Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5772157 | Journal of Functional Analysis | 2017 | 20 Pages |
Abstract
We investigate properties of compositions of conditional expectations on a non-atomic probability space (Ω,F,μ). Let 1â¤p<â and X,YâLp(Ω,F,μ). If for any convex f:RâR we have Ef(X)â¥Ef(Y), then for each ε>0 there exist conditional expectations P1,P2,P3 and E1,â¦,Enâ{P1,P2,P3} such that âYâEnâ¦E1Xâp<ε. This theorem seems particularly surprising if one simplifies the assumption by taking X and Y to have the same distribution since, intuitively, it seems to contradict the second law of thermodynamics. We use this theorem to build the following counterexample for the Amemiya-Andô conjecture: There exist fâL1(R)â©L2(R), conditional expectations P1,â¦,P5 on (R,Borel(R),λ) and E1,E2,â¦â{P1,â¦,P5} such that the sequence of iterations (Enâ¦E2E1f) diverges in L2(R).
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Andrzej Komisarski,