Convergences of Random Variables with Respect to Coherent Upper Probabilities Defined by Hausdorff Outer Measures

Abstract

Given a non-empty set \( \Omega\ \)and a partition B of \( \Omega\ \) let L be the class of all subsets of Ω. Upper conditional probabilities \(\overline{P}(A|B)\) are defined on L × B by a class of Hausdorff outer measures when the conditioning event B has positive and finite Hausdorff measure in its dimension; otherwise they are defined by a 0-1 valued finitely additive (but not countably additive) probability. The unconditional upper probability is obtained as a particular case when the conditioning event is Ω. Relations among different types of convergence of sequences of random variables are investigated with respect to this upper probability. If Ω has finite and positive Hausdorff outer measure in its dimension the given upper probability is continuous from above on the Borel σ-field. In this case we obtain that the pointwise convergence implies the μ-stochastic convergence. Moreover, since the outer measure is subadditive then stochastic convergence with respect to the given upper probability implies convergence in μ-distribution.KeywordsHausdorff DimensionHausdorff MeasureConditioning EventPointwise ConvergenceOuter MeasureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.