6.3. Slutsky’s Theorem

Definition

A sequence of random variables \(\{X_n,n\ge 1\}\) is bounded in probability if their associated distributions are tignt, i.e., for any \(\epsilon>0\), there exists a constant \(\mathcal{M}>0\) s.t. \[ P(|X_n|\le \mathcal{M})>1-\epsilon\mbox{ }\mbox{ }\mbox{ for all }n\ge 1. \]

• 즉 measure에서의 tight가 random variable에서는 bounded in probability로 불린다

• Note : measure의 notation인 \(P_{X_n}([-\mathcal{M},\mathcal{M}])\)이 결국 \(P(|X_n|\le \mathcal{M})\)과 같다; \[ P(|X_n|\le \mathcal{M})= \int_\Omega I_{\{|X_n(\omega)|\le \mathcal{M}\}}dP(\omega)=\int_\mathbb{R}I_{\{|x|\le\mathcal{M}\}}dP_{X_n} = P_{X_n}([-\mathcal{M},\mathcal{M}])\stackrel{\text{Def of tight}}{>} 1-\epsilon. \]

Lemma

If \(\{X_n\}\) is bounded in probability and \(Y_n\leadsto 0\), then \(X_nY_n\leadsto 0\).

• 증명: \(\{X_n\}\)이 bounded in prob이기 때문에 \(\exists\) \(\delta>0\) and \(\mathcal{M}\)>0 s.t. \[ P(|X_n|>\mathcal{M})\le \delta. \] 또한 \(Y_n\leadsto 0\implies Y_n\stackrel{\text{Pr}}\rightarrow 0\)이다. 때문에 \[\begin{eqnarray*} P(|X_nY_n|>\epsilon)&=& P(|X_nY_n|>\epsilon, |X_n|\le \mathcal{M})+P(|X_nY_n|>\epsilon, |X_n|> \mathcal{M})\\ &\le&P(|Y_n|>\frac{\epsilon}{\mathcal{M}})+P(|X_n|> \mathcal{M})\\ &\le&P(|Y_n|>\frac{\epsilon}{\mathcal{M}})+\delta \longrightarrow \delta \mbox{ }\mbox{ }\mbox{ as }n\rightarrow \infty. \end{eqnarray*}\] Therefore, \(X_nY_n\stackrel{\text{Pr}}\rightarrow 0\implies X_nY_n\leadsto 0\).

Slutsky 이론 증명 위해 필요

Definition

If \(X_n\leadsto 0(\)or equivalently, \(X_n\stackrel{\text{Pr}}\rightarrow 0)\), then we write \(X_n=o_p(1)\).

If \(\{X_n,n\ge 1\}\) is bounded in probability, then we write \(X_n=O_p(1)\).

• 이전 Lemma : \(X_n=O_p(1)\), \(Y_n=o_p(1)\), \(X_nY_n=o_p(1)\)이므로 \(O_p(1)o_p(1)=o_p(1)\)이다.

• \(X_n \stackrel{\text{Pr}}\rightarrow 0\), \(Y_n \stackrel{\text{Pr}}\rightarrow 0\implies X_n+Y_n \stackrel{\text{Pr}}\rightarrow 0\), i.e., \(o_p(1)+o_p(1)=o_p(1)\)이다.

• \(O_p(1)+O_p(1)=O_p(1)\) :

  \(P(|X_n|>\mathcal{M}_x)<\epsilon/2\), \(P(|Y_n|>\mathcal{M}_y)<\epsilon/2\)

  \(\implies P(|X_n+Y_n|>\mathcal{M}_x\vee\mathcal{M}_y )\le P(|X_n|>\mathcal{M}_x\vee\mathcal{M}_y)+P(|Y_n|>\mathcal{M}_x\vee\mathcal{M}_y)<\epsilon\);

• \(o_p(1)=O_p(1)\) (weak convergence \(\implies\) tightness);

• \(O_p(1)\ne o_p(1)\) (모든 subsequence가 하나의 measure로 weakly converge한다는 내용이 추가로 필요하다).

매우 중요하다

Lemma (Converging Together Lemma)

\(X_n\leadsto X\) and \(Y_n\leadsto Y \implies Y_n\leadsto X\)

• 즉 \(Y_n-X_n\leadsto 0\)이라면 \(X_n\)과 \(Y_n\)은 같은 곳으로 weakly converge한다.

매우 중요하다

Theorem (Slutsky’s Theorem)

If for all \(n\ge 1\), \(X_n,A_n\), and \(B_n\) are random variables defined on the same probability space, with \(X_n\leadsto X\), \(A_n\leadsto a\), and \(B_n\leadsto b\), \(a,b\in \mathbb{R}\), then \[ A_n X_n+ B_n \leadsto aX+b. \]

• 증명

0. \(X_n\leadsto X\implies \{X_n\}\) is tight(bounded in probability);

1. \(A_n-a\leadsto a-a =0\), and \(B_n-b\leadsto b-b =0\);

2. \((A_nX_n+B_n)-(aX_n+b)=(A_n-a)X_n+(B_n-b)= o_p(1)O_p(1)+o_p(1)\leadsto 0\)

3. \(aX_n+b\leadsto aX+b\).

• Converging Together Lemma에 의해 증명된다.

정리

1. Bounded in Prob\((O_p(1))\) : \(P(|X_n|\le \mathcal{M})>1-\epsilon\mbox{ }\mbox{ }\mbox{ }\mbox{ }\forall\mbox{ }n\).

2. \(O_p(1)\), \(o_p(1)\): \(o_p(1)\)은 \(X_n\leadsto 0\)를, \(X_n=O_p(1)\) 은 bounded in probability를 의미.

   • \(o_p(1)O_p(1)=o_p(1)\); \(X_n\leadsto 0\), \(Y_n\) bounded in prob \(\implies\) \(X_nY_n\leadsto 0\).

   • \(o_p(1)+o_p(1)=o_p(1)\); \(X_n\leadsto 0\), \(Y_n\leadsto 0\implies X_n+Y_n\leadsto 0\).

   • \(O_p(1)+O_p(1)=O_p(1)\); \(X_n, Y_n\) bounded in prob\(\implies X_n+Y_n\) bounded in prob.

   • \(o_p(1)=O_p(1)\); Weak convergence implies bounded in prob.

   • \(O_p(1)\ne o_p(1)\); bounded in prob does not implies weak convergence.

   
   

3. 함께수렴(Converging Together) : \(X_n\leadsto X\) and \(Y_n\leadsto Y \implies Y_n\leadsto X\)

4. 슬러츠키 : \(X_n\leadsto X\), \(A_n\leadsto a\), \(B_n\leadsto b\implies A_n X_n+ B_n \leadsto aX+b\).

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