Working Paper 02
The finite horizon: variance dominance, the description–inference distinction, and why published odds describe the process, not the participant
September 2026
Abstract
This paper distinguishes probabilities defined for a stochastic process from outcomes observed over a finite horizon. Simulations cover seven processes at up to 500,000 replications per point and show that the wrong-side outcome rate approaches Φ(−√(n/n*)) when sample size is normalized by the variance horizon n* = (σ/μ)², with documented small-sample lattice deviations. In a 20,000-player cohort, 26% of players finish ahead after 200 bets, while the predicted leading result is about +48 units; a closed result such as 5/10 is exact as description, and uncertainty enters only when it is used to infer a broader process.