Let L be a linear space of real bounded random variables on the probability space (Ω,,P0) . A finitely additive probability P on such that P∼P0 and EP(X)=0 for each X∈L is called EMFA (equivalent martingale finitely additive probability). In this note, EMFA’s are investigated in case P 0 is atomic. Existence of EMFA’s is characterized and various examples are given. Given y ∈ ℝ and a bounded random variable Y, it is also shown that Xn+y⟶a.s.Y , for some sequence (X n ) ⊂ L, provided EMFA’s exist and E P (Y) = y for each EMFA P.
Finitely Additive FTAP under an Atomic Reference Measure / Berti, Patrizia; Pratelli, Luca; Rigo, Pietro. - STAMPA. - 14th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2012, Catania, Italy, July 9-13, 2012, Proceedings, Part IV:(2012), pp. 114-123. [10.1007/978-3-642-31724-8_13]
Finitely Additive FTAP under an Atomic Reference Measure
BERTI, Patrizia;
2012
Abstract
Let L be a linear space of real bounded random variables on the probability space (Ω,,P0) . A finitely additive probability P on such that P∼P0 and EP(X)=0 for each X∈L is called EMFA (equivalent martingale finitely additive probability). In this note, EMFA’s are investigated in case P 0 is atomic. Existence of EMFA’s is characterized and various examples are given. Given y ∈ ℝ and a bounded random variable Y, it is also shown that Xn+y⟶a.s.Y , for some sequence (X n ) ⊂ L, provided EMFA’s exist and E P (Y) = y for each EMFA P.
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