Riv. Mat. Univ. Parma, Vol. 12, No. 2, 2021

Azat Miftakhov [a]

Modulus of continuity for a martingale sequence
Pages: 319-326
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.
Abstract: 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.

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