# Some Developments in Clustering Analysis on Stochastic Processes

## TL;DR

This paper reviews recent advances in clustering stochastic processes, emphasizing conditions for asymptotic consistency such as ergodicity and metric properties, with examples across various process types.

## Contribution

It identifies key conditions for consistent clustering of stochastic processes and provides specific examples including distribution ergodic, covariance ergodic, and locally asymptotically self-similar processes.

## Key findings

- Asymptotically consistent clustering requires ergodicity and a metric dissimilarity measure.
- Examples include distribution ergodic, covariance ergodic, and locally asymptotically self-similar processes.
- Triangle inequality is crucial for the dissimilarity measure.

## Abstract

We review some developments on clustering stochastic processes and come with the conclusion that asymptotically consistent clustering algorithms can be obtained when the processes are ergodic and the dissimilarity measure satisfies the triangle inequality. Examples are provided when the processes are distribution ergodic, covariance ergodic and locally asymptotically self-similar, respectively.