How can you tell if a Markov chain is ergodic?

Defn: A Markov chain with finite state space is regular if some power of its transition matrix has only positive entries. P(going from x to y in n steps) > 0, so a regular chain is ergodic.

What makes a Markov chain ergodic?

A Markov chain is said to be ergodic if there exists a positive integer such that for all pairs of states in the Markov chain, if it is started at time 0 in state then for all , the probability of being in state at time is greater than .

What is Markov chain analysis give the properties of Markov process?

A Markov chain is a Markov process with discrete time and discrete state space. So, a Markov chain is a discrete sequence of states, each drawn from a discrete state space (finite or not), and that follows the Markov property.

What do you mean by Markov chains give any 2 examples?

The term Markov chain refers to any system in which there are a certain number of states and given probabilities that the system changes from any state to another state. The probabilities for our system might be: If it rains today (R), then there is a 40% chance it will rain tomorrow, and 60% chance of no rain.

Is a Markov chain ergodic?

A Markov chain is called an ergodic chain if it is possible to go from every state to every state (not necessarily in one move). In many books, ergodic Markov chains are called . A Markov chain is called a chain if some power of the transition matrix has only positive elements.

What is a transition probability?

The one-step transition probability is the probability of transitioning from one state to another in a single step. The Markov chain is said to be time homogeneous if the transition probabilities from one state to another are independent of time index .

How do you define a Markov chain?

A Markov chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC).

What is the difference between Markov chain and Markov process?

A Markov chain is a discrete-time process for which the future behaviour, given the past and the present, only depends on the present and not on the past. A Markov process is the continuous-time version of a Markov chain. Many queueing models are in fact Markov processes.

What are Markov chains used for?

They are stochastic processes for which the description of the present state fully captures all the information that could influence the future evolution of the process. Predicting traffic flows, communications networks, genetic issues, and queues are examples where Markov chains can be used to model performance.

Where is Markov chains used?

Predicting traffic flows, communications networks, genetic issues, and queues are examples where Markov chains can be used to model performance. Devising a physical model for these chaotic systems would be impossibly complicated but doing so using Markov chains is quite simple.

What is Markov chain example?

Definition: The state of a Markov chain at time t is the value of Xt. For example, if Xt = 6, we say the process is in state 6 at time t. Definition: The state space of a Markov chain, S, is the set of values that each Xt can take. For example, S = {1,2,3,4,5,6,7}.

When do you call a Markov chain an ergodic chain?

A second important kind of Markov chain we shall study in detail is an ergodic Markov chain, defined as follows. A Markov chain is called an ergodic chain if it is possible to go from every state to every state (not necessarily in one move). In many books, ergodic Markov chains are called .

When does an irreducible Markov chain have a stationary distribution?

An irreducible Markov chain has a stationary distribution if and only if the Markov chain is ergodic. If the Markov chain is ergodic, the stationary distribution is unique. Many probabilities and expected values can be calculated for ergodic Markov chains by modeling them as absorbing Markov chains with one absorbing state.

Is the random walk with reflection an ergodic chain?

q = 1 – p q =1−p, the random walk with reflection is an ergodic Markov chain A Markov chain that is aperiodic and positive recurrent is known as ergodic. Ergodic Markov chains are, in some senses, the processes with the “nicest” behavior. An ergodic Markov chain is an aperiodic Markov chain, all states of which are positive recurrent.

Which is the nicest behavior of a Markov chain?

Ergodic Markov chains are, in some senses, the processes with the “nicest” behavior. An ergodic Markov chain is an aperiodic Markov chain, all states of which are positive recurrent. An irreducible Markov chain has a stationary distribution if and only if the Markov chain is ergodic.