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

LLN for exchangeable sequences
( ) n≥0
( P)
( )
n σ Ω n ⊂ n+
1
Let Y be a sequence of -algebras on , Y Y , . Y Y Y
Y
∀n ≥ 0. Then, a sequence of r.v.s X is called a martingale if,
{ n }
( i) E X < ∞,
each n;
( ii) X is Y -measurable, each n;
n n
{ Y }
n n≥0
( iii )E X | Y = X a . s ., each m ≤ n .
n m m
Martingale Convergence Theorem:
( X n ) { X n }
Let be a martingale s.t. sup E < ∞.
n≥1
n
1
Then lim X n =
X exists a. s. ( and is finite a. s.). Moreover, X is in L .
n→∞