[機率論] 隨機漫步- Wald's equation

這次要介紹 Random Walk + stopping time 的重要的結果，介紹之前我們需要一些先導概念

考慮 $T$ 為 stopping time，則 $ET =?$
注意到幾件事實：
1. 若 $P(T=\infty) >0$，則 $ET = \infty$
2. 若 $P(T = \infty)=0$ 且 $n \sum_n P(T=n) < \infty$ 若且唯若 $ET < \infty$
3. $P(T < \infty) = \sum_n^\infty P(T=n)$

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Theorem: Wald's Equation
令 $X_1, X_2,...$ 為 i.i.d. 且 $E|X_i| < \infty$，令 $S_n:=\sum_m^n X_m$。若 $N$ 為 stopping time 且 $EN < \infty $ 則 $ES_N = EX_1 EN$
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Proof:
首先假設 $X_i \ge 0$，觀察
\[E{S_N} = \sum\limits_{n = 1}^\infty  {E\left[ {{S_n}{1_{N = n}}} \right]}  = \sum\limits_{n = 1}^\infty  {E\left( {\sum\limits_{k = 1}^n {{X_k}{1_{N = n}}} } \right)} \]由於 $X_n \ge 0$ 故 利用 Fubini Theorem 我們可對調 summation 順序：
\[\sum\limits_{n = 1}^\infty  {\sum\limits_{k = 1}^n {E{X_k}{1_{N = n}}} }  = \sum\limits_{k = 1}^\infty  {\sum\limits_{n = k}^\infty  {E{X_k}{1_{N = n}}} } \]現在注意到 ${1_{\left\{ {N \ge k} \right\}}} = {1_{{{\left\{ {N \le k - 1} \right\}}^c}}} \in {F_{k - 1}}$ 由於 $X_k$ 為 independent，故 $X_k \perp F_{k-1}$ ，亦即可將其改寫為
\[\begin{array}{l}
\sum\limits_{k = 1}^\infty  {E{X_k}{1_{N \ge k}}}  = \sum\limits_{k = 1}^\infty  {E{X_k}E{1_{N \ge k}}} \\
 \Rightarrow E{S_N} = E{X_1}\sum\limits_{k = 1}^\infty  {E{1_{N \ge k}}}  = E{X_1}\sum\limits_{k = 1}^\infty  {P\left( {N \ge k} \right)} \ \ \ \ \ (*)
\end{array}\]現在注意到
\[\begin{array}{l}
\sum\limits_{k = 1}^\infty  {P\left( {N \ge k} \right)}  = P\left( {N \ge 1} \right) + P\left( {N \ge 2} \right) + P\left( {N \ge 3} \right) + ...\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \left[ \begin{array}{l}
P\left( {N = 1} \right) + P\left( {N = 2} \right) + P\left( {N = 3} \right) + ... + \\
\begin{array}{*{20}{c}}
{}&{}
\end{array} + P\left( {N = 2} \right) + P\left( {N = 3} \right) + ... + \\
\begin{array}{*{20}{c}}
{}&{}
\end{array} + P\left( {N = 3} \right) + P\left( {N = 4} \right) + ... + \\
\begin{array}{*{20}{c}}
{}&{}
\end{array} + ...
\end{array} \right]\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = P\left( {N = 1} \right) + 2P\left( {N = 2} \right) + 3P\left( {N = 3} \right) + ...\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \sum\limits_{k = 1}^\infty  {kP\left( {N = k} \right)}  = E\left[ N \right]
\end{array}\]故 $(*)$ 可表為
\[E{S_N} = E{X_1}\sum\limits_{k = 1}^\infty  {P\left( {N \ge k} \right)}  = E{X_1}ET\]