数理統計学ノート

Kolmogorovの不等式

\(\{X_i\}_{1 \leq i \leq n}\)を\((\Omega,\mathcal{F},P)\)上の独立な確率変数列とする. \(E[X_i] = 0,{\rm var}(X_i)<\infty\)とすると,任意の\(\lambda > 0\)に対して, \[ P(\max _{1 \leq k \leq n}| \sum^{k} _{i=1} X _i| \geq \lambda) \leq \frac{1}{\lambda^2} \sum^{n} _{i=1} {\rm var}(X_i) \] が成立する.

• 証明
  \(A_n=\{\omega \in \Omega| \max _{1 \leq k \leq n} | \sum^{k} _{i=1} X _i| \geq \lambda \} \) とする. \[ \begin{align} B _k &= \left\{ \begin{array}{ll} \emptyset & (k = 0) \\ \{\omega \in \Omega| \max _{1 \leq j \leq k} | \sum^{j} _{i=1} X _i| \geq \lambda \} & (k > 0) \end{array}\right. \\ B' _k &= \{\omega \in \Omega| |\sum^{k} _{i=1} X _i| \geq \lambda \} \end{align} \] とし \[ \begin{align} A' _k &= (\Omega \backslash B _{k-1} ) \cap B' _k\\ &= \{\omega \in \Omega| \max _{1 \leq j \leq k-1} | \sum^{j} _{i=1} X _i| < \lambda \} \cap \{\omega \in \Omega| |\sum^{k} _{i=1} X _i| \geq \lambda \} \\ &= \{\omega \in \Omega| \max _{1 \leq j \leq k-1} | \sum^{j} _{i=1} X _i| < \lambda \land |\sum ^{k} _{i=1} X _i| \geq \lambda \} \end{align} \] 構成すると,\(A' _{k-1} \)までで\(\max _{1 \leq j \leq k-1} | \sum^{j} _{i=1} X _i| \geq \lambda\)を満たさなかった見本点\(\omega\)の中から, \(|\sum ^{k} _{i=1} X _i| \geq \lambda\)を満たす\(\omega\)を集めて\(A'_k\)を作るので\(A'_k\)は互いに排他で,\(A_n=\bigcup^{n} _{k=1} A'_k\)となる.
  \[ P(A _n) = \sum ^{n} _{k=1} P(A' _k) \] ここで,\(I _{A' _k}:\Omega \mapsto \{0,1\}\)なる指示関数を用意する. \(\{A' _k\}\)は互いに排他なので,\(I _{A' _j}(\omega)=1\)ならば,\(k \not = j\)の\(k\)については,\(I _{A' _k}(\omega)=0\).
  Markovの不等式から\(A _k\)は\(|\sum ^{k} _{i=1} X _i| \geq \lambda\)なので, \[ P(A' _k) \leq \frac{1}{\lambda^2} E[ (\sum ^{k} _{i=1} X _i)^2 I _{A' _k}] \] \[ \sum ^{n} _{k=1} P(A' _k) \leq \sum ^{n} _{k=1} \frac{1}{\lambda^2} E[ (\sum ^{k} _{i=1} X _i)^2 I _{A' _k}] \] \(E[ (\sum ^{k} _{i=1} X _i)^2 I _{A' _k}]\)は \[ \begin{align} E[(\sum ^{n} _{i=1} {X _i})^2 I _{A' _k}] &= \int _{\Omega} (\sum ^{n} _{i=1} X _i(\omega))^2 I _{A' _k}(\omega) P(d\omega) \\ &= \int _{A' _k} (\sum ^{n} _{i=1} X _i(\omega))^2 P(d\omega) \\ &= E[(\sum ^{n} _{i=1} X _i)^2:A'_k] \end{align} \]

  \(E[(\sum ^{n} _{i=1} X _i(\omega))^2:A'_k]\)は互いに排他な\(\{A' _k\}\)事象で制限された期待値で,\( \Omega \supset A _n \supset A' _k \)なので, \[ \begin{align} E[(\sum ^{n} _{i=1} X _i)^2] &= \int _{\Omega} (\sum ^{n} _{i=1} {X _i(\omega)})^2 P(d\omega)\\ & \geq \int _{A_n} (\sum ^{n} _{i=1} {X _i(\omega)})^2 P(d\omega) \\ & = \sum ^{n} _{k=1} \int _{A' _k} (\sum ^{n} _{i=1} {X _i(\omega)})^2 P(d\omega) \\ & = \sum ^{n} _{k=1}E[(\sum ^{n} _{i=1} {X _i})^2:A' _k] \\ & = \sum ^{n} _{k=1} E[(\sum ^{n} _{i=1} {X _i})^2 I _{A' _k}] \end{align} \] \[ \begin{align} (\sum ^{n} _{i=1} {X _i})^2 &= (\sum ^{n} _{i=k+1} {X _i}+\sum ^{k} _{i=1} {X _i})^2 \\ &= (\sum ^{n} _{i=k+1} {X _i})^2 + 2(\sum ^{n} _{i=k+1} {X _i})(\sum ^{k} _{i=1} {X _i}) + (\sum ^{k} _{i=1} {X _i})^2 \\ &\geq 2(\sum ^{n} _{i=k+1} {X _i})(\sum ^{k} _{i=1} {X _i}) + (\sum ^{k} _{i=1} {X _i})^2 \end{align} \] より, \[ E[(\sum ^{n} _{i=1} {X _i})^2 I _{A' _k}] - E[(\sum ^{k} _{i=1} {X _i})^2 I _{A' _k}] \geq E[2(\sum ^{n} _{i=k+1} {X _i})(\sum ^{k} _{i=1} {X _i})I _{A' _k}] \] ここで,\(\sum ^{n} _{i=k+1} {X _i}\)と\((\sum ^{k} _{i=1} {X _i})I _{A' _k}\)は独立しているので \[ E[(\sum ^{n} _{i=1} {X _i})^2 I _{A' _k}] - E[(\sum ^{k} _{i=1} {X _i})^2 I _{A' _k}] \geq 2E[(\sum ^{n} _{i=k+1} {X _i})]E[(\sum ^{k} _{i=1} {X _i})I _{A' _k}] \] \(E[X _i]=0\)なので, \[ E[(\sum ^{n} _{i=1} {X _i})^2 I _{A' _k}] - E[(\sum ^{k} _{i=1} {X _i})^2 I _{A' _k}] \geq 0 \]

\[ \begin{align} P(A _n) &= \sum ^{n} _{k=1} P(A' _k) \\ &\leq \sum ^{n} _{k=1} \frac{1}{\lambda^2} E[ (\sum ^{k} _{i=1} X _i)^2 I _{A' _k}] \\ &\leq \sum ^{n} _{k=1} \frac{1}{\lambda^2} E[(\sum ^{n} _{i=1} {X _i})^2 I _{A' _k}] \\ & = \frac{1}{\lambda^2} E[(\sum ^{n} _{i=1} {X _i})^2] \end{align} \] \(\{X_i\}\)は互いに独立しているので \[ P(\max _{1 \leq k \leq n}| \sum^{k} _{i=1} X _i| \geq \lambda) \leq \frac{1}{\lambda^2} \sum^{n} _{i=1} {\rm var}(X_i) \] 証明終わり.