# 累积量

## 简介

$\sum_{n=1}^\infty \kappa_n \frac{t^n}{n!}=\log \mathbb{E} e^{t X}=:g(t).$

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

\begin{align}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{align}

\begin{align} \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{align}

$h(t)=\sum_{n=1}^\infty \kappa_n \frac{(it)^n}{n!}=\log(\operatorname{E} (e^{i t X}))=\mu it - \sigma^2 \frac{ t^2}{2} + \cdots.\,$

## 统计数学中的应用

\begin{align} 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{align}

## 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)