Limit theorems for runs containing two types of contaminations. Paper with detailed proofs

TL;DR

This paper derives limit theorems for the distribution of runs with at most one failure of each of two types in sequences of trials with three outcomes, supported by detailed proofs and simulations.

Contribution

It introduces new limit theorems for the distribution of contaminated runs with two failure types, including detailed proofs and simulation validation.

Findings

Limiting distribution of first hitting time derived

Distribution of longest contaminated run characterized

Simulation results support theoretical theorems

Abstract

In this paper, sequences of trials having three outcomes are studied. The outcomes are labelled as success, failure of type I and failure of type II. A run is called at most 1+1 contaminated, if it contains at most 1 failure of type I and at most 1 failure of type II. The limiting distribution of the first hitting time and the accompanying distribution for the length of the longest at most 1+1 contaminated run are obtained. This paper contains the detailed mathematical proofs. Simulation results supporting the theorems are also presented.