Stability in stochastic language change models

Exemplar models are a popular class of models used to describe language change. Exemplars are detailed memories of stimuli people are exposed to, and when modelling language change are represented as vectors where each component is a phonetic variable. Each exemplar is given a category label, representing what that sound is identified as. New sounds are categorized based on how close they are to the exemplars in each category. Newly categorized exemplars become a part of the system and affect how the future sounds are produced and perceived. It is possible in certain situations in language for a category of sound to become extinct, such as a pronunciation of a word. One of the successes of exemplar models has been to model extinction of sound categories. The focus of this dissertation will be to determine whether categories become extinct in certain exemplar models and why. The first model we look at is an exemplar model which is an altered version of a k-means clustering algorithm by MacQueen. It models how the category regions in phonetic space vary over time among a population of language users. For this particular model, we show that the categories of sound will not become extinct: all categories will be maintained in the system for all time. Furthermore, we show that the boundaries between category regions fluctuate and we quantitatively study the fluctuations in a simple instance of the model. The second model we study is a simple exemplar model which can be used to model direct competition between categories of sound. Our aim in investigating this model is to determine how limiting the memory capacity of an individual in exemplar models affects whether categories become extinct. We will prove for this model that all the sound categories but one will always become extinct, whether memory storage is limited or not. Lastly, we create a new model that implements a bias which helps align all the categories in the phonetic space, using the framework of an earlier exemplar model. We make an argument that this exemplar model does not have category extinction.