다변량 확률 변수의 확률 수렴
📂수리통계학 다변량 확률 변수의 확률 수렴 정의 p p p 랜덤 벡터 X \mathbf{X} X 시퀀스 { X n } \left\{ \mathbf{X}_{n} \right\} { X n } n → ∞ n \to \infty n → ∞ X n \mathbf{X}_{n} X n X \mathbf{X} X 확률 수렴 convergence in Probability 한다고 말하고, X n → P X \mathbf{X} _ {n} \overset{P}{\to} \mathbf{X} X n → P X ∀ ε > 0 , lim n → ∞ P [ ∥ X n − X ∥ < ε ] = 1
\forall \varepsilon > 0 , \lim_{n \to \infty} P \left[ \left\| \mathbf{X}_{n} - \mathbf{X} \right\| < \varepsilon \right] = 1
∀ ε > 0 , n → ∞ lim P [ ∥ X n − X ∥ < ε ] = 1
∥ ⋅ ∥ \| \cdot \| ∥ ⋅ ∥ 유클리드 놈 으로써, ∥ ( x 1 , ⋯ , x n ) ∥ = x 1 2 + ⋯ + x n 2 \left\| \left( x_{1} , \cdots , x_{n} \right) \right\| = \sqrt{ x_{1}^{2} + \cdots + x_{n}^{2}} ∥ ( x 1 , ⋯ , x n ) ∥ = x 1 2 + ⋯ + x n 2 정리 p p p X = ( X 1 , ⋯ , X p ) \mathbf{X} = \left( X_{1} , \cdots , X_{p} \right) X = ( X 1 , ⋯ , X p ) X n → P X ⟺ X n k → P X k , ∀ k = 1 , ⋯ , p
\mathbf{X}_{n} \overset{P}{\to} \mathbf{X} \iff X_{nk} \overset{P}{\to} X_{k} \qquad, \forall k = 1, \cdots, p
X n → P X ⟺ X nk → P X k , ∀ k = 1 , ⋯ , p
증명 ( ⇒ ) (\Rightarrow) ( ⇒ )
X n → P X \mathbf{X}_{n} \overset{P}{\to} \mathbf{X} X n → P X ε > 0 \varepsilon > 0 ε > 0 ε ≤ ∣ X n k − X k ∣ ≤ ∥ X n k − X k ∥
\varepsilon \le \left| X_{nk} - X_{k} \right| \le \left\| \mathbf{X}_{nk} - \mathbf{X}_{k} \right\|
ε ≤ ∣ X nk − X k ∣ ≤ ∥ X nk − X k ∥ lim sup n → ∞ P [ ∣ X n k − X k ∣ ≥ ε ] ≤ lim sup n → ∞ P [ ∥ X n k − X k ∥ ≥ ε ] = 0
\limsup_{n \to \infty} P \left[ \left| X_{nk} - X_{k} \right| \ge \varepsilon \right] \le \limsup_{n \to \infty} P \left[ \left\| \mathbf{X}_{nk} - \mathbf{X}_{k} \right\| \ge \varepsilon \right] = 0
n → ∞ lim sup P [ ∣ X nk − X k ∣ ≥ ε ] ≤ n → ∞ lim sup P [ ∥ X nk − X k ∥ ≥ ε ] = 0
( ⇐ ) (\Leftarrow) ( ⇐ )
X n k → P X k , ∀ k = 1 , ⋯ , p X_{nk} \overset{P}{\to} X_{k} , \forall k = 1, \cdots, p X nk → P X k , ∀ k = 1 , ⋯ , p ε > 0 \varepsilon > 0 ε > 0 ε ≤ ∥ X n − X ∥ ≤ ∑ k = 1 p ∣ X n k − X k ∣
\varepsilon \le \left\| \mathbf{X}_{n} - \mathbf{X} \right\| \le \sum_{k=1}^{p} \left| X_{nk} - X_{k} \right|
ε ≤ ∥ X n − X ∥ ≤ k = 1 ∑ p ∣ X nk − X k ∣ lim sup n → ∞ P [ ∥ X n − X ∥ ≥ ε ] ≤ lim sup n → ∞ P [ ∣ X n k − X k ∣ ≥ ε ] ≤ ∑ k = 1 p lim sup n → ∞ P [ ∣ X n k − X k ∣ ≥ ε ] = 0
\begin{align*}
& \limsup_{n \to \infty} P \left[ \left\| \mathbf{X}_{n} - \mathbf{X} \right\| \ge \varepsilon \right]
\\ \le & \limsup_{n \to \infty} P \left[ \left| X_{nk} - X_{k} \right| \ge \varepsilon \right]
\\ \le & \sum_{k=1}^{p} \limsup_{n \to \infty} P \left[ \left| X_{nk} - X_{k} \right| \ge \varepsilon \right]
\\ =& 0
\end{align*}
≤ ≤ = n → ∞ lim sup P [ ∥ X n − X ∥ ≥ ε ] n → ∞ lim sup P [ ∣ X nk − X k ∣ ≥ ε ] k = 1 ∑ p n → ∞ lim sup P [ ∣ X nk − X k ∣ ≥ ε ] 0
■
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