Asymptotics for relative frequency when population is driven by arbitrary evolution

TL;DR

This paper establishes strong consistency and asymptotic properties for relative frequency estimates of a time-varying probability in a nonstationary dichotomous urn model, using advanced probabilistic theorems.

Contribution

It introduces a novel approach to estimate and analyze the probability of an event in a nonstationary setting with arbitrary evolution, extending classical results to time-dependent functions.

Findings

Proves strong consistency of relative frequency estimates in nonstationary contexts.

Develops a Riemann-Dini type theorem for SLLN applicable to nonstationary processes.

Provides a method for estimating the mean function of unknown nonstationary processes.

Abstract

Strongly consistent estimates are shown, via relative frequency, for the probability of "white balls" inside a dichotomous urn when such a probability is an arbitrary continuous time dependent function over a bounded time interval. The asymptotic behaviour of relative frequency is studied in a nonstationary context using a Riemann-Dini type theorem for SLLN of random variables with arbitrarily different expectations; furthermore the theoretical results concerning the SLLN can be applied for estimating the mean function of unknown form of a general nonstationary process.