\documentclass[10pt,a4paper]{article}

% Packages
\usepackage{fancyhdr}           % For header and footer
\usepackage{multicol}           % Allows multicols in tables
\usepackage{tabularx}           % Intelligent column widths
\usepackage{tabulary}           % Used in header and footer
\usepackage{hhline}             % Border under tables
\usepackage{graphicx}           % For images
\usepackage{xcolor}             % For hex colours
%\usepackage[utf8x]{inputenc}    % For unicode character support
\usepackage[T1]{fontenc}        % Without this we get weird character replacements
\usepackage{colortbl}           % For coloured tables
\usepackage{setspace}           % For line height
\usepackage{lastpage}           % Needed for total page number
\usepackage{seqsplit}           % Splits long words.
%\usepackage{opensans}          % Can't make this work so far. Shame. Would be lovely.
\usepackage[normalem]{ulem}     % For underlining links
% Most of the following are not required for the majority
% of cheat sheets but are needed for some symbol support.
\usepackage{amsmath}            % Symbols
\usepackage{MnSymbol}           % Symbols
\usepackage{wasysym}            % Symbols
%\usepackage[english,german,french,spanish,italian]{babel}              % Languages

% Document Info
\author{andreyhorta35762}
\pdfinfo{
  /Title (procesos-estocasticos.pdf)
  /Creator (Cheatography)
  /Author (andreyhorta35762)
  /Subject (Procesos Estocasticos Cheat Sheet)
}

% Lengths and widths
\addtolength{\textwidth}{6cm}
\addtolength{\textheight}{-1cm}
\addtolength{\hoffset}{-3cm}
\addtolength{\voffset}{-2cm}
\setlength{\tabcolsep}{0.2cm} % Space between columns
\setlength{\headsep}{-12pt} % Reduce space between header and content
\setlength{\headheight}{85pt} % If less, LaTeX automatically increases it
\renewcommand{\footrulewidth}{0pt} % Remove footer line
\renewcommand{\headrulewidth}{0pt} % Remove header line
\renewcommand{\seqinsert}{\ifmmode\allowbreak\else\-\fi} % Hyphens in seqsplit
% This two commands together give roughly
% the right line height in the tables
\renewcommand{\arraystretch}{1.3}
\onehalfspacing

% Commands
\newcommand{\SetRowColor}[1]{\noalign{\gdef\RowColorName{#1}}\rowcolor{\RowColorName}} % Shortcut for row colour
\newcommand{\mymulticolumn}[3]{\multicolumn{#1}{>{\columncolor{\RowColorName}}#2}{#3}} % For coloured multi-cols
\newcolumntype{x}[1]{>{\raggedright}p{#1}} % New column types for ragged-right paragraph columns
\newcommand{\tn}{\tabularnewline} % Required as custom column type in use

% Font and Colours
\definecolor{HeadBackground}{HTML}{333333}
\definecolor{FootBackground}{HTML}{666666}
\definecolor{TextColor}{HTML}{333333}
\definecolor{DarkBackground}{HTML}{A3A3A3}
\definecolor{LightBackground}{HTML}{F3F3F3}
\renewcommand{\familydefault}{\sfdefault}
\color{TextColor}

% Header and Footer
\pagestyle{fancy}
\fancyhead{} % Set header to blank
\fancyfoot{} % Set footer to blank
\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
    \SetRowColor{DarkBackground}
    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
    }
\end{tabulary}
\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Procesos Estocasticos Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{andreyhorta35762} via \textcolor{DarkBackground}{\uline{cheatography.com/159146/cs/36544/}}}
\end{tabulary}
\end{multicols}}

\fancyfoot[L]{ \footnotesize
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}andreyhorta35762 \\
  \uline{cheatography.com/andreyhorta35762} \\
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 16th January, 2023.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
  \SetRowColor{white}
  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
\end{multicols}}




\begin{document}
\raggedright
\raggedcolumns

% Set font size to small. Switch to any value
% from this page to resize cheat sheet text:
% www.emerson.emory.edu/services/latex/latex_169.html
\footnotesize % Small font.

