Doob Random-Walk Maximal Bound
Bound the chance of ever crossing a threshold and distinguish it from the final-time event.
Free theory articleBound the chance of ever crossing a threshold and distinguish it from the final-time event.
Concept & interpretation
A random walk with independent ± 1 increments forms a simple discrete martingale. Doob's maximal inequality bounds the probability that the absolute value of the walk ever exceeds a given threshold, contrasting the event of crossing at any time with merely being above the threshold at the final time.
Exploring Random Walks
Use the interactive controls to visualize and verify Doob's bounds for different path lengths and thresholds.
- Set the steps parameter to explore longer horizons.
- Adjust the threshold to see how the tail bound becomes sharper or trivial (1.0).
- Observe the difference between the "Ever crossing" probability and the "Terminal crossing" probability.
- Note how the expected maximum square remains strictly bounded by 4 times the expected terminal square.