Sensitivity of Conditions for Lumping Finite Markov Chains

摘要

Markov chains with large transition probability matrices occur in many applications such as manpower models. Under certain conditions the state space of a stationary discrete parameter finite Markov chain may be partitioned into subsets, each of which may be treated as a single state of a smaller chain that retains the Markov property. Such a chain is said to be lumpable and the resulting lumped chain is a special case of more general functions of Markov chains. There are several reasons why one might wish to lump. First, there may be analytical benefits, including relative simplicity of the reduced model and the development of a new model which inherits known or assumed strong properties of the original model (the Markov property). Second, there may be statistical benefits, such as increased robustness of the smaller chain as well as improved estimates of transition probabilities. Finally, the identification of lumps may provide new insights about the process under investigation. However, a problem arises in connection with practical applications of Markov chain models is to determine whether the chain is lumpable. This thesis examines the sensitivity of lumping conditions.