Suggested Exercises III:
1. Deduce the Weil equidistribution theorem from the ergodic theorem (fill in the details
of the argument given in class) :
If α ∈ [0, 1] is irrational, then {α, 2α, 3α, . . . , } is uniformly distributed mod 1 in [0, 1]:
for any subinterval (a, b) ⊂ [0, 1]
|{i ∈ [0, n − 1] : iα ∈ (a, b)}|
→b−a
n
2. Show that mixing is an invariant of isomorphism of MPT’s.
3. Complete the proof that a stochastic matrix is primitive iff it is irreducible and aperiodic.
4. Complete the geometric proof (using convexity ideas, given in class) that if P is a
primitive stochastic matrix with stationary vector π, then
(P n )ij → πj
5. Show that if α is a (finite, measurable) partition and Bn , B are sigma-algebras s.t.
Bn ↑ B, then
H(α|Bn ) ↓ H(α|B).
In the following problems, let α and β be finite measurable partitions and E, F subsigma-algebras, and σ(·) be the sigma-algebra generated by ·
6. Show that Iα∨β|F = Iα|F + Iβ|σ(α∪F ) a.e.
7. Show that H(α ∨ β|F) = H(α|F) + H(β|σ(α ∪ F)) ≤ H(α|F) + H(β|F)
8. Show that H(β|σ(F ∪ E)) ≤ H(β|F) with equality iff E ⊥F β
9. Does β α imply:
(a) Iβ|F ≤ Iα|F a.e. –or–
(b) Iβ|F ≥ Iα|F a.e. –or–
(c) neither
10. Does E ⊆ F imply
(a) Iα|F ≤ Iα|E a.e. –or–
(b) Iα|F ≥ Iα|E a.e. –or–
(c) neither
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