Kac Positive Return Identity

Relate a state’s stationary weight to its first strictly positive return time.

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.

Worked Example

Consider the Asymmetric chain starting at State 0. The stationary probability for State 0 is exactly 0.25. Kac's identity tells us the mean positive return time is simply 1 / 0.25 = 4. Let's calculate the truncated mean up to Horizon 4. First, we find the first-return probabilities (PMF) by starting with all mass at State 0 and removing arrivals to State 0 at each step.

Chain: Asymmetric
Row 0: [0.5,  0.5,  0]
Row 1: [0.25, 0.5,  0.25]
Row 2: [0,    0.5,  0.5]
Stationary distribution: pi = [0.25, 0.5, 0.25]

Starting at State 0 (v = [1, 0, 0]):
t=1: next_v = [0.5, 0.5, 0]. Arrival to 0 is 0.5. Remaining v = [0, 0.5, 0]. P(T>0) = 1.
t=2: next_v = [0.125, 0.25, 0.125]. Arrival to 0 is 0.125. Remaining v = [0, 0.25, 0.125]. P(T>1) = 0.5.
t=3: next_v = [0.0625, 0.1875, 0.09375]. Arrival to 0 is 0.0625. Remaining v = [0, 0.1875, 0.09375]. P(T>2) = 0.375.
t=4: next_v = [0.046875, 0.140625, 0.09375]. Arrival to 0 is 0.046875. Remaining v = [0, 0.140625, 0.09375]. P(T>3) = 0.3125.

First-return CDF up to horizon 4:
0.5 + 0.125 + 0.0625 + 0.046875 = 0.734375

Truncated mean up to horizon 4 (sum of P(T>t) for t=0 to 3):
1 + 0.5 + 0.375 + 0.3125 = 2.1875

Remaining mean (Exact - Truncated):
4.0 - 2.1875 = 1.8125

Experiment

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

Select the Chain control to choose "Asymmetric".
Set the State to "0" and verify the mean positive return time is 4.
Change the Horizon control to see how the truncated return mean and first-return CDF grow towards their theoretical limits.
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.

Limitations

This demonstration is restricted to finite, irreducible, discrete-time Markov chains with exactly 3 states. It calculates exact arithmetic up to the given short horizons. For infinite state spaces or highly complex chains, finding the stationary distribution or computing exact first-passage times requires generalized numerical solvers or simulation. Additionally, continuous-time Markov chains require a different formulation for return times, often relying on exponential holding times.

References

For a formal treatment of Discrete-Time Markov Chains and Kac's Identity, consult:

Lecture 3: Discrete-Time Markov Chains (Part 1)

Kac Positive Return Identity