Can Markov chains be continuous?
Can Markov chains be continuous?
A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a stochastic matrix. …
Can a Markov chain have infinite States?
Markov chains with a countably infinite state space exhibit some types of behavior not possible for chains with a finite state space. Figure 5.1 helps explain how these new types of behavior arise.
Can Markov chains be independent?
A series of independent events (for example, a series of coin flips) satisfies the formal definition of a Markov chain. However, the theory is usually applied only when the probability distribution of the next step depends non-trivially on the current state.
What is irreducible Markov chain?
A Markov chain in which every state can be reached from every other state is called an irreducible Markov chain. If a Markov chain is not irreducible, but absorbable, the sequences of microscopic states may be trapped into some independent closed states and never escape from such undesirable states.
How do you find the limiting distribution of a Markov chain?
How do we find the limiting distribution? The trick is to find a stationary distribution. Here is the idea: If π=[π1,π2,⋯] is a limiting distribution for a Markov chain, then we have π=limn→∞π(n)=limn→∞[π(0)Pn]. Similarly, we can write π=limn→∞π(n+1)=limn→∞[π(0)Pn+1]=limn→∞[π(0)PnP]=[limn→∞π(0)Pn]P=πP.
What makes something a Markov chain?
A Markov chain is a mathematical system that experiences transitions from one state to another according to certain probabilistic rules. The defining characteristic of a Markov chain is that no matter how the process arrived at its present state, the possible future states are fixed.
What is an absorbing state in Markov chain?
In the mathematical theory of probability, an absorbing Markov chain is a Markov chain in which every state can reach an absorbing state. An absorbing state is a state that, once entered, cannot be left. Like general Markov chains, there can be continuous-time absorbing Markov chains with an infinite state space.