Arcsine Last-Zero Random Walk

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).

Worked Example

Let's explicitly calculate the probability that the last zero occurs at time 2k for a random walk of length 2n=8 (i.e., half-length n=4), specifically when the last-zero half-index is k=2. The probability is given by the formula:

P(L=2k) = [ binom(2k,k) * binom(2n-2k,n-k) ] / 4^n

Identify the parameters: n = 4, k = 2. The total number of steps is 2n = 8.
Calculate the first binomial coefficient for the path up to the last zero: binom(4, 2) = 6.
Calculate the second binomial coefficient for the path after the last zero (which must not return to zero): binom(4, 2) = 6.
Calculate the denominator, which is the total number of paths of length 8: 4^4 = 256.
Multiply and divide: P(L=4) = (6 * 6) / 256 = 36 / 256 = 0.140625.

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.

Limitations

The theoretical U-shaped arcsine distribution is asymptotic. For finite n, the distribution is discrete and only exists at even times 2k. While the continuous Beta(1/2, 1/2) distribution provides a good approximation for the shape as n grows large, it is a density that diverges to infinity at the endpoints 0 and 1. The discrete model correctly assigns finite probabilities (atoms) to the endpoints. This model specifically describes the last return to zero, not the first passage time or the fraction of time spent positive, which have different but related distributions.

References

This module uses the asymptotic continuous density approach to model random walks.

Random Walks - Provides the mathematical foundation and derivation for the arcsine law for the last visit to the origin.

Arcsine Last-Zero Random Walk