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Two Kinds of Invariance of Full Conditional Probabilities

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Pruss A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2013

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tom 61

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nr 3

277-283

EN

Let G be a group acting on Ω and ℱ a G-invariant algebra of subsets of Ω. A full conditional probability on ℱ is a function P: ℱ × (ℱ∖{∅}) → [0,1] satisfying the obvious axioms (with only finite additivity). It is weakly G-invariant provided that P(gA|gB) = P(A|B) for all g ∈ G and A,B ∈ ℱ, and strongly G-invariant provided that P(gA|B) = P(A|B) whenever g ∈ G and A ∪ gA ⊆ B. Armstrong (1989) claimed that weak and strong invariance are equivalent, but we shall show that this is false and that weak G-invariance implies strong G-invariance for every Ω, ℱ and P as above if and only if G has no non-trivial left-orderable quotient. In particular, G = ℤ provides a counterexample to Armstrong's claim.

2

Linear extensions of orders invariant under abelian group actions

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Pruss A.

Colloquium Mathematicum

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2014

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tom 137

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nr 1

117-125

EN

Let G be an abelian group acting on a set X, and suppose that no element of G has any finite orbit of size greater than one. We show that every partial order on X invariant under G extends to a linear order on X also invariant under G. We then discuss extensions to linear preorders when the orbit condition is not met, and show that for any abelian group acting on a set X, there is a linear preorder ≤ on the powerset 𝓟X invariant under G and such that if A is a proper subset of B, then A < B (i.e., A ≤ B but not B ≤ A).

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On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables

100%

Pruss A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2013

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tom 61

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nr 2

161-168

EN

Let Ω be a countable infinite product $Ω₁^{ℕ}$ of copies of the same probability space Ω₁, and let {Ξₙ} be the sequence of the coordinate projection functions from Ω to Ω₁. Let Ψ be a possibly nonmeasurable function from Ω₁ to ℝ, and let Xₙ(ω) = Ψ(Ξₙ(ω)). Then we can think of {Xₙ} as a sequence of independent but possibly nonmeasurable random variables on Ω. Let Sₙ = X₁ + ⋯ + Xₙ. By the ordinary Strong Law of Large Numbers, we almost surely have $E_{*}[X₁] ≤ lim inf Sₙ/n ≤ lim sup Sₙ/n ≤ E*[X₁]$, where $E_{*}$ and E* are the lower and upper expectations. We ask if anything more precise can be said about the limit points of Sₙ/n in the nontrivial case where $E_{*}[X₁] < E*[X₁]$, and obtain several negative answers. For instance, the set of points of Ω where Sₙ/n converges is maximally nonmeasurable: it has inner measure zero and outer measure one.

Ograniczanie wyników

2 Bulletin of the Polish Academy of Sciences. Mathematics

1 Colloquium Mathematicum

3 Pruss A.

1 2014

2 2013

Two Kinds of Invariance of Full Conditional Probabilities

Linear extensions of orders invariant under abelian group actions

On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables