Non-Ergodic Random Walks
Ergodicity breaking in heavy-tailed distributions at .
Abstract
For stochastic processes with heavy-tailed jump distributions (Lévy flights with tail index ), we demonstrate the failure of ergodic convergence. Unlike well-behaved processes where time averages converge to ensemble averages, these systems exhibit persistent non-stationarity: extreme events dominate the trajectory indefinitely, preventing any normalization scheme from achieving stable limiting behavior.
This work analyzes the failure of the Law of Large Numbers for heavy-tailed distributions. We show evidence that no renormalization function can satisfy universality, continuity, and record insensitivity simultaneously. The result has implications for financial modeling, turbulence theory, and any domain where heavy-tailed phenomena arise.
Conjecture A: Absence of Intrinsic Ergodicity
Argument 1: Infinite Total Reset Record Events
Argument 2: Path Surgery & Connectivity Disconnect
Observation 3: Inevitable Oscillation ()
Computational Verification
While the theorem is negative, we computed the theoretical Ensemble Average (the value a multiverse observer would see) versus the Time Average (what a single observer sees).
References
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