Strong laws of large numbers for generalizations of Fréchet mean sets
Christof Schötz
Abstract
A Fréchet mean of a random variable Y with values in a metric space (Q,d) is an element of the metric space that minimizes q↦Ed(Y,q)2. This minimizer may be non-unique. We study strong laws of large numbers for sets of generalized Fréchet means. Following generalizations are considered: the minimizers of Ed(Y,q)α for α>0, the minimizers of EH(d(Y,q)) for integrals H of non-decreasing functions, and the minimizers of Ec(Y,q) for a quite unrestricted class of cost functions c. We show convergence of empirical versions of these sets in outer limit and in one-sided Hausdorff distance. The derived results require only minimal assumptions.
Topics & Concepts
MathematicsHausdorff distanceHausdorff spaceLimit (mathematics)Limit of a functionLaw of large numbersMetric spaceSpace (punctuation)Convergence (economics)Element (criminal law)Metric (unit)CombinatoricsRandom variableWeak convergenceClass (philosophy)Mathematical analysisPure mathematicsLawStatisticsArtificial intelligenceEconomicsOperations managementLinguisticsComputer scienceEconomic growthPolitical sciencePhilosophyComputer securityAsset (computer security)Optimization and Variational AnalysisFuzzy Systems and OptimizationFixed Point Theorems Analysis