架构已经搭好，需要李昕闻补充

document.tex

@@ -75,6 +75,7 @@ wangwenhao11@nudt.edu.cn).
 \input{main/introduction}
 \input{main/background}
 \input{main/nonergodic}
+\input{main/2048isNonergodic}
 
 %\input{main/nonergodicity}
 %\input{main/paradox}

main/2048isNonergodic.tex

+\section{Non-ergodicity of 2048}
+
+
+\begin{theorem}
+2048 game is non-ergodic between non-absorbing states.
+\end{theorem}
+\begin{IEEEproof}
+
+\end{IEEEproof}
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main/background.tex

@@ -54,9 +54,9 @@ reaching the leftmost or rightmost node where it terminates.
 The terminal states are usually called absorbing states.
 The transition probobility matrix
 of random walk with absorbing states 
-$P_{\text{absorbing}}$ is defined as follows:
+$P_{\text{ab}}$ is defined as follows:
 \[
-P_{\text{absorbing}}\dot{=}\begin{array}{c|ccccccc}
+P_{\text{ab}}\dot{=}\begin{array}{c|ccccccc}
 &\text{T} & \text{A} & \text{B} & \text{C} & \text{D} & \text{E}  \\\hline
 \text{T} & 1 & 0 & 0 & 0 & 0 & 0 \\
 \text{A} & \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 & 0 \\
@@ -122,6 +122,7 @@ where $Q$ is the matrix of transition probabilities between
   is
   \begin{equation}
   N\dot{=} \sum_{i=0}^{\infty}Q^i=(I_{n-1}-Q)^{-1},
+  \label{definitionN}
   \end{equation}
   where $I_{n-1}$ is the $(n-1)\times(n-1)$ identity matrix.
 It is now easy to define whether the non-absorbing states 
@@ -139,15 +140,15 @@ Assume that $N$  exists for any policy $\pi$
 
 For random walk with absorbing states,
 \[
-P_{\text{absorbing}} =
+P_{\text{ab}} =
 \begin{bmatrix}
-Q_{\text{absorbing}} & R_{\text{absorbing}} \\
-0 & I_{\text{absorbing}}
+Q_{\text{ab}} & R_{\text{ab}} \\
+0 & I_{\text{ab}}
 \end{bmatrix},
 \]
 where
 \[
-Q_{\text{absorbing}}\dot{=}\begin{array}{c|ccccc}
+Q_{\text{ab}}\dot{=}\begin{array}{c|ccccc}
  & \text{A} & \text{B} & \text{C} & \text{D} & \text{E}  \\\hline
 \text{A}  & 0 & \frac{1}{2} & 0 & 0 & 0 \\
 \text{B} & \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\
@@ -175,7 +176,7 @@ Q_{\text{absorbing}}\dot{=}\begin{array}{c|ccccc}
 
 Then,
 \[
-N_{\text{absorbing}}=(I_5-Q_{\text{absorbing}})^{-1}=\begin{array}{c|ccccc}
+N_{\text{ab}}=(I_5-Q_{\text{ab}})^{-1}=\begin{array}{c|ccccc}
  & \text{A} & \text{B} & \text{C} & \text{D} & \text{E}  \\\hline
 \text{A}  & \frac{5}{3} & \frac{4}{3} & 1 & \frac{2}{3} & \frac{1}{3} \\
 \text{B} & \frac{4}{3} & \frac{8}{3} & 2 & \frac{4}{3} & \frac{2}{3} \\

main/nonergodic.tex

@@ -62,7 +62,7 @@ Figure \ref{TruncatedPetersburg} is a truncated version
 of the St. Petersburg paradox. The transition probabilities between
 non-absorbing states are as follows:
 \[
-Q_{\text{truncated}}\dot{=}\begin{array}{c|ccccc}
+Q_{\text{st}}\dot{=}\begin{array}{c|ccccc}
  & \text{S}_1 & \text{S}_2 & \text{S}_3 & \text{S}_4 & \text{S}_5  \\\hline
 \text{S}_1  & 0 & \frac{1}{2} & 0 & 0 & 0 \\
 \text{S}_2 & 0 & 0 & \frac{1}{2} & 0 & 0 \\
@@ -73,7 +73,7 @@ Q_{\text{truncated}}\dot{=}\begin{array}{c|ccccc}
 \]
 Then,
 \[
-N_{\text{truncated}}=(I_5-Q_{\text{truncated}})^{-1}=\begin{array}{c|ccccc}
+N_{\text{st}}=(I_5-Q_{\text{st}})^{-1}=\begin{array}{c|ccccc}
 & \text{S}_1 & \text{S}_2 & \text{S}_3 & \text{S}_4 & \text{S}_5  \\\hline
 \text{S}_1 & 1 & \frac{1}{2} & \frac{1}{4} & \frac{1}{8} & \frac{1}{16} \\
 \text{S}_2 & 0 & 1 & \frac{1}{2} & \frac{1}{4} & \frac{1}{8} \\
@@ -86,6 +86,71 @@ Bases on Definition \ref{definition3},
 the truncated  St. Petersburg paradox
 is non-ergodic between non-absorbing states.
 
+\subsection{A sufficient condition for non-ergodicity between non-absorbing states}
+Based on the truncated  St. Petersburg paradox, 
+it is easy to provide a sufficient condition for non-ergodicity between non-absorbing states.
+\begin{theorem}[A sufficient condition for non-ergodicity between non-absorbing states]
+Given a Markov chain with absorbing states, 
+suppose the size of the non-absorbing states $|S\setminus\{\text{T}\}|\geq 2$.
+If the transition probabilities $Q$ between non-absorbing states satifies,
+$\forall i,j \in S\setminus\{\text{T}\}$,
+\begin{equation}
+Q_{i,j}=\begin{cases}
+0, & \text{if } i\leq j; \\
+\geq 0, & \text{otherwise.}  
+\end{cases}
+\label{condition}
+\end{equation}
+Then, the Markov chain is non-ergodic between non-absorbing states.
+\end{theorem}
+\begin{IEEEproof}
+Based on the assumption, the $Q$ matrix is an upper triangular matrix.
+The product of two upper triangular matrices is still an upper triangular matrix.
+Furthermore, the sum of two upper triangular matrices 
+is still an upper triangular matrix.
+Based on Definition \ref{definitionN}, 
+the $N$ matrix is product and sum of upper triangular matrices.
+Then, the $N$ matrix is an upper triangular matrix.
+The claim now follows based on Definition \ref{definition3}.
+\end{IEEEproof}
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