Summary
In a probability space (Ω, σ,P), for α⊂σ a sub-σ-field, in general the best approximation inL ∞ by elements ofL ∞(α) has not an unique solution. For the election between these, we prove the convergenceP-almost surely of the conditionalr-means, whenr→∞, to one solution, which we call conditional mid-range. This is characterized for each ω∈Ω by the mid-range, of one regular conditional distributionQ (ω, •):
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Albertos, JA.C., Bea, C.M. Semi-recorrido condicionado (Expresion asintotica de la r-esperanza condicionada). Trabajos de Estadistica y de Investigacion Operativa 34, 54–60 (1983). https://doi.org/10.1007/BF02888791
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DOI: https://doi.org/10.1007/BF02888791