In probability theory, the arcsine laws are a collection of results for one-dimensional random walks and Brownian motion (the Wiener process). The best known of these is attributed to .
All three laws relate path properties of the Wiener process to the arcsine distribution. A random variable X on [0,1] is arcsine-distributed if
\Pr\left[X\leqx\right]=
| 2 | |
| \pi |
\arcsin\left(\sqrt{x}\right), \forallx\in[0,1].
Throughout we suppose that (Wt)0 ≤ t ≤ 1 ∈ R is the one-dimensional Wiener process on [0,1]. Scale invariance ensures that the results can be generalised to Wiener processes run for t ∈[0,∞). ===First (Lévy's) arcsine law=== The first arcsine law states that the proportion of time that the one-dimensional Wiener process is positive follows an arcsine distribution. Let : <math> T_+ = \left| \{\, t \in [0,1] \, \colon \, W_t > 0 \,\}\right|
be the measure of the set of times in [0,1] at which the Wiener process is positive. Then
T+
The second arcsine law describes the distribution of the last time the Wiener process changes sign. Let
L=\sup\left\{t\in[0,1]\colonWt=0\right\}
be the time of the last zero. Then L is arcsine distributed.
The third arcsine law states that the time at which a Wiener process achieves its maximum is arcsine distributed.
The statement of the law relies on the fact that the Wiener process has an almost surely unique maxima,[1] and so we can define the random variable M which is the time at which the maxima is achieved. i.e. the unique M such that
WM=\sup\{Ws\colons\in[0,1]\}.
Then M is arcsine distributed.
Defining the running maximum process Mt of the Wiener process
Mt=\sup\{Ws\colons\in[0,t]\},
then the law of Xt = Mt - Wt has the same law as a reflected Wiener process |Bt| (where Bt is a Wiener process independent of Wt).
Since the zeros of B and |B| coincide, the last zero of X has the same distribution as L, the last zero of the Wiener process. The last zero of X occurs exactly when W achieves its maximum. It follows that the second and third laws are equivalent.