Probability on Solarized Sublime Sekai

Finite Population Sampling without Replacement

Fri, 26 Apr 2019 09:46:29 +0200

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 $

Borel Cantelli Exercise 2019

Mon, 04 Feb 2019 16:24:40 +0100

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 $\sum_i P(X_i = 0)<+\infty$.

Weak LLN Practice

Sun, 02 Dec 2018 08:11:56 +0100

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,

Mesurabilité des réalisations trajectorielles

Thu, 11 Oct 2018 23:20:37 +0200

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.

$$ \Bor{\R}{\OXT} \cap \CO = \Bor{\CO} $$

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.

2018-10-04 Seminar Notes

Thu, 04 Oct 2018 02:16:44 +0200

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”.

Measures Are Regular

Wed, 03 Oct 2018 22:28:34 +0200

Problem

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

outer regularity: approximation by inner closed sets
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

Espace de trajectoires

Fri, 28 Sep 2018 10:41:32 +0200

Tribu produit

source	symbole	engendrée par
Prof	$\Er{\OXT}$	$\mathcal{C}_0 = \Big\lbrace \lbrace f \in E^\Bbb{T} \mid f(t) \in B \rbrace \bigm\vert t \in \Bbb{T}, B \in \Er \Big\rbrace$ $\mathcal{C}_1 = \Big\lbrace \lbrace f \in E^\Bbb{T} \mid f(t_i) \in B_i \forall i \in \lbrace 1,\dots,n \rbrace \rbrace \newline \bigm\vert t_j \in \Bbb{T}, B_j \in \Er \forall j \in \lbrace 1,\dots,n \rbrace, n \in \N^* \Big\rbrace$
Meyre	$\bigotimes_{t \in \Bbb{T}} \Er$	des cylindres $C = \prod_{t \in \Bbb{T}} A_t$ d’ensembles mesurables $A_t \in \Er$ de dimension finie $\card{\lbrace t \in \Bbb{T} \mid A_t \neq E \rbrace} < \infty$

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

$\pi$–$\lambda$ Theorem

Wed, 19 Sep 2018 15:56:35 +0200

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.

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

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.