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.