# 累积量

（重定向自累積量

## 定义

${\displaystyle K(t)=\log \mathbb {E} e^{tX}=\sum _{n=1}^{\infty }\kappa _{n}{\frac {t^{n}}{n!}}=:g(t).}$

{\displaystyle {\begin{aligned}\kappa _{1}&=g'(0)=\mu '_{1}=\mu ,\\\kappa _{2}&=g''(0)=\mu '_{2}-{\mu '_{1}}^{2}=\sigma ^{2},\\&{}\ \ \vdots \\\kappa _{n}&=g^{(n)}(0),\\&{}\ \ \vdots \end{aligned}}}

${\displaystyle \mathbb {E} (e^{tX})=1+\sum _{m=1}^{\infty }\mu '_{m}{\frac {t^{m}}{m!}}=e^{g(t)}.}$

{\displaystyle {\begin{aligned}g(t)&=\log(\operatorname {E} (e^{tX}))=-\sum _{n=1}^{\infty }{\frac {1}{n}}\left(1-\operatorname {E} (e^{tX})\right)^{n}=-\sum _{n=1}^{\infty }{\frac {1}{n}}\left(-\sum _{m=1}^{\infty }\mu '_{m}{\frac {t^{m}}{m!}}\right)^{n}\\&=\mu '_{1}t+\left(\mu '_{2}-{\mu '_{1}}^{2}\right){\frac {t^{2}}{2!}}+\left(\mu '_{3}-3\mu '_{2}\mu '_{1}+2{\mu '_{1}}^{3}\right){\frac {t^{3}}{3!}}+\cdots .\end{aligned}}}

${\displaystyle h(t)=\sum _{n=1}^{\infty }\kappa _{n}{\frac {(it)^{n}}{n!}}=\log(\operatorname {E} (e^{itX}))=\mu it-\sigma ^{2}{\frac {t^{2}}{2}}+\cdots .\,}$

## 统计数学中的应用

{\displaystyle {\begin{aligned}g_{X+Y}(t)&=\log(\operatorname {E} (e^{t(X+Y)}))=\log(\operatorname {E} (e^{tX})\operatorname {E} (e^{tY}))\\&=\log(\operatorname {E} (e^{tX}))+\log(\operatorname {E} (e^{tY}))=g_{X}(t)+g_{Y}(t).\end{aligned}}}

## 一些具体概率分布的累积量

• 常量${\displaystyle X=\mu }$ 的累积生成函数是 ${\displaystyle K(t)=\mu t}$ 。 一阶累积量是${\displaystyle \kappa _{1}=K'(0)=\mu }$ ,其他阶的累积量均为0， ${\displaystyle \kappa _{2}=\kappa _{3}=\kappa _{4}=...=0}$
• 服从伯努利分布的随机变量的累积生成函数是 ${\displaystyle K(t)=log(1-p+pe^{t})}$ 。一阶累积量是${\displaystyle \kappa _{1}=K'(0)=p}$ ，二阶累积量是${\displaystyle \kappa _{2}=K''(0)=p(1-p)}$ ,累积量满足递推公式
${\displaystyle \kappa _{n+1}=p(1-p){\frac {d\kappa _{n}}{dp}}.}$
• 服从几何分布的随机变量的累积生成函数是${\displaystyle K(t)=log({\frac {p}{1+(p-1)e^{t}}})}$ 。 一阶累积量是${\displaystyle \kappa _{1}=K'(0)=p^{-1}-1}$ ，二阶累积量是${\displaystyle \kappa _{2}=K''(0)=\kappa _{1}p^{-1}}$
• 服从泊松分布的随机变量的累积生成函数是${\displaystyle K(t)=\mu {e^{t}-1}}$ 。所有的累积量军等于参数${\displaystyle \mu }$ : ${\displaystyle \kappa _{1}=\kappa _{2}=\kappa _{3}=...=\kappa _{n}=\mu }$
• 服从二项分布的随机变量的累积生成函数是${\displaystyle K(t)=nlog(1-p+pe^{t})}$ 。 一阶累积量是${\displaystyle \kappa _{1}=K'(0)=np}$ ，二阶累积量是${\displaystyle \kappa _{2}=K''(0)=\kappa _{1}(1-p)}$
• 服从负二项分布的随机变量的累积生成函数的导数是${\displaystyle K'(t)={\frac {n}{{\frac {1}{(1-p)e^{t}}}-1}}}$ 。一阶累积量是${\displaystyle \kappa _{1}=K'(0)=n({\frac {1}{p}}-1)}$ ，二阶累积量是${\displaystyle \kappa _{2}=K''(0)=\kappa _{1}p^{-1}}$

## 参考来源

1. ^ Kendall, M.G., Stuart, A. (1969) The Advanced Theory of Statistics, Volume 1 (3rd Edition). Griffin, London. (Section 3.12)
2. ^ Lukacs, E. (1970) Characteristic Functions (2nd Edition). Griffin, London. (Page 27)
3. ^ Lukacs, E. (1970) Characteristic Functions (2nd Edition). Griffin, London. (Section 2.4)
4. ^ Aapo Hyvarinen, Juha Karhunen, and Erkki Oja (2001) Independent Component Analysis, John Wiley & Sons. (Section 2.7.2)

## 外部链接

• （英文）累积量：一些数学术语的早期使用