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7 Oct 2019: Fix integers a,b 1 and setT = min{n 0 : Xn {a,b}}. ... Use (b) to find random variables (Mn)n3 such that, forall n 0,.

16 Sep 2019: Proof. Results 1 to 3 are immediate from the definition of conditional probability.For result 4, note that A B B, so P (AB) P (B) and thus P (A | B) ... 1.P (B | B) = 1 (obviously), so it just remains to show the Axiom III.

5 Dec 2019: a) G G3 has a transitive graph height function, (Theorem 3.1(b)),(b) G is the Cayley graph of the Grigorchuk group with three generators, (Theorem 8.1),(c) G ... ad2,b} and R = {a21,a. 22,. ,a. 2d2,b. g}. The proofs follow.

4 Dec 2019: a) P(ρ > 0) = 1,. (b) P̃(A) = E(ρ1A) for all A F. ... Hence (b) holds.Write Rd as an orthogonal direct sum E0 E1, where.