# On discrete-time self-similar processes with stationary increments

## TL;DR

This paper investigates discrete-time self-similar processes with stationary increments, revealing that their scaling functions can differ from the classic power law and identifying a new unique type with specific properties.

## Contribution

It introduces a new type of discrete-time self-similar process with stationary increments, distinct from continuous-time models, and provides spectral representations and properties for this class.

## Key findings

- Scaling functions can be non-power in discrete-time processes
- Identification of a new unique class of processes
- Spectral representation results for the new process type

## Abstract

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 $b(a)=a^H$. 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.