\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Definiciones}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Proceso Estocástico}}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Colección Infinita de variables aleatorias sobre un espacio de probabilidad común {\emph{(Ω, F, P)}} que esta indexada por un parametro (i.e. tiempo)} \tn 
% Row Count 4 (+ 3)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Estado}}} \tn 
% Row Count 5 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Valores que toman las variables aleatorias {\bf{X`tn`}}} \tn 
% Row Count 7 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Espacio de Estados del Proceso}}} \tn 
% Row Count 8 (+ 1)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Conjunto de todos los posibles estados. Discreto {\emph{\{X`n`, n = 0, 1, 2...\}}} o Continuo {\emph{\{X`tn`  , t ∈ T\} y \{X`t`, t ∈ T\}.}}} \tn 
% Row Count 11 (+ 3)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Trayectoria}}} \tn 
% Row Count 12 (+ 1)
% Row 7
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Recorrido de un evento específico. Dado {\emph{ω∈Ω}}, la trayectoria de {\emph{ω}} es {\emph{ƒ: t→Xt(ω)}}} \tn 
% Row Count 14 (+ 2)
% Row 8
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Distribución n-dimencional del proceso}}} \tn 
% Row Count 15 (+ 1)
% Row 9
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\emph{ {\bf{F`t1`...`tn` (x`1`,..., x`n`)}} := P(X`t1` ≤ x`1`,..., X`tn`≤ x`n`)}}, donde {\emph{X}}un P.E. real y {\emph{\{t`1`, t`2`,... , t`n`\} ⊂ T}}  donde {\emph{t`1` \textless{} t`2` \textless{}...\textless{} t`n`}}} \tn 
% Row Count 19 (+ 4)
% Row 10
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{P.E. con Incrementos Independientes}}} \tn 
% Row Count 20 (+ 1)
% Row 11
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Cuando las variables aleatorias {\emph{X`t2`− X`t1`, X`t3` − X`t2`, ..., X`tn` − X`tn-1`}} son independientes dado que {\emph{∀ t`1`, t`2`,..., t`n`}} y {\emph{t`1`\textless{}t`2`\textless{} ... \textless{}t`n`}}} \tn 
% Row Count 24 (+ 4)
% Row 12
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{P.E. Estacionario de orden n}} Se puede trasladar en el tiempo y la distribucion no cambia} \tn 
% Row Count 26 (+ 2)
% Row 13
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Sea {\emph{∀t1, t2,..., tn}} y las distribuciones conjuntas de {\emph{X`t1`, X`t2`..., X`tn`)}} y {\emph{(X`t1+h` + h), X`t2+h`..., X`tn+h`)}} son iguales para todo {\emph{h\textgreater{}0}}} \tn 
% Row Count 30 (+ 4)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Definiciones (cont)}}  \tn
% Row 14
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{P.E. Estacionario de orden n}}} \tn 
% Row Count 1 (+ 1)
% Row 15
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Si el proceso es estacionario de orden n para todo {\emph{n∈N}},} \tn 
% Row Count 3 (+ 2)
% Row 16
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{P.E. con Incrementos Estacionarios}}} \tn 
% Row Count 4 (+ 1)
% Row 17
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Si para cualesquiera {\emph{0≤s≤t}} y {\emph{0≤h}}, se cumple que {\emph{X`t`-X`s`}} tiene la misma distribucion que {\emph{X`t+h`-X`s+h`}}} \tn 
% Row Count 7 (+ 3)
% Row 18
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{P.E. de segundo orden o regular}}} \tn 
% Row Count 8 (+ 1)
% Row 19
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Si {\emph{E {[}X`t`{]} 2 \textless{} ∞}} para todo t ∈ T. Ademas Las funciones de media {\emph{mX (t) = E (X`t`)}} y de covarianza {\emph{CX (s, t) = Cov (X`s` , X`t`)}}} \tn 
% Row Count 11 (+ 3)
% Row 20
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{P.E. ortogonal}}} \tn 
% Row Count 12 (+ 1)
% Row 21
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Si es P.E. regular y {\emph{E {[}X`t`X`s`{]}=0}}, {\emph{∀t,s, ∈ T}} y {\emph{t ≠s.}}} \tn 
% Row Count 14 (+ 2)
% Row 22
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{P.E. Estacionario}}} \tn 
% Row Count 15 (+ 1)
% Row 23
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Si es P.E. regular y su función de media {\emph{mX(t)}} es independiente de t y si su función de covarianza {\emph{CX(s, t)}} es una función que depende sólo de |t − s|, para todo t,s,esto {\emph{C(s, t) = Cov(X(s), X(t)) = f(t − s).}}} \tn 
% Row Count 20 (+ 5)
% Row 24
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{P.E. Evolutivo}}} \tn 
% Row Count 21 (+ 1)
% Row 25
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Que no es Estacionario} \tn 
% Row Count 22 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Ejemplos Especiales de P.E.}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Martingala}} El valor esperado del {\emph{X\_\{n+1\}}} dado que paso por n valores dados, es el ultimo valor observado {\emph{x\_\{n\}}}} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Si {\emph{E(X\_\{n+1\} | X\_\{0\}=x\_\{0\},...,X\_\{n\}=x\_\{n\})=x\_\{n\}}}} \tn 
