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 . 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.
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.
"On discrete-time self-similar processes with stationary increments." Electron. J. Probab. 26 1 - 24, 2021. https://doi.org/10.1214/21-EJP689