# Nonparametric estimation of the kernel function of symmetric stable moving average random functions

## TL;DR

This paper introduces a nonparametric estimator for the kernel function of symmetric alpha stable moving average processes, demonstrating its consistency and good finite-sample performance through simulations.

## Contribution

It proposes a novel estimator based on the empirical normalized periodogram for symmetric alpha stable moving average functions, with proven consistency under high-frequency observations.

## Key findings

- Estimator is weakly consistent for positive definite kernels.
- Performs well in finite samples with stable and other infinitely divisible integrators.
- Validated through simulation studies.

## Abstract

We estimate the kernel function of a symmetric alpha stable ($S\alpha S$) moving average random function which is observed on a regular grid of points. The proposed estimator relies on the empirical normalized (smoothed) periodogram. It is shown to be weakly consistent for positive definite kernel functions, when the grid mesh size tends to zero and at the same time the observation horizon tends to infinity (high frequency observations). A simulation study shows that the estimator performs well at finite sample sizes, when the integrator measure of the moving average random function is $S\alpha S$ and for some other infinitely divisible integrators.