Markov chains: period of a state

In summary, The period of a state in a Markov chain refers to the number of steps it takes for the state to return to itself. In the example given, the gcd (greatest common divisor) of the number of steps it takes to return is one, indicating that the state is aperiodic. This means the state does not have a specific repeating pattern and can occur at any point in the chain. However, if the gcd is a larger number, such as 2, then the state has a periodicity of 2, meaning it will only occur at every 2nd, 4th, 6th step, and so on. This concept is further explained in various sources, including Wikipedia and a post on
  • #1
mmarkov
1
0
Hello,

I am trying to understand the intuition of the definition of the period of a state in a Markov chain.

Say for example we can go from state i to state i in 3,5,7,9,11... and so on steps.
The gcd here is one. So is this aperiodic state or one with periodicity of 2.

Thanks
 
Physics news on Phys.org
  • #2
Last edited:

What is a Markov chain?

A Markov chain is a mathematical model used to describe the movement of a system between different states. It is based on the principle of "memorylessness", meaning that the probability of transitioning from one state to another depends only on the current state and not on any previous states.

What is the period of a state in a Markov chain?

The period of a state in a Markov chain is the greatest common divisor (GCD) of all possible numbers of steps it takes to return to the state from itself. In other words, it is the number of steps it takes for the state to repeat itself.

How is the period of a state calculated?

The period of a state is calculated by finding the GCD of all possible numbers of steps it takes to return to the state from itself. This can be done by listing out all possible paths from the state back to itself and finding the shortest path length, which represents the period.

What does a period of 1 mean in a Markov chain?

A period of 1 in a Markov chain means that the state is aperiodic, meaning that it does not follow a specific pattern or cycle. This indicates that the state has an equal chance of being visited at any point in time.

Can a state have a period of 0 in a Markov chain?

No, a state cannot have a period of 0 in a Markov chain. This is because a state with a period of 0 would imply that it is impossible to return to the state from itself, which contradicts the definition of a Markov chain where all states must be reachable from each other.

Similar threads

MHB "Two step" Markov chain is also a Markov chain.
    • Set Theory, Logic, Probability, Statistics
Replies
4
Views
2K
A Martingale, calculation of probability - Markov chain needed?
    • Set Theory, Logic, Probability, Statistics
Replies
3
Views
1K
MHB Markov Chain - Is state 2 periodic?
    • Set Theory, Logic, Probability, Statistics
Replies
4
Views
2K
A Markov Chain as a function of dimensions
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
Markov chains: period of a state
    • Set Theory, Logic, Probability, Statistics
Replies
12
Views
16K
General first-order markov chain of 2 states
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
2K
Computer Communication Theory & Information Theory - A name change?
    • Electrical Engineering
Replies
3
Views
259
Simulating a discrete time markov chain
    • Set Theory, Logic, Probability, Statistics
Replies
4
Views
5K
I A seemingly simple problem about probability
    • Set Theory, Logic, Probability, Statistics
Replies
13
Views
1K
Conditional Probability - Markov chain
    • Set Theory, Logic, Probability, Statistics
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top