Exchangeable Sequences, Laws of Large Numbers, and the ...

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$University of Colorado Denver Math 1070 Topics Updated: Fall 2009 ...$

$Math 3000 Running Glossary$

Exchangeability Definition: An infinite sequence of X , L , X , L of random variables is said to be exchangeable if ∀ n ≥ 2, D ( ) ( ) 1 L n (1) L ( n ) X , , X = X , , X ∀ ∈ S ( n), 1 π π π n { L } where S ( n) is the group of permutations of 1, , n .
Exchangeability Example: Polya’s urn An urn has initially r red and b black balls. Draw a ball at random and note its color, and replace the ball back and add another ball of the same color. th Let X =1 if the i draw yields a red ball and X =0 otherwise. i Then, r r + 1 b r + 2 P( 1,1,0,1 ) = ⋅ ⋅ ⋅ b + r b + r + 1 b + r + 2 b + r + 3 r b r + 1 r + 2 = ⋅ ⋅ ⋅ = P b + r b + r + 1 b + r + 2 b + r + 3 Similarly for other cases. The sequence X , L, X , L is exchangeable. 1 n i ( ) 1,0,1,1 .
Page 1 and 2: Exchangeable Sequences, Laws of Lar
Page 3 and 4: Independence of random variables De
Page 5: Laws of Large Numbers Strong Law of
Page 9 and 10: LLN for exchangeable sequences ( )
Page 11 and 12: LLN for exchangeable sequences proo
Page 13 and 14: De Finetti’s Theorem Definition :
Page 15 and 16: De Finetti’s Theorem Proof : ( co
Page 17: References Aldous, D. (1985). Excha

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