Abstract
We state a number of daring conjectures concerning the boundedness of random series of functions with non-negative summands are the special setting of selector processes.
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Notes
- 1.
In other words, we could not think of any other way to bound δ +(T).
- 2.
- 3.
It would be an astonishing fact if it were true that S(T) ≤ Lδ +(T), and proving it would be a sensational result.
- 4.
We do not reproduce this proof here because it uses the rather complicated Theorem 11.1 of [131], and we hope that a creative reader will invent a better argument.
- 5.
Far more so than Problem 13.2.3 itself
References
Talagrand, M.: Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. I.H.E.S. 81, 73–205 (1995)
Talagrand, M.: Selector processes on classes of sets. Probab. Theory Relat. Fields 135(4), 471–486 (2006)
Talagrand, M.: Are many small sets explicitly small? In: STOC’10–Proceedings of the 2010 ACM International Symposium on Theory of Computing, pp. 13–35. ACM, New York (2010). http://research.microsoft.com/apps/video/dl.aspx?id=137091
Talagrand, M.: Upper and Lower Bounds for Stochastic Processes, 1st edn. Springer, Berlin (2014)
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Talagrand, M. (2021). Unfulfilled Dreams. In: Upper and Lower Bounds for Stochastic Processes. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 60. Springer, Cham. https://doi.org/10.1007/978-3-030-82595-9_13
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