Summary
If μ 1, ... , μ η are non-atomic probability measures on the same measurable space (S, ℱ), then there is an ℱ-measurable partition {A i } n i = 1 of S so that μ i (A i )≧(n − 1 + m)−1 for all i=1, ..., n, where \(m = \left\| {\mathop \Lambda \limits_{i = 1}^n \mu _i } \right\|\) is the total mass of the largest measure dominated by each of the μ i ’S; moreover, this bound is attained for all n≧1 and all m in [0, 1]. This result is an analog of the bound (n+1-M) -1of Elton et al. [5] based on the mass M of the supremum of the measures; each gives a quantative generalization of a well-known cake-cutting inequality of Urbanik [10] and of Dubins and Spanier [2].
References
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Research partly supported by NSF Grants DMS-84-01604 and DMS-86-01608
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Hill, T.P. A sharp partitioning-inequality for non-atomic probability measures based on the mass of the infimum of the measures. Probab. Th. Rel. Fields 75, 143–147 (1987). https://doi.org/10.1007/BF00320087
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DOI: https://doi.org/10.1007/BF00320087
Keywords
- Stochastic Process
- Probability Measure
- Probability Theory
- Total Mass
- Statistical Theory