Calculator Input
Enter a square transition matrix for a finite-state Markov chain. Use commas or spaces between values, and a new line for each row.
Example Data Table
Use this example if you want to test the calculator quickly.
| State | To A | To B | To C | Initial Probability |
|---|---|---|---|---|
| A | 0.60 | 0.30 | 0.10 | 0.50 |
| B | 0.20 | 0.50 | 0.30 | 0.30 |
| C | 0.25 | 0.25 | 0.50 | 0.20 |
Formula Used
μₙ = μ₀ Pⁿ μₙ₊₁ = μₙ P π = πP, with Σπᵢ = 1 Convergence test: max |μₙ₊₁(i) − μₙ(i)| < toleranceHere, P is the transition matrix, μ₀ is the starting distribution, μₙ is the distribution after n steps, and π is the stationary or limiting distribution when convergence occurs.
The calculator applies repeated vector-matrix multiplication. It also builds Pⁿ, checks how stable the probabilities become, and reports a stationary residual to measure how close the final vector is to satisfying π = πP.
How to Use This Calculator
- Choose the number of states in your Markov chain.
- Enter state labels, or leave them blank for automatic labels.
- Paste the transition matrix, one row per line.
- Enter an initial probability distribution, or leave it blank for a uniform start.
- Set the tolerance, maximum iterations, and display frequency.
- Enable auto-normalization if your rows or initial vector do not sum to one exactly.
- Press the calculate button.
- Review the limiting distribution, power matrix, iteration history, and convergence chart.
Frequently Asked Questions
1) What is a limiting distribution?
A limiting distribution is the long-run probability pattern approached by a Markov chain after many transitions. When it exists, the probabilities stabilize and stop changing meaningfully from one iteration to the next.
2) Will every transition matrix converge?
No. Some chains are periodic or reducible, so the distribution may oscillate or depend on the starting state. This calculator reports whether stabilization was observed under your chosen tolerance and iteration limit.
3) Why must each row sum to one?
Each row represents transition probabilities leaving a state. Because one of the possible next states must occur, the row total must equal one. Otherwise, the matrix is not a valid stochastic transition matrix.
4) What does the initial distribution mean?
The initial distribution gives the probability of starting in each state before any transition occurs. It is your step-zero probability vector and should also sum to one.
5) What does the residual show?
The residual measures how closely the final vector satisfies the stationary equation π = πP. Smaller values indicate that the reported vector behaves more like a true stationary or limiting distribution.
6) Why is P^n displayed?
P^n shows n-step transition probabilities between states. It helps you inspect how the chain behaves after many moves and whether the rows are approaching a common long-run pattern.
7) Can the starting distribution affect the result?
For regular chains, different starting distributions usually converge to the same limiting distribution. For non-regular chains, the start can matter, and the long-run result may fail to settle uniquely.
8) Why does the chart matter?
The chart makes convergence easier to judge visually. Flat, stable lines suggest limiting behavior, while repeating waves or persistent movement can signal slow mixing, periodicity, or non-convergence.