
\section{Proof of the Theorems}\label{sec:proof}

\input{sections/thm1}
\input{sections/thm2}
\input{sections/thm3}
\input{sections/thm4}


\section{Simulation Details and Additional Simulations}\label{sec:exsim}
\input{sections/Extrasim_new}

\input{sections/compare}
