Arcsine Law of Occupation Time, explained.
The arcsine occupation law describes a limiting distribution for the fraction of time a symmetric random process spends positive, with substantial mass near the extremes.
Why it happens
Batches of one-hundred-step fair walks record positive occupation using a stated return-to-zero convention. A histogram distinguishes the symmetric ensemble average from individual paths’ uneven occupation.
The arcsine occupation law describes a limiting distribution for the fraction of time a symmetric random process spends positive, with substantial mass near the extremes.
Read the result
The average occupation fraction and the distribution of individual occupation fractions answer different questions. Fair increments do not enforce equal time in each realised path.
A worked example
Look beyond a fair walk’s average sign
Generate a batch of twenty one-hundred-step symmetric walks and inspect occupation bins.
Increase the batch size to reduce the roughness of the histogram without changing each walk’s horizon.
Compare the mean near one-half with the number of walks spending below twenty or above eighty percent positive.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Batches of one-hundred-step fair walks record positive occupation using a stated return-to-zero convention. A histogram distinguishes the symmetric ensemble average from individual paths’ uneven occupation.
Where this idea is useful
A practical use
Avoid assuming a fair process must divide each realised finite history evenly between its two sides.
A common misconception
“Fair random walks spend about half of every finite path above zero.”
Symmetry governs ensemble averages, while individual histories can remain on one side for much of the horizon.
What this explanation leaves out
- The finite discrete histogram is not claimed to equal the continuous limiting density. Sampling noise, step count and zero conventions affect the displayed distribution.
Is the plotted histogram the exact continuous arcsine density?
No. It records finite discrete walks under a declared zero convention; the continuous law is a limiting reference.
Have you mistaken a symmetric average across histories for a typical pattern within each history?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Explore the original research or the teaching reference behind this experiment.