09Stoch3

# 09Stoch3 - 20 CHAPTER 1 BASIC PROBABILITY THEORY 1.3 Markov...

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Unformatted text preview: 20 CHAPTER 1. BASIC PROBABILITY THEORY 1.3 Markov Property Let A be an index set and let {F α : α ∈ A } be family of sub- σ-fields of F . We say that the family of F α ’s are conditionally independent relative to G if for any Λ i ∈ F α i i = 1 , ··· ,n , P ( n \ j =1 Λ j |G ) = n Y j =1 P (Λ i |G ) . (1.3.1) Proposition 1.3.1. For α ∈ A , let F ( α ) denote the sub- σ-field generated by F β , β ∈ A \{ α } . Then the family {F α } α are conditionally independent relative to G if and only if P (Λ | F ( α ) ∨ G ) = P (Λ | G ) , Λ ∈ F α where F ( α ) ∨ G is the sub- σ-field generated by F ( α ) and G . Proof . We only prove the case A = { 1 , 2 } , i.e., P (Λ | F 2 ∨ G ) = P (Λ | G ) , Λ ∈ F 1 . (1.3.2) The general case follows from the same argument. To prove the sufficiency, we assume (1.3.2). To check (1 . 3 . 1), let Λ ∈ F 1 , then for M ∈ F 2 , P (Λ ∩ M |G ) = E ( P (Λ ∩ M |F 2 ∨ G ) |G ) = E ( P (Λ | F 2 ∨ G ) χ M |G ) = E ( P (Λ | G ) χ M |G ) (by (1 . 3 . 2)) = P (Λ | G ) P ( M | G ) . Hence F 1 and F 2 are G-independent. To prove the necessity, suppose (1.3.1) holds, we claim that for Δ ∈ G , Λ ∈ F 1 and M ∈ F 2 , Z M ∩ Δ P (Λ |G ) dP = Z M ∩ Δ P (Λ | F 2 ∨ G ) dP 1.3. MARKOV PROPERTY 21 Since the sets of the form M ∩ Δ generate G∨F 2 , we have P (Λ |G ) = P (Λ | F 2 ∨ G ). i.e., (1.3.2) holds. The claim follows from the following: let Λ ∈ F 1 , M ∈ F 2 , then E ( P (Λ |G ) χ M |G ) = P (Λ |G ) P ( M |G ) = P (Λ ∩ M |G ) (by (1 . 3 . 1)) = E ( P (Λ |F 2 ∨ G ) χ M |G ) / Corollary 1.3.2. Let { X α } α ∈ A be a family of r.v. and let F α be the sub- σ- field generated by X α . Then the X α ’s are independent if and only if for any Borel set B , P ( X α ∈ B |F ( α ) ) = P ( X α ∈ B ) . Moreover the above condition can be replaced by: for any integrable Y ∈ F α , E ( Y |F ( α ) ) = E ( Y ) . Proof . The first identity follows from Proposition 1.3.1 by taking G as the trivial σ-field. The second one follows from an approximation by simple func- tion and use the first identity. / To consider the Markov property, we first consider an important basic case. Theorem 1.3.3. Let { X n } ∞ n =1 be a sequence of independent r.v. and each X n has a distribution μ n on R . Let S n = ∑ n j =1 X j . Then for B ∈ B , P ( S n ∈ B | S 1 , ··· ,S n- 1 ) = P ( S n ∈ B | S n- 1 ) = μ n ( B- S n- 1 ) (Hence S n is independent of S 1 , ··· ,S n- 2 given S n- 1 .) 22 CHAPTER 1. BASIC PROBABILITY THEORY Proof . We divide the proof into two steps. Step 1 . We show that P ( X 1 + X 2 ∈ B | X 1 ) = μ 2 ( B- X 1 ) Let Λ ∈ F X 1 , then Λ = X- 1 1 ( A ) for some A ∈ B , and Z Λ μ 2 ( B- X 1 ) dP = Z A μ 2 ( B- x 1 ) dμ 1 ( x 1 ) = Z A ‡ Z x 1 + x 2 ∈ B dμ 2 ( x 2 ) · dμ 1 ( x 1 ) = ZZ x 1 ∈ A, x 1 + x 2 ∈ B d ( μ 1 × μ 2 )( x 1 ,x 2 ) = P ( X 1 ∈ A, X 1 + X 2 ∈ B ) = Z Λ P ( X 1 + X 2 ∈ B | F...
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09Stoch3 - 20 CHAPTER 1 BASIC PROBABILITY THEORY 1.3 Markov...

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