)$, the direct way $\forall A \in \mathcal{B}(S), \dots$ won’t work. We have to use the indirect way: denote $$\mathcal{C} = \lbrace A \in \mathcal{B}(S) \mid \mathinner{\text{desired properties}} \dots \rbrace.$$ Show that

Tribu produit

Je trouve $\Er{\OXT}$ plus court à écrire, tandis que $\bigotimes_{t \in \Bbb{T}} \Er$ est plus flexible.

Statement

Slogan version

$$\sigma = \pi + \lambda$$

$\sigma$	$\pi$	$\lambda$
“sum”	“product”	“limit”
universe	nonempty	universe
complement		complement
countable union	finite intersection	disjoint countable union

A $\sigma$-algebra is a $\pi$-system and a $\lambda$-system, and vice versa.

Wiki version

A $\lambda$-system is a synonym of a Dykin system.

$$\mathcal{P} \subseteq \mathcal{D} \Longrightarrow \sigma(\mathcal{P}) \subseteq \mathcal{D}$$

Given a $\pi$-system contained in a $\lambda$-system. Then the $\sigma$-algebra generated by the $\pi$-system is also contained in the $\lambda$-system.

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