# Gaussian self-similar random fields with distinct stationary properties of their rectangular increments

## TL;DR

This paper introduces two classes of Gaussian self-similar random fields with different stationary properties of their rectangular increments, providing explicit representations and characterizations, including new spectral forms for fractional Brownian motion.

## Contribution

It defines and characterizes two distinct classes of Gaussian self-similar fields with stationary rectangular increments, offering explicit spectral and moving average representations.

## Key findings

- Both classes include fractional Brownian sheets.
- Explicit spectral and moving average representations are derived.
- A new spectral representation for fractional Brownian motion is obtained.

## Abstract

We describe two classes of Gaussian self-similar random fields: with strictly stationary rectangular increments and with mild stationary rectangular increments. We find explicit spectral and moving average representations for the fields with strictly stationary rectangular increments and characterize fields with mild stationary rectangular increments by the properties of covariance functions of their Lamperti transformations as well as in terms of their spectral densities. We establish that both classes contain not only fractional Brownian sheets and we provide corresponding examples. As a by-product, we obtain a new spectral representation for the fractional Brownian motion.