A unifying view on some problems in probability and statistics

Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$L$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>L</mi> </math> </EquationSource> </InlineEquation> be a linear space of real random variables on the measurable space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$(\varOmega ,\mathcal {A})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="italic">Ω</mi> <mo>,</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation>. Conditions for the existence of a probability <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$P$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>P</mi> </math> </EquationSource> </InlineEquation> on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\mathcal {A}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="script">A</mi> </math> </EquationSource> </InlineEquation> such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$E_P|X|>\infty $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>E</mi> <mi>P</mi> </msub> <mrow> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>∞</mi> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$E_P(X)=0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>E</mi> <mi>P</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math> </EquationSource> </InlineEquation> for all <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$X\in L$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>X</mi> <mo>∈</mo> <mi>L</mi> </mrow> </math> </EquationSource> </InlineEquation> are provided. Such a <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$P$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>P</mi> </math> </EquationSource> </InlineEquation> may be finitely additive or <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$\sigma $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">σ</mi> </math> </EquationSource> </InlineEquation>-additive, depending on the problem at hand, and may also be requested to satisfy <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$P\sim P_0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>P</mi> <mo>∼</mo> <msub> <mi>P</mi> <mn>0</mn> </msub> </mrow> </math> </EquationSource> </InlineEquation> or <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$$P\ll P_0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>P</mi> <mo>≪</mo> <msub> <mi>P</mi> <mn>0</mn> </msub> </mrow> </math> </EquationSource> </InlineEquation> where <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$$P_0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>P</mi> <mn>0</mn> </msub> </math> </EquationSource> </InlineEquation> is a reference measure. As a motivation, we note that a plenty of significant issues reduce to the existence of a probability <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$$P$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>P</mi> </math> </EquationSource> </InlineEquation> as above. Among them, we mention de Finetti’s coherence principle, equivalent martingale measures, equivalent measures with given marginals, stationary and reversible Markov chains, and compatibility of conditional distributions. Copyright Springer-Verlag Berlin Heidelberg 2014