中心極限定理

試行によって得られた事象の平均を得る作業を繰り返しおこなったとき, その平均の分布は正規分布に近づくことを示したのが,中心極限定理(Central limit theorem)である.

\(\{X _i\} _{\overline{\mathbb{N} _+}},\; {\scriptsize \mathit i.i.d.} \sim (\mu,\sigma^2)\)である確率変数列に対して, \[ \displaystyle \frac{\sqrt{n}}{\sigma} \left( \frac{\sum ^n _{i=1} X _i}{n} - \mu \right) \xrightarrow{d} \mathcal{N}(0, 1) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \] が成立する.

証明
\(\displaystyle Z_n = \frac{\sqrt{n}}{\sigma} \left( \frac{\sum^n _{i=1} X_i}{n} -\mu \right)\)とおく, \[ \begin{align} E[{Z _n}^2] &= E\left[ \left( \frac{\sqrt{n}}{\sigma} \left( \frac{\sum^n _{i=1} X_i}{n} -\mu \right) \right)^2 \right] \\ &= \frac{1}{n \sigma^2 }E\left[(\sum^n _{i=1} X_i - n\mu )^2 \right] \\ &= \frac{1}{n \sigma^2 }E\left[\left( \sum^n _{i=1} (X_i - \mu) \right)^2 \right] \\ &= \frac{1}{n \sigma^2 } E\left[\sum ^n _{i=1}(X_i - \mu)^2 + \sum ^n _{i=1} \sum ^n _{j=1, i \not= j} (X_i - \mu) (X_j - \mu) \right] \\ &= \frac{1}{n \sigma^2 } \sum ^n _{i=1} E\left[(X_i - \mu)^2 + \sum ^n _{j=1, i \not= j} (X_i - \mu) (X_j - \mu) \right] \\ &= \frac{1}{n \sigma^2 } \sum ^n _{i=1} \left( E[(X_i - \mu)^2] + E\left [\sum ^n _{j=1, i \not= j} (X_i - \mu) (X_j - \mu) \right] \right) \\ &= \frac{1}{n \sigma^2 } \sum ^n _{i=1} \left( E[(X_i - \mu)^2] + \sum ^n _{j=1, i \not= j} E[ (X_i - \mu) (X_j - \mu) ] \right) \\ &= \frac{1}{n \sigma^2 } \sum ^n _{i=1} \left( E[(X_i - \mu)^2] + \sum ^n _{j=1, i \not= j} E[ X_i X_j - \mu X_i - \mu X_j + \mu^2 ] \right) \\ &= \frac{1}{n \sigma^2 } \sum ^n _{i=1} \left( E[(X_i - \mu)^2] + \sum ^n _{j=1, i \not= j} ( E[ X_i ] E[ X_j ] - \mu E[X_i] - \mu E[X_j] + \mu^2 ) \right) \\ &= \frac{1}{n \sigma^2 } \sum ^n _{i=1} \left( E[(X_i - \mu)^2] + \sum ^n _{j=1, i \not= j} E[ X_i ] E[ X_j ] - \mu E[X_i] - \mu E[X_j] + \mu^2 ) \right) \\ &= \frac{1}{n \sigma^2 } \sum ^n _{i=1} E[(X_i - \mu)^2] \; \; ( E[ X_i ] = \mu より)\\ &= \frac{1}{n \sigma^2 } n \sigma^2 = 1 \end{align} \] \(Z_n\)の特性関数は, \[ \begin{align} \varphi _{Z _n}(t) &= E[\exp({it{Z_n}})] \\ &= E\left[ \exp \left(it \frac{\sqrt{n}}{\sigma} \left( \frac{\sum^n _{i=1} X _i}{n} - \mu \right) \right) \right] \\ &= E\left[ \exp \left(it \frac{\sqrt{n}}{\sigma} \frac{1}{n} \left( \sum^n _{i=1} X _i - n \mu \right) \right) \right] \\ &= E\left[ \exp \left(it \frac{\sqrt{n}}{\sigma} \frac{1}{n} \sum^n _{i=1} (X _i - \mu) \right) \right] \\ &= E\left[ \exp \left(it \frac{\sqrt{n}}{\sigma} \frac{1}{n} \sum^n _{i=1} (X _i - \mu) \right) \right] \\ &= E\left[ \exp \left(\sum^n _{i=1} i \frac{t}{\sqrt{n}} \frac{(X _i - \mu)}{\sigma} \right) \right] \\ &= \prod ^n _{i=1} E\left[ \exp \left( i \frac{t}{\sqrt{n}} \frac{(X _i - \mu)}{\sigma} \right) \right] \\ &= \prod ^n _{i=1} \varphi _{Z_i} \left( \frac{t}{\sqrt{n}} \right) \end{align} \] ここで\(\displaystyle Z_i = \frac{(X _i - \mu)}{\sigma}\).

\(\varphi _{Z_i} \left( \frac{t}{\sqrt{n}} \right)\)をマクローリン展開すると, \[ \varphi _{Z _i} \left( \frac{t}{\sqrt{n}} \right) = \varphi _{Z _i}(0) + {\varphi _{Z _i}}'(0)\left(\frac{t}{\sqrt{n}}\right) + {\varphi _{Z _i}}''(0)\frac{1}{2!}\left(\frac{t}{\sqrt{n}}\right)^2 + {\varphi _{Z _i}}'''(0)\frac{1}{3!}\left(\frac{t}{\sqrt{n}}\right)^3 + \cdots \] \[ \begin{align} \varphi _{Z _i}(0) &= \left. E\left[ \exp \left( i \frac{t}{\sqrt{n}} Z _i \right) \right] \right| _{\frac{t}{\sqrt{n}}=0} = \left. \int _{\Omega} \exp \left( i \frac{t}{\sqrt{n}} z _i \right) dP(z_i) \right| _{\frac{t}{\sqrt{n}}=0} = 1 \\ \varphi _{Z _i}'(0) &= \left. \frac{d}{d(\frac{t}{\sqrt{n}})} \left\{ E\left[ \exp \left( i \frac{t}{\sqrt{n}} Z _i \right) \right] \right\} \right| _{\frac{t}{\sqrt{n}}=0} \\ &= \left. \frac{d}{d(\frac{t}{\sqrt{n}})} \int _{\Omega} \exp \left( i \frac{t}{\sqrt{n}} z _i \right) dP(z_i) \right| _{\frac{t}{\sqrt{n}}=0} \\ &= \int _{\Omega} \left. \frac{\partial}{\partial (\frac{t}{\sqrt{n}})} \exp \left( i \frac{t}{\sqrt{n}} z _i \right) \right| _{\frac{t}{\sqrt{n}}=0} dP(z_i) \\ &= \int _{\Omega} \left. i z_i \exp \left( i \frac{t}{\sqrt{n}} z _i \right) \right| _{\frac{t}{\sqrt{n}}=0} dP(z_i) \\ &= i \int _{\Omega} z_i dP(z_i) = i E[Z_i] = i E\left[\frac{X_i - \mu}{\sigma}\right] = \frac{i}{\sigma}(E[X_i] - \mu) = 0 \\ \varphi _{Z _i}''(0) &= \left. \frac{d^2}{d^2(\frac{t}{\sqrt{n}})} \left\{ E\left[ \exp \left( i \frac{t}{\sqrt{n}} Z _i \right) \right] \right\} \right| _{\frac{t}{\sqrt{n}}=0} \\ &= \left. \frac{d}{d(\frac{t}{\sqrt{n}})} \int _{\Omega} \exp \left( i \frac{t}{\sqrt{n}} z _i \right) dP(z_i) \right| _{\frac{t}{\sqrt{n}}=0} \\ &= \int _{\Omega} \left. \frac{\partial^2}{\partial^2 (\frac{t}{\sqrt{n}})} \exp \left( i \frac{t}{\sqrt{n}} z _i \right) \right| _{\frac{t}{\sqrt{n}}=0} dP(z_i) \\ &= \int _{\Omega} \left. \frac{\partial}{\partial (\frac{t}{\sqrt{n}})} i z_i \exp \left( i \frac{t}{\sqrt{n}} z _i \right) \right| _{\frac{t}{\sqrt{n}}=0} dP(z_i) \\ &= -\int _{\Omega} \left. z_i^2 \exp \left( i \frac{t}{\sqrt{n}} z _i \right) \right| _{\frac{t}{\sqrt{n}}=0} dP(z_i) \\ &= - \int _{\Omega} z_i^2 dP(z_i) = - E\left[\left(\frac{X_i - \mu}{\sigma}\right)^2\right] = - (\frac{1}{\sigma^2}E[(X_i - \mu)^2]) = - \frac{1}{\sigma^2} \sigma^2 = -1\\ \end{align} \] から, \[ \begin{align} \varphi _{Z _i} \left( \frac{t}{\sqrt{n}} \right) &= \varphi _{Z _i}(0) + {\varphi _{Z _i}}'(0)\left(\frac{t}{\sqrt{n}}\right) + {\varphi _{Z _i}}''(0)\frac{1}{2!}\left(\frac{t}{\sqrt{n}}\right)^2 + {\varphi _{Z _i}}'''(0)\frac{1}{3!}\left(\frac{t}{\sqrt{n}}\right)^3 + \cdots \\ &= 1 - \frac{t^2}{2n} + {\varphi _{Z _i}}'''(0) \frac{t^3}{6n^{\frac{3}{2}}} + \cdots \end{align} \] なので,\(\varphi _{Z _n}(t)\)は, \[ \varphi _{Z _n}(t) = \prod ^n _{i=1} \varphi _{Z_i} \left( \frac{t}{\sqrt{n}} \right) = \left(1 - \frac{t^2}{2n} + {\varphi _{Z _i}}'''(0) \frac{t^3}{6n^{\frac{3}{2}}} + \cdots \right)^n \] \(n \to \infty\)とした時,第3項以降は,\(\frac{1}{n}\)より早く\(0\)に近づくので, \[ \begin{align} \lim _{n \to \infty} \varphi _{Z _n}(t) &= \lim _{n \to \infty} \left(1 - \frac{t^2}{2n} + {\varphi _{Z _i}}'''(0) \frac{t^3}{6n^{\frac{3}{2}}} + \cdots \right)^n \\ &= \lim _{n \to \infty} \left(1 - \frac{t^2}{2n} \right)^n = e^{-\frac{t^2}{2}} \end{align} \] Levyの収束定理より, \[ \lim _{n \to \infty} E[\exp(itZ_n)] = E[\exp(itZ)] \Rightarrow Z_n \xrightarrow{d} Z \] 特性関数\(\displaystyle e^{-\frac{t^2}{2}}\)に対応する確率変数は正規分布\(\mathcal{N}(0, 1)\)なので \[ \displaystyle \frac{\sqrt{n}}{\sigma} \left( \frac{\sum ^n _{i=1} X _i}{n} - \mu \right) \xrightarrow{d} \mathcal{N}(0, 1) \] となる.