09Stoch3

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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Page1 / 11

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online