\newcommand{\Wlower}{\widetilde{W}^*}
\newcommand{\Zbar}{\bar{Z}}
\newcommand{\sideinfo}{\mathbb{S}}
\newcommand{\kelly}{\texttt{ApproxKelly}\xspace}
\newcommand{\Ztilde}{\widetilde{Z}\xspace}
\newcommand{\Wtilde}{\widetilde{W}\xspace}
\newcommand{\Mtilde}{\widetilde{A}\xspace}
\newcommand{\Vtilde}{\widetilde{V}\xspace}
\newcommand{\Wprocesstilde}{\widetilde{\mathbb{W}}}
\newcommand{\Wprocess}{\mathbb{W}}
\newcommand{\wor}{WoR\xspace}

\newcommand{\var}{\mathbb{V}}
\newcommand{\cov}{\mathrm{Cov}}

\newcommand{\mhat}{\widehat{m}}
\newcommand{\vhat}{\widehat{v}}
\newcommand{\red}[1]{\textcolor{red}{#1}}
\newcommand{\iid}{i.i.d.}
\newcommand{\rlfa}{(\varepsilon, \delta)\text{-}\mathrm{RLFA}\xspace}
\newcommand{\blue}[1]{{#1}}