the arcsine laws
The arcsine laws are three beautifully counterintuitive results about how Brownian motion spends its time over an interval [0, 1]. Naively one expects a fair coin-flip-like process to spend about half its time positive and to be 'in the lead' near the middle. The truth is the opposite: the time spent positive, the time of the last zero, and the time of the maximum are all most likely to be near the ENDPOINTS 0 or 1, almost never near 1/2.
Each of the three statistics has the same arcsine distribution, with density f(x) = 1 / (pi sqrt(x (1 - x))) on (0, 1) and CDF P(X <= x) = (2/pi) arcsin(sqrt(x)). Specifically: (i) A_1 = integral over [0,1] of 1{B_s > 0} ds, the time spent positive, is arcsine-distributed (Levy). (ii) The time of the last zero, L = sup{ s <= 1 : B_s = 0 }, is arcsine-distributed. (iii) The time theta at which B attains its maximum over [0,1] is arcsine-distributed. The density blows up at 0 and 1 and is minimal at 1/2, so the most probable outcomes are spending nearly all the time positive (or nearly all negative), the last zero occurring very early or very late, and the peak occurring near the start or end. The proofs flow from the reflection principle and excursion structure; the discrete analogue is the ballot/random-walk arcsine theorem of Levy and of Sparre Andersen.
Why it matters: the arcsine laws are the canonical warning that 'fairness' at each step does not give 'balance' over time — a symmetric game produces long one-sided streaks as the rule, not the exception, with deep consequences for how we (mis)judge winning and losing streaks, fund manager track records, and gambler's-ruin intuition. The honest caveat: arcsine is the law of these particular functionals of the WHOLE path; it does not say B_t itself is arcsine (B_t is normal). And the result is special to processes with the strong Markov property and a recurrent zero set of measure zero — it does not hold for a drifted or otherwise asymmetric process.
By the arcsine CDF, the probability that Brownian motion spends LESS than 10 percent of [0,1] positive is (2/pi) arcsin(sqrt(0.1)) approximately 0.205 — more than one chance in five. By symmetry, spending less than 10 percent NEGATIVE is equally likely; staying near a 50/50 split is the rare case.
The arcsine density peaks at the endpoints: lopsided time-on-one-side is typical, balance is rare.
Arcsine is the law of path FUNCTIONALS (time positive, last zero, argmax), not of B_t itself, and it requires the strong Markov property and a recurrent measure-zero zero set; a drift destroys it.