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 Paul Lévy (1939).
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
![{\displaystyle \Pr \left[X\leq x\right]={\frac {2}{\pi }}\arcsin \left({\sqrt {x}}\right),\qquad \forall x\in [0,1].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a41666490f133970aabfcb0c625b69312641126)
Statement of the laws edit
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 edit
The first arcsine law states that the proportion of time that the one-dimensional Wiener process is positive follows an arcsine distribution. Let
![{\displaystyle T_{+}=\left|\{\,t\in [0,1]\,\colon \,W_{t}>0\,\}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72770db491b42a0dc18d13b5b2bb88f1bbb88aa3)
be the measure of the set of times in [0,1] at which the Wiener process is positive. Then is arcsine distributed.
Second arcsine law edit
The second arcsine law describes the distribution of the last time the Wiener process changes sign. Let
![{\displaystyle L=\sup \left\{t\in [0,1]\,\colon \,W_{t}=0\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d2a610d1008a43cd635a25214d0150cacf3bcf6)
be the time of the last zero. Then L is arcsine distributed.
Third arcsine law edit
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
![{\displaystyle W_{M}=\sup\{W_{s}\,\colon \,s\in [0,1]\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62f82c59efa23a4a5e021b1a023a617214ee5451)
Then M is arcsine distributed.
Equivalence of the second and third laws edit
Defining the running maximum process Mt of the Wiener process
![{\displaystyle M_{t}=\sup\{W_{s}\,\colon \,s\in [0,t]\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/184d233f636a0392105747cbc6c3bbf6a0f4f941)
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).[1]
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.[1] It follows that the second and third laws are equivalent.
Notes edit
References edit
- Lévy, Paul (1939), "Sur certains processus stochastiques homogènes", Compositio Mathematica, 7: 283–339, ISSN 0010-437X, MR 0000919
- Morters, Peter & Peres, Yuval (2010). Brownian motion. Vol. 30. Cambridge University Press.
- Rogozin, B. A. (2001) [1994], "Arcsine law", Encyclopedia of Mathematics, EMS Press