$\newcommand{\R}{\mathcal R}\newcommand{\C}{\mathcal C}\newcommand{\Rbb}{\mathbb R}\newcommand{\0}{\mathbf 0}\newcommand{\one}{\mathbb 1}\newcommand{\span}{\operatorname{span}}\newcommand{\rank}{\operatorname{rank}}$In this post I’m going to prove the Steinitz exchange lemma, that the row rank…
$\newcommand{\Om}{\Omega}\newcommand{\F}{\mathscr F}\newcommand{\one}{\mathbf 1}\newcommand{\R}{\mathbb R}\newcommand{\e}{\varepsilon}\newcommand{\E}{\operatorname{E}}\newcommand{\Var}{\operatorname{Var}}\newcommand{\convas}{\stackrel{\text{a.s.}}{\to}}\newcommand{\w}{\omega}\newcommand{\N}{\mathbb N}\newcommand{\convp}{\stackrel{\text{p}}{\to}}$In this post I’m going to introduce almost sure convergence for sequences of random variables, compare…