
\usepackage{amsmath}
\usepackage{amssymb}
\newcommand{\E}{\mathbb{E}}
\newcommand{\Var}[1]{\mathrm{op}
\operatorname{Var}\left[#1\right]}
\newcommand{\Pro}[1]{\mathbb{P}\left(#1\right)}
\newcommand{\Ind}{\mathbf{1}}
\newcommand{\di}{\operatorname{diag}}
\newcommand{\tr}{\operatorname{Tr}}
\newcommand{\thetapop}{\theta^*_{\rm{pop}}}
\newcommand{\thetaperfok}{\theta_0^{\text{perfo}}}
\newcommand{\thetas}{\theta^*}
\newcommand{\thetainf}{\theta^{\infty}}
\newcommand{\N}{\mathcal{N}}
\newcommand{\R}{\mathbb{R}}

\newcommand{\D}{\mathcal{D}}
\usepackage{amsthm} 
\usepackage{graphicx}
\usepackage{subcaption}
\newcommand{\fixedthetaperf}{\theta^\infty_{\mathrm{perf}}}
\newcommand{\fixedthetaeq}{\theta^\infty_{\mathrm{eq}}}
\newcommand{\fixedthetaeqtop}{\theta^{\infty \top}_{\mathrm{eq}}}
\newcommand{\thetaeq}{\theta_{\mathrm{eq}}}

\newcommand{\yeq}{y_{\mathrm{eq}}}
\newcommand{\thetapopT}{(\thetapop)^{\top}}
\newcommand{\fixedthetaeqT}{{\fixedthetaeq}^{\top}}
\newcommand{\fixedthetaperfT}{{\fixedthetaperf}^{\top}}
\newcommand{\fixedriskeq}{\mathcal R_{\mathrm{eq}}}
\newcommand{\ofixedriskeq}{\overline{\mathcal R}_{\mathrm{eq}}}
\newcommand{\fixedriskeqz}{\mathcal R_{\mathrm{eq}, D=0}}
\newcommand{\fixedriskeqs}{\mathcal R_{\mathrm{eq}}^*}
\newcommand{\lambdaeqs}{\lambda^*_{\mathrm{eq}}}
\newcommand{\fixedriskeqsz}{\mathcal R_{\mathrm{eq}, D=0}^*}
\newcommand{\lambdaeqsz}{\lambda^*_{\mathrm{eq}, D=0}}
\newcommand{\fixedriskeqa}{\mathcal R_{\mathrm{eq}, 1}}
\newcommand{\fixedriskeqb}{\mathcal R_{\mathrm{eq}, 2}}
\newcommand{\tfixedriskeqa}{\widetilde{\mathcal R}_{\mathrm{eq}, 1}}
\newcommand{\tfixedriskeqb}{\widetilde{\mathcal R}_{\mathrm{eq}, 2}}
\newcommand{\bfixedriskeq}{{\widehat{\mathcal R}}_{\mathrm{eq}}}
\newcommand{\tfixedriskeq}{{\widetilde{\mathcal R}}_{\mathrm{eq}}}
