Riv. Mat. Univ. Parma, Vol. 12, No. 2, 2021
Azat Miftakhov [a]
Modulus of continuity for a martingale sequence
Received: 23 March 2021
Accepted in revised form: 27 May 2021
Mathematics Subject Classification: 60G42, 60G17, 60B05.
Keywords: Martingales, Hölder property, subadditivity.
[a]: Moscow State University, Faculty of Mechanics and Mathematics, 1 Leninskiye Gory 119991, Moscow, Russia.
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The author is supported by the RFBR grants 14-01-90406, 14-01-00237 and the SFB 701 at Bielefeld University.
Given a martingale sequence of random fields that satisfies a natural assumption of boundedness,
it is shown that the pointwise limit of this sequence can be modified in such a way that a certain
class of moduli of continuity is preserved. That is, if every element of the sequence admits a given
modulus of continuity, one can construct a modification of the limiting random field so that this new
field also admits the same modulus of continuity. Additionally, it is shown that requiring further
smoothness and a stronger notion of boundedness for the original sequence guarantees further smoothness
of the limiting field and a stronger mode of convergence to this limit. Moreover, the modulus of continuity
is also preserved for the derivatives.
K. L. Chung,
A course in probability theory,
Harcourt, Brace \& World, New York, 1968.
J. Diestel and J. J. Uhl,
The Radon-Nikodym theorem for Banach space valued measures,
Rocky Mountain J. Math. 6 (1976), 1-46.
Martingales in Banach spaces,
Cambridge Stud. Adv. Math., 155,
Cambridge Univ. Press, Cambridge, 2016.
J. J. Uhl,
The Radon-Nikodym theorem and the mean convergence of Banach space valued martingales,
Proc. Amer. Math. Soc. 21 (1969), 139-144.