RANDOM SEQUENCES
Michiel van Lambalgen
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Abstract
References 166
Key takeaways
- In other words, the usual version of the strong law can be derived from the version acceptable to von Mises if we take the collection of probability distributions (P n ), induced by the Kollektiv x ∈ 2 ω , to define a single σ-additive measure µ p on 2 ω .
- In the second stage of the construction, Ville temporarily adopts von Mises' viewpoint and interprets probability measures on 2 ω as in effect being induced by Kollektivs ξ ∈ (2 ω ) ω ; so that µ A = 1 must mean:
- Speaking mathematically, a distribution (1-p,p) on {0,1} determines a measure µ p = (1-p,p) ω on 2 ω , but this measure is a probability only if it is induced by a Kollektiv ξ ∈ (2 ω ) ω .
- This section addresses the following question: if N ⊆ 2 ω ×2 ω is a (total) recursive sequential test with respect to µ×ν, is it possible to construct a (total) recursive sequential test M with respect to µ such that {x| νN x > 0} ⊆ M?
- For this, it suffices to show that the mapping π 2 : 2 ω ×2 ω → 2 ω defined by π 2 <x,y> = y is such that for any recursive sequential test N with respect to λ, π 2 -1 N is a recursive sequential test with respect to µ, for in that case, <x,y> ∈ R(µ) implies y ∈ R(λ).
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