\pdfstringdefDisableCommands{\def\varepsilon{\textepsilon}}

\newtheorem{theorem}{Theorem}                % Theorems
\newtheorem{proposition}{Proposition}        % Propositions
\newtheorem{corollary}{Corollary}  % Corollary
\newtheorem{lemma}{Lemma}                     % Lemmas


\theoremstyle{definition}                               % definitions
\newtheorem{definition}{Definition}[section]
\newtheorem{remark}{Remark}[]
\newtheorem{question}{Question}[]


\newenvironment{claim}[1]{\par\noindent\underline{Claim:}\space#1}{}
\newenvironment{claimproof}[1]{\par\noindent\underline{Proof:}\space#1}{\hfill $\blacksquare$}

 {
	\theoremstyle{plain}
	\newtheorem{assumption}{Assumption}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
                        % Notation 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\CErrwi}{\mathcal{G}_{{\omega, i}}} % Expected interventional error when intervened on the $i$th component of the vector $X_{t-\omega}$

\newcommand{\CErrw}{\mathcal{G}_{do_{\omega}}} % Expected interventional error when intervened on the scalar time series $X_{t-\omega}$

\newcommand{\SErrw}{\mathcal{S}_{\omega}} % Expected Statistical error when the past of $X_{t-\omega}$ is observed.

\newcommand{\SpecRad}{\rho} %Spectral radius. We denote this by $\rho$

\newcommand{\EigVal}[1]{\lambda_{#1}} %Eigenvalues a matrix denoted by $\lambda_i$ 

\newcommand{\CompMat}{A} % Companion or Multi-companion matrix

\newcommand{\SCovn}[1]{\Sigma_{#1}} % Population covariance matrix of the observational distribution of $Y_t = (X_t, X_{t-1}, \cdots X_{t-n+1})^T$ for any $n \in \N$ given by $\E (Y_t Y^T_t)$

\newcommand{\ICovn}[1]{\Gamma_{#1}} % Population covariance matrix of the Interventional distribution of $Y_t = (X_t, X_{t-1}, \cdots X_{t-n+1})^T$ for any $n \in \N$ given by $\E (Y_t Y^T_t)$


\newcommand{\Covhkl}[3]{\Sigma(#2, #1d + #3)} % Covariance matrix between $k$(th) component of $X_t$ and $l$th component of $X_{t+h}$ for any $h > 0$

\newcommand{\CovNoise}{\Sigma_{\epsilon}} % Covariance matrix of the noise vector$

\newcommand{\VarNoise}{\sigma^2_{\epsilon}} % Variance of the noise.

\newcommand{\NoiseVec}[2]{\E_[#1, #2]} %Noise vector at time t. 

\newcommand{\NoiseVal}[1]{\epsilon_[#1]} %Noise scalar at time t. 

\newcommand{\NoiseVecL}[1]{(\epsilon_{#1,1}, \epsilon_{#1,2}, \cdots, \epsilon_{#1, d})} %Noise vector of dimension $d$ at time $t$. 

\newcommand{\PastFullX}[1]{\mathbb{X}_{#1}} %The entire past of $X_{t-\omega}$ for both vector-valued and scalar-valued variables.

\newcommand{\PastnX}[2]{\mathbb{X}^{#2}_{#1}} % $\mathbb{X}^n_{t-\omega} = \myCurls{X_{t-\omega-n}, \cdots, X_{t-\omega - 1}, X_{t-\omega}} $ denotes the set of $X_{t-\omega}$ and $n$ observations in the past of $X_{t-\omega}$.

\newcommand{\Identity}[1]{\mathbb{I}_{#1}} %\mathbb{I}_{p}$ is the identity matrix of size $p$

\newcommand{\dow}{do_{\omega}}   %Intervention $do(X_{t-\omega} = x_{t-\omega})$

\newcommand{\dowi}[1]{do_{\omega, #1}} %Intervention $do(X_{t-\omega,i} = x_{t-\omega},i)$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
                                        %Math shortcuts
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\abs}[1]{\vert #1 \vert}  %Absolute value
\newcommand{\myabs}[1]{\left \vert #1 \right \vert} %Scaled absolute value
\newcommand{\SumLimits}{\sum \limits}  %Sum followed by Limits
\newcommand{\myBracs}[1]{\left ( #1 \right )} %Round brackets scaled
\newcommand{\myCurls}[1]{\left \{ #1 \right \}} %Curly brackets scaled
\newcommand{\norm}[1]{\left\lVert#1\right\rVert} %Norm

\newcommand{\indep}{\perp \!\!\! \perp} %Independence

\newcommand{\ZeroVec}{\mathbf{0}}

%%%%%% Nicer looking empty set
\let\oldemptyset\emptyset
\let\emptyset\varnothing
%%%%%%


\DeclareMathOperator*{\E}{\mathbb{E}} % Expectation
\DeclareMathOperator*{\Z}{\mathbb{Z}} % Integers
\DeclareMathOperator*{\N}{\mathbb{N}} % Natural numbers
\DeclareMathOperator*{\R}{\mathbb{R}} % Reals
\DeclareMathOperator*{\C}{\mathbb{C}} % Complex numbers
\DeclareMathOperator*{\argmax}{arg\,max} %argmax
\DeclareMathOperator*{\arginf}{arg\,inf} %arginf
\DeclareMathOperator*{\argsup}{arg\,sup} %argsup
\DeclareMathOperator*{\argmin}{arg\,min} %argmin

\newcommand{\prob}{\mathbb{P}} %Probability
\newcommand{\Xcal}{\mathcal{X}}
\newcommand{\Ycal}{\mathcal{Y}}

\newcommand{\bsp}{\boldsymbol{\phi}}






















