How to increase the width of one column and decrease the width of the other one by the same length?

Here's a solution that uses a tabularx environment with two variable-width columns of type X. The requirement is that the individual relative widths -- here: 0.75 and 1.25 -- sum to the number of columns of type X -- here, 2. (If you would rather force the usual widths to have a ratio of 1:2, use 0.6667 and 1.3333 as the relative widths of the X columns.)

\documentclass[12pt]{extreport}
\usepackage[margin = 0.5in]{geometry}
\usepackage{mathtools,amssymb,tabularx,ragged2e}
\DeclarePairedDelimiter\abs\lvert\rvert
\newcolumntype{L}[1]{>{\RaggedRight\arraybackslash%
    \hsize=#1\hsize\linewidth=\hsize}X}

\begin{document}

\noindent
\begin{tabularx}{\textwidth}{@{} L{0.75}|L{1.25} @{}} % 0.75+1.25=2.0
\textbf{Theorem 1} \quad The distribution of
\[
\mathcal{F}_n \coloneqq \sup_{x\in\mathcal X}\abs{F_n(x)-F(x)}
\] 
is known and the same for any DGP whose $F$ is \textbf{continuous}.
&
\textbf{Theorem 2} \quad If $F$ is \textbf{continuous} then for all $t > 0$, we have 
\[
\mathbb{P} [\sqrt{n} \mathcal F_n \le t ] \longrightarrow H(t)
    = 1- 2\sum_{j=1}^{\infty} (-1)^{j-1} e^{-2 j^2 t^2}
\] 
where $H$ is the cdf of the K-S distribution.
\end{tabularx}

\end{document}

Another possibility uses the paracol package and real theorem environments:

\documentclass[12pt]{extreport}
\usepackage[left = 0.5in, right = 0.5in, top = 0.5in, bottom = 0.5in]{geometry}
\usepackage{multicol}
\usepackage{paracol}
\usepackage{amsmath}
\usepackage{amsthm, amsfonts}
\newtheorem{theorem}{Theorem}

\setlength{\columnseprule}{1pt}

\begin{document}

\columnratio{0.4}
\begin{paracol}{2}
\begin{theorem}
The distribution of
\[ \mathcal F_n: = \sup_{x \in \mathcal X} |F_n (x) - F(x)| \]
 is known and the same for any DGP whose $F$ is \textbf{continuous}.
\end{theorem}
\switchcolumn \stepcounter{theorem}
\begin{theorem}
 If $F$ is \textbf{continuous} then for all $t > 0$, we have
 \[ \mathbb P \left [\sqrt{n} \mathcal F_n \le t \right ] \longrightarrow H(t)= 1-2\smash[t]{ \sum_{j=1}^{\infty}}(-1)^{j-1} e^{-2 j^{2} t^2}, \]
  where $H$ is the c.d.f of K-S distribution.
\end{theorem}
\end{paracol}

\end{document}