# 鞅中心极限定理

## 定理内容

${\displaystyle \mathbb {E} [X_{t+1}-X_{t}\vert X_{1},\dots ,X_{t}]=0\,,}$ （鞅的定义）

${\displaystyle |X_{t+1}-X_{t}|\leq k}$

${\displaystyle \sigma _{t}^{2}=\mathbb {E} [(X_{t+1}-X_{t})^{2}|X_{1},\ldots ,X_{t}],}$

${\displaystyle \sum _{t=1}^{\infty }\sigma _{t}^{2}=\infty }$

${\displaystyle \tau _{\nu }=\min \left\{t:\sum _{i=1}^{t}\sigma _{i}^{2}\geq \nu \right\}.}$

${\displaystyle {\frac {X_{\tau _{\nu }}}{\sqrt {\nu }}}}$

## 随机增量的条件方差之和必须发散

${\displaystyle \sum _{t=1}^{\infty }\sigma _{t}^{2}=\infty }$

${\displaystyle \tau _{v}<\infty ,\forall v\geq 0}$

## 定理的直观理解

${\displaystyle {\frac {X_{\tau _{v}}}{\sqrt {v}}}={\frac {X_{1}}{\sqrt {v}}}+{\frac {1}{\sqrt {v}}}\sum _{i=1}^{\tau _{v}-1}(X_{i+1}-X_{i}),\forall \tau _{v}\geq 1}$

${\displaystyle \mathbb {E} [(X_{i+1}-X_{i})(X_{i+m+1}-X_{i+m})]=\mathbb {E} [\mathbb {E} [(X_{i+1}-X_{i})(X_{i+m+1}-X_{i+m})|X_{1},\ldots ,X_{i+m}]]=0}$

## 参考文献

• Hall, Peter; C. C. Heyde. Martingale Limit Theory and Its Application. New York: Academic Press. 1980. ISBN 0-12-319350-8. Hall, Peter; C. C. Heyde. Martingale Limit Theory and Its Application. New York: Academic Press. 1980. ISBN 0-12-319350-8. Hall, Peter; C. C. Heyde. Martingale Limit Theory and Its Application. New York: Academic Press. 1980. ISBN 0-12-319350-8.
• 有关定理5.4的讨论以及推论5.3（ii）的正确形式，请参见Bradley, Richard. On some results of MI Gordin: a clarification of a misunderstanding. Journal of Theoretical Probability (Springer). 1988, 1 (2): 115–119. doi:10.1007/BF01046930.