Kac Positive Return Identity

Relate a state’s stationary weight to its first strictly positive return time. For a finite, irreducible discrete-time Markov chain, Kac's theorem establishes an elegant and fundamental reciprocal relationship between the steady-state probability of being in a state and the expected time it takes to return to that state.

Experiment

You can use the interactive controls to explore how the return times differ between chains and states:

1. Select the Chain control to choose "Asymmetric".
2. Set the State to "0" and verify the mean positive return time is 4.
3. Change the Horizon control to see how the truncated return mean and first-return CDF grow towards their theoretical limits.
4. Try the "Cycle" chain to observe a deterministic return pattern where the return time is always exactly 3, despite the chain being periodic.

Explore the Kac Positive Return Identity experiment.