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2021 On discrete-time self-similar processes with stationary increments
Yi Shen, Zhenyuan Zhang
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Electron. J. Probab. 26: 1-24 (2021). DOI: 10.1214/21-EJP689


In this paper we study the self-similar processes with stationary increments in a discrete-time setting. Different from the continuous-time case, it is shown that the scaling function of such a process may not take the form of a power function b(a)=aH. More precisely, its scaling function can belong to one of three types, among which one type is degenerate, one type has a continuous-time counterpart, while the other type is new and unique for the discrete-time setting. We then focus on this last type of processes, construct two classes of examples, and prove a special spectral representation result for the processes of this type. We also derive basic properties of discrete-time self-similar processes with stationary increments of different types.

Funding Statement

Yi Shen acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada (RGPIN-2014-04840).


The authors would like to thank Gennady Samorodnitsky, Wanchun Shen, Ruodu Wang and Yimin Xiao for their valuable inputs.


Download Citation

Yi Shen. Zhenyuan Zhang. "On discrete-time self-similar processes with stationary increments." Electron. J. Probab. 26 1 - 24, 2021.


Received: 20 June 2019; Accepted: 19 August 2021; Published: 2021
First available in Project Euclid: 14 September 2021

Digital Object Identifier: 10.1214/21-EJP689

Primary: 60G10 , 60G18

Keywords: discrete-time , Self-similar , Stationary increments


Vol.26 • 2021
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