Almost sure central limit theorems for random ratios and applications to LSE for fractional Ornstein-Uhlenbeck processes

TL;DR

This paper establishes an almost sure limit theorem for ratios of random variables, with applications to the almost sure convergence of least squares estimators in fractional Ornstein-Uhlenbeck processes.

Contribution

It introduces a new almost sure limit theorem for ratios where the numerator satisfies ASCLT and the denominator converges to 1, applying it to fractional Ornstein-Uhlenbeck processes.

Findings

Proves an ASCLT for ratios of random variables.

Demonstrates ASCLT for least squares estimators in fractional Ornstein-Uhlenbeck processes.

Provides theoretical foundation for statistical inference in fractional stochastic models.

Abstract

We investigate an almost sure limit theorem (ASCLT) for sequences of random variables having the form of a ratio of two terms such that the numerator satisfies the ASCLT and the denominator is a positive term which converges almost surely to 1. This result leads to the ASCLT for least square estimators for Ornstein-Uhlenbeck process driven by fractional Brownian motion.