Arcsine Last-Zero Random Walk
Discover why a fair random walk often makes its last return near an endpoint.
Free theory articleDiscover why a fair random walk often makes its last return near an endpoint.
Concept & interpretation
In a simple symmetric random walk of length 2n, the probability distribution of the time of the last return to zero follows a discrete arcsine distribution. Instead of clustering around the mean as typical random variables do, the last return to zero is most likely to occur near the endpoints (near 0 or near 2n).
Experiment
You can interact with the Arcsine Last-Zero Random Walk module to see this U-shaped distribution in action.
- Select the Half-length to set the total duration of the random walk (length is 2n).
- Select the Last-zero half-index to choose the specific return time 2k you want to calculate.
- Observe the computed probabilities and notice how the mass is concentrated at the edges (k=0 and k=n) and reaches a minimum in the middle.