A positive recurrent CTMC (continuous-time Markov chain) is a stochastic process that evolves over time in a continuous manner, where the transitions between states are governed by a set of transition rates. It is considered positive recurrent if the chain eventually returns to every state with a positive probability, regardless of the initial state.
A null recurrent embedded DTMC (discrete-time Markov chain) is a stochastic process that evolves in discrete time steps, where the transitions between states are governed by a set of transition probabilities. It is considered null recurrent if the chain eventually returns to every state with a probability of zero, regardless of the initial state. This means that it may take an infinite amount of time for the chain to return to a particular state.
CTMCs and DTMCs are related in the sense that a DTMC can be embedded within a CTMC. This means that the DTMC can be seen as a special case of the CTMC, where the continuous-time process is sampled at discrete time intervals. This relationship is useful in analyzing the behavior of complex systems, as it allows us to model both continuous and discrete events.
The main difference between positive and null recurrence is the probability of returning to a state. In a positive recurrent process, the probability of returning to a state is positive, while in a null recurrent process, the probability of returning to a state is zero. This means that for a positive recurrent process, the chain will eventually visit all states with a positive probability, while for a null recurrent process, it may take an infinite amount of time for the chain to visit a particular state.
Understanding positive and null recurrence is important because it allows us to analyze the behavior of stochastic processes over time. It helps us to determine the long-term behavior of a system and whether it will eventually reach a steady state. Additionally, it allows us to make predictions about the future behavior of a system and assess the impact of different parameters on the system's behavior.