# Towards a general theory for non-linear locally stationary processes

## TL;DR

This paper develops a comprehensive theoretical framework for non-linear locally stationary processes, establishing fundamental probabilistic results and applying them to nonlinear Markov models and parameter estimation.

## Contribution

It introduces a general theory based on stationary approximation and derivatives, extending to nonlinear non-stationary Markov models and asymptotic analysis of maximum likelihood estimates.

## Key findings

- Laws of large numbers and central limit theorems are proved for these processes.
- Bias expansions are derived for deterministic and stochastic cases.
- Asymptotic properties of MLE in nonlinear non-stationary models are established.

## Abstract

In this paper some general theory is presented for locally stationary processes based on the stationary approximation and the stationary derivative. Laws of large numbers, central limit theorems as well as deterministic and stochastic bias expansions are proved for processes obeying an expansion in terms of the stationary approximation and derivative. In addition it is shown that this applies to some general nonlinear non-stationary Markov-models. In addition the results are applied to derive the asymptotic properties of maximum likelihood estimates of parameter curves in such models.