鞅中心极限定理的基本内容可陈述如下：令随机变量
X
1
,
X
2
,
…
{\displaystyle X_{1},X_{2},\dots \,}
鞅 ，即满足条件：
E
[
X
t
+
1
−
X
t
|
X
1
,
…
,
X
t
]
=
0
,
{\displaystyle \mathbb {E} [X_{t+1}-X_{t}\vert X_{1},\dots ,X_{t}]=0\,,}
进一步假设这个鞅是有限增量的，即：存在一个固定常数
k
>
0
{\displaystyle k>0}
|
X
t
+
1
−
X
t
|
≤
k
{\displaystyle |X_{t+1}-X_{t}|\leq k}
对所有
t
{\displaystyle t}
|
X
1
|
≤
k
{\displaystyle |X_{1}|\leq k}
σ
t
2
=
E
[
(
X
t
+
1
−
X
t
)
2
|
X
1
,
…
,
X
t
]
,
{\displaystyle \sigma _{t}^{2}=\mathbb {E} [(X_{t+1}-X_{t})^{2}|X_{1},\ldots ,X_{t}],}
并假设所有条件方差之和发散，即下式以概率1成立：
∑
t
=
1
∞
σ
t
2
=
∞
{\displaystyle \sum _{t=1}^{\infty }\sigma _{t}^{2}=\infty }
据此，对任意给定的常数
ν
>
0
{\displaystyle \nu >0}
τ
ν
=
min
{
t
:
∑
i
=
1
t
σ
i
2
≥
ν
}
.
{\displaystyle \tau _{\nu }=\min \left\{t:\sum _{i=1}^{t}\sigma _{i}^{2}\geq \nu \right\}.}
在所有上述假设成立的条件下，鞅中心极限定理做出如下结论：标准化的鞅随机变量：
X
τ
ν
ν
{\displaystyle {\frac {X_{\tau _{\nu }}}{\sqrt {\nu }}}}
随着
ν
→
+
∞
{\displaystyle \nu \to +\infty \!}
依分布收敛 于标准正态分布 。
随机增量的条件方差之和必须发散 编辑
上述定理假设了所有随机增量的条件方差之和为无穷大，即以下条件以概率1成立：
∑
t
=
1
∞
σ
t
2
=
∞
{\displaystyle \sum _{t=1}^{\infty }\sigma _{t}^{2}=\infty }
这样可以确保以概率1，下式成立：
τ
v
<
∞
,
∀
v
≥
0
{\displaystyle \tau _{v}<\infty ,\forall v\geq 0}
并不是所有鞅都满足这个条件，例如恒为零的平凡鞅。
定理的直观理解 编辑
可以通过将
X
τ
ν
ν
{\displaystyle {\frac {X_{\tau _{\nu }}}{\sqrt {\nu }}}}
X
τ
v
v
=
X
1
v
+
1
v
∑
i
=
1
τ
v
−
1
(
X
i
+
1
−
X
i
)
,
∀
τ
v
≥
1
{\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}
右边的第一项渐近收敛于零，可以忽略。第二项在形式上，与独立同分布随机增量的经典中心极限定理 相似，虽然其中被求和项
X
i
+
1
−
X
i
{\displaystyle X_{i+1}-X_{i}}
E
[
(
X
i
+
1
−
X
i
)
(
X
i
+
m
+
1
−
X
i
+
m
)
]
=
E
[
E
[
(
X
i
+
1
−
X
i
)
(
X
i
+
m
+
1
−
X
i
+
m
)
|
X
1
,
…
,
X
i
+
m
]
]
=
0
{\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}
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 
![{\displaystyle \mathbb {E} [X_{t+1}-X_{t}\vert X_{1},\dots ,X_{t}]=0\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/210b03bd0f888602209f786c88a19487170cf14f)
![{\displaystyle \sigma _{t}^{2}=\mathbb {E} [(X_{t+1}-X_{t})^{2}|X_{1},\ldots ,X_{t}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3dafe9a18f074d89de0f8f0c6c6f1330e13dfc)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6aaf329e1203d6fbf6de5ca05547566bdd629f3)