Azuma Predictable Bets
Bound adapted betting outcomes without assuming independent increments.
Free theory articleBound adapted betting outcomes without assuming independent increments.
Concept & interpretation
In many financial and machine learning settings, decisions are made sequentially. A bet size or step size is chosen based on past information before the next random event occurs. Azuma's inequality provides a powerful tool to bound the probability that the final wealth deviates significantly from its expected value, even when the increments are not independent, provided they form a martingale difference sequence.
Experiment
In the Azuma Predictable Bets module, you can explore how different predictable policies affect the tail probabilities.
- Set Policy to "Unit stakes", Steps to 6, and Threshold to 2.
- Observe that the terminal variance is 6.0 and the exact upper tail probability is 0.34375.
- Change the Policy to "Stop at two".
- Observe that the terminal variance drops to 4.75 and the exact tail probability changes to 0.453125. The deterministic squared bound budget remains 6, keeping the Azuma bound at ≈ 0.7165.