# A Study of the Allan Variance for Constant-Mean Non-Stationary Processes

## TL;DR

This paper extends the Allan Variance to non-stationary processes, providing a theoretical framework that improves understanding and interpretation of AV in practical, real-world signals with non-stationary characteristics.

## Contribution

It generalizes the Allan Variance to non-stationary processes, enabling more accurate analysis of signals with evolving covariance structures.

## Key findings

- New theoretical form of AV for non-stationary processes
- Simulation results demonstrate improved process differentiation
- Enhanced interpretation of AV in applied signal analysis

## Abstract

The Allan Variance (AV) is a widely used quantity in areas focusing on error measurement as well as in the general analysis of variance for autocorrelated processes in domains such as engineering and, more specifically, metrology. The form of this quantity is widely used to detect noise patterns and indications of stability within signals. However, the properties of this quantity are not known for commonly occurring processes whose covariance structure is non-stationary and, in these cases, an erroneous interpretation of the AV could lead to misleading conclusions. This paper generalizes the theoretical form of the AV to some non-stationary processes while at the same time being valid also for weakly stationary processes. Some simulation examples show how this new form can help to understand the processes for which the AV is able to distinguish these from the stationary cases and hence allow for a better interpretation of this quantity in applied cases.