Computability, Unsolvability, Randomness
Math 497A: Homework #9
Stephen G. Simpson
Due Monday, October 29, 2007
1. Hoeffding’s Inequality says that the probability space 2Nwith the fair
coin probability measure satisfies
Prob Pn−1
i=0 X(i)
n−1
2
>
!<2
exp 2n2.
Use Hoeffding’s Inequality to prove that if a point X∈2Nis random
(i.e., random in the sense of Martin-L¨of), then Xobeys the Strong Law
of Large Numbers:
Pn−1
i=0 X(i)
n→1
2as n→∞.
2. Prove that there exist weakly 1-random points in 2Nwhich do not obey
the Strong Law of Large Numbers.
Hint: Use finite approximation.
3. In problem 1, can you say anything about the rate of convergence to
1/2?
4. Prove that if X⊕Y∈2Nis random (i.e., random in the sense of
Martin-L¨of), then XTYand YTX.
5. Prove that there exist points X, Y ∈2Nsuch that X⊕Yis weakly
1-random yet X≡TY.
6. A set B⊆Nis said to be biimmune if both Band its complement
N\Bare immune. Prove that if X∈2Nis weakly 1-random then X
is the characteristic function of a biimmune set.
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