Limit distribution of the least square estimator with observations sampled at random times driven by standard Brownian motion

TL;DR

This paper investigates the asymptotic distribution of the least squares estimator in a regression model with observations sampled at random times driven by Brownian motion, establishing asymptotic normality through theoretical analysis and simulations.

Contribution

It provides the first rigorous proof of the asymptotic normality of the least squares estimator under random sampling times driven by Brownian motion.

Findings

Establishes asymptotic normality of the estimator

Provides simulation results supporting theoretical findings

Analyzes the limit behavior of the characteristic function

Abstract

In this article, we study the limit distribution of the least square estimator, properly normalized, from a regression model in which observations are assumed to be finite ($\alpha N$) and sampled under two different random times. Based on the limit behavior of the characteristic function and convergence result we prove the asymptotic normality for the least square estimator. We present simulation results to illustrate our theoretical results.