# Non-parametric estimation of time varying AR(1)--processes with local stationarity and periodicity

## TL;DR

This paper develops a kernel-based non-parametric method for estimating time-varying AR(1) processes with local stationarity and periodicity, providing theoretical guarantees and minimax rates under mild conditions.

## Contribution

It introduces a novel estimation approach for a new class of periodic, locally stationary AR(1) processes with proven asymptotic properties.

## Key findings

- Kernel estimators reach classical minimax rates.
- Establishment of central limit theorems for the estimators.
- Method requires only second-order moments of noise.

## Abstract

Extending the ideas of [7], this paper aims at providing a kernel based non-parametric estimation of a new class of time varying AR(1) processes (Xt), with local stationarity and periodic features (with a known period T), inducing the definition Xt = at(t/nT)X t--1 + $\xi$t for t $\in$ N and with a t+T $\not\equiv$ at. Central limit theorems are established for kernel estima-tors as(u) reaching classical minimax rates and only requiring low order moment conditions of the white noise ($\xi$t)t up to the second order.