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Claim 1: y^ is independent of (x^,c^).
This follows by noting that under Hq and using the
exchangeability property of type 2 and 3 pairs (as was
shown in Lemma 3.2.2) that
P{yi = l | (xi,ci)} = P{X1i=xi,X2i = ci| (x ct) }
= P{ X1 i = ci X21 = xi | (xi c ) } = P{ Y^O | (xj. c )}
Since P{ Yi = l | ci ) } + P { y = 0 | ( x c ) } = 1, Claim 1
follows. Now define y = (Yn + i Yn +2>**>Yn +n
1 1 1 c
Claim 2: y is a vector of n i.i.d. Bernoulli random
variables which are independent of
^xnL + l cn1 + l ^ ^xnL + 2 cn1 + 2 ^ * (xn1 + nc cni+nc^
This follows from Claim 1 and the fact that
{ (xnj + l cn1 + l ^ ^xn:+2 cni+2 ^ * ('Xn1+nc Cn1 + n(;
)}
are i.i.d.
Claim 3: y is independent of
* ^xnj + l cn1 + l ) (xni+2 cni+2 ^ * (xn1 + n(, Cn1+n(.)
x and x where
~nl ~n4
~nl {Ull,X12)(xl2x22)>---.