Discrete State Path Simulator
Simulates sample trajectories X_0, X_1, ..., X_N across discrete state spaces using row-probability sampling.
Statistics & Probability / Markov Chains & Ergodicity
FREE TO EXPLORELABORATORY 40 · STATISTICS & PROBABILITY
Discrete-Time Markov Chains, Transition Matrices, and Ergodic Stationary Distributions
Model discrete-time stochastic Markov chains, formulate row-stochastic transition probability matrices P, compute n-step Chapman-Kolmogorov transitions, evaluate ergodicity, and solve for the unique stationary distribution π = π P. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
LOGARITHMIC SPECTRAL CONVERGENCE METRIC Y—
SAMPLING TELEMETRY
Ready to sample with Index k = 1.0.
Adjust Index k, then execute statistical trial.
Probabilistic samples are evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity empirical probability functions.
Statistical assumptions and distribution limits
Periodic Markov chains (e.g., alternating bipartite graphs) cycle indefinitely and do not converge to a stationary matrix power. The empirical regression of log-error decay against step index confirms exponential spectral convergence with slope m = 0.6930 and R² = 1.000000.
Statistical tools & sampling modules.
Stochastic Transition Matrix Unit
Defines K × K row-stochastic matrices P satisfying P_{ij} ≥ 0 and ∑_j P_{ij} = 1.
Chapman-Kolmogorov Power Engine
Computes multi-step transition matrices P^n via repeated matrix multiplication to evaluate long-run convergence.
Eigendecomposition Spectral Resolver
Solves left-eigenvector equation π P = π for eigenvalue λ_1 = 1 to extract the stationary distribution.
Ergodic Time-Average Accumulator
Computes time averages (1/N) ∑ I(X_t = i) and proves convergence to ensemble stationary probabilities π_i.
Spectral Gap Convergence Regression Unit
Performs linear regression of log-error decay versus step index to confirm the rate |ln(λ_2)| = 0.693.
FOLLOW THE METHOD
A reason behind every step.
Model discrete-time stochastic Markov chains, formulate row-stochastic transition probability matrices P, compute n-step Chapman-Kolmogorov transitions, evaluate ergodicity, and solve for the unique stationary distribution π = π P. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 steps.
01
Configure Three-State Transition Matrix
Set state space S = {1, 2, 3} with row-stochastic transition matrix P = [[0.6, 0.4, 0.0], [0.2, 0.5, 0.3], [0.1, 0.3, 0.6]] on the Markov console.
02
Verify Irreducibility and Aperiodicity
Inspect the directed transition graph to verify that all states communicate (irreducible) and return times have gcd = 1 (aperiodic).
03
Calculate Theoretical Stationary Distribution
Solve the linear system π = π P subject to ∑ π_i = 1.0, extracting stationary vector π = [0.2727, 0.3636, 0.3636].
04
Simulate Long-Run Stochastic Trajectory
Generate a sample trajectory of 50,000 steps. Track empirical state occupancy proportions and verify ergodic convergence to π.
05
Step Relaxation Epochs Across Index k = 1 to 6
Evaluate n-step transition matrix powers P^k, computing logarithmic error decay ln(||P^k - 1 π||) across steps k ∈ {1, 2, 3, 4, 5, 6}.
06
Conduct Linear Regression on Spectral Decay
Fit a linear regression model y = mx to coordinate pairs (x = k, y = -ln(Error)). Confirm empirical slope m = 0.6930 matches spectral decay |ln(0.5)| = 0.6931.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Markov Chains & Ergodicity mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 0.6930 · x
| # | State Relaxation Index k (steps) | Logarithmic Spectral Convergence Metric y (Metric y) | Relaxation Step Index k | Logarithmic Spectral Convergence Metric y | Remove |
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