% Row Count 5 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Proceso de Levy}}} \tn 
% Row Count 6 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Cuando los incrementos son independientes y estacionarios} \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Proceso Gausiano}}} \tn 
% Row Count 9 (+ 1)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Si para cualesquiera n tiempos crecientes, se tiene que {\emph{(X(t1), X(t2)..., X(tn))}} tiene una distribucion normal multivariada} \tn 
% Row Count 12 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{p{0.54747 cm} x{4.42953 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Tipos de P.E.}}  \tn
% Row 0
\SetRowColor{LightBackground}
PDED & Tiempo Discreto + Espacio Discreto \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
PDEC & Tiempo Discreto + Espacio Continuo \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
PCED & Tiempo Continuo + Espacio Discreto \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
PCEC & Tiempo Continuo +Espacio Continuo \tn 
% Row Count 4 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Propiedades P.E}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\emph{X=Y}} sii {\emph{X`t`(ω) = Y`t`(ω)}} con {\emph{∀t∈ T}} y {\emph{∀ω ∈ Ω}}} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{X y Y son {\bf{estocásticamente equivalentes}} si {\emph{P(X`t` = Y`t`) =1}} y {\emph{∀t∈ T}}.} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{X y Y son {\bf{estocásticamente equivalentes en el sentido amplio}} si {\emph{∀ n∈N}}, {\emph{∀ \{t`1`,..., t`n`\}⊂T}} y {\emph{\{B`1`,..., B`n`\} ⊂ 2\textasciicircum{}Ω\textasciicircum{}}} se satisface:  {\emph{P(X`t1` ∈ B`1`, ..., X`tn` ∈ B`n`) = P(Y`t1` ∈ B`1`,..., Y`tn` ∈ B`n`)}}} \tn 
% Row Count 9 (+ 5)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{X y Y se dicen {\bf{indistinguibles}} si casi todas sus trayectorias coinciden, esto es, {\emph{P(X`t` = Y`t` , t ∈ T) = 1}}.} \tn 
% Row Count 12 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Aclaraciones}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Sea {\emph{X = \{Xt, t∈ T\}}} un proceso estocástico} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\emph{X}} es real si las v.a. {\emph{Xt}} son de valor real para todo {\emph{t∈T}}} \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\emph{X}} es complejo si las v.a. {\emph{Xt}} son de valor complejo para todo t ∈ T} \tn 
% Row Count 5 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Si {\emph{T}} es finito o contable, entonces es proceso estocástico con parámetro de tiempo discreto.} \tn 
% Row Count 7 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Si {\emph{T}} es un intervalo de la recta real entonces es proceso con parámetro de tiempo continuo.} \tn 
% Row Count 9 (+ 2)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Si {\emph{T⊆Rn}} con {\emph{n\textgreater{}1}} entonces el proceso se denomina campo aleatorio.} \tn 
% Row Count 11 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Cadena de Markov}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Un P.E. Real {\emph{X}} si para todo n tiempos crecientes y {\emph{a, b ∈ R con a \textless{} b,}} se satisface que {\bf{P (X`t` ∈ (a, b{]} | X`t1`= x1, X`t2` = x2, ,..., X`tn` ) = P (X`t` ∈ (a, b{]} | X`tn` )}}} \tn 
% Row Count 4 (+ 4)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Es decir, solo importa el estado inmediatamente anterior {\emph{X`tn`}} para determinar el estado siguiente {\emph{X`tn+1`}}} \tn 
% Row Count 7 (+ 3)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Equivalente a {\emph{P (X`0` = i`0` , ..., X`n+1` = i`n+1`) = P (X`0` = i`0`) P (X`1` = i`1` | X`0` = i`0`) ...P (X`n+1` = i`n+1` | X`n` = i`n`)}}} \tn 
% Row Count 10 (+ 3)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Distribucion Inicial}}} \tn 
% Row Count 11 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Distribucion de la variable X`0`, es decir \{P (X`0` = 0), P(X`0` = 1), ...\}} \tn 
% Row Count 13 (+ 2)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Probabilidad de transicion}}} \tn 
% Row Count 14 (+ 1)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Probabilidad de que pase de {\emph{i}} a {\emph{j}} en un paso (del tiempo {\emph{n}} al tiempo {\emph{n+1}}) es {\emph{P(X`n+1`=j| X`n`=i)=P`ij`(n,n+1)=P`ij`(1)}}} \tn 
% Row Count 17 (+ 3)
% Row 7
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Matriz de transicion}} en un paso} \tn 
% Row Count 18 (+ 1)
% Row 8
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Sea {\emph{P={[}p`ij`{]}=P`ij`(1)}}. Es la matriz estocastica. El {\emph{i}} es la salida y el {\emph{j}} es la llegada} \tn 
% Row Count 20 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}