Quasi stationary distributions in Markov Chains with absorbing subchains

For some absorbing chains the time to absorption may be sufficiently long to allow a certain type of stationarity (called quasi-stationary) to set in transient states. This paper derives the quasi-stationary distribution of absorbing Markov Chains with absorbing subchains and studies the different types of unconditional quasi-stationary distributions when the absorbing subchain is not regarded as a single absorbing state, using the method of limiting conditional probabilities. The problem of quasi-stationarity have been motivated by some biological research.