Abstract
Let μ be any probability measure onR with λ |x|dμ(x)<∞, and let μ* denote its associated Hardy and Littlewood maximal p.m. It is shown that for any p.m.v for which μ<ν<μ* in the usual stochastic order, there is a martingale (X t)0≦t≦1 for which sup0≦t≦1 X t andX 1 have respective p.m. 'sv and μ. The proof uses induction and weak convergence arguments; in special cases, explicit martingale constructions are given. These results provide a converse to results of Dubins and Gilat [6]; applications are made to give sharp martingale and ‘prophet’ inequalities.
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Supported in part by NSF grants DMS-86-01153 and DMS-88-01818.
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Kertz, R.P., Rösler, U. Martingales with given maxima and terminal distributions. Israel J. Math. 69, 173–192 (1990). https://doi.org/10.1007/BF02937303
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DOI: https://doi.org/10.1007/BF02937303