# Arcsine law

A limit theorem describing the fluctuations of a random walk on the real line, which results in an arcsine distribution or a generalized arcsine distribution. The following feature of a Brownian motion $\{ {\xi _ {t} } : {t \geq 0, \xi _ {0} =0 } \}$ was noted in 1939 by P. Lévy. Let $\tau _ {t}$ be the Lebesgue measure of the set $\{ {u } : {\xi _ {u} > 0, 0 \leq u \leq t } \}$ or, in other words, the time spent by a Brownian particle on the positive semi-axis during the interval of time $[0, t]$. The ratio $\tau _ {t} / t$ will then have the arcsine distribution:

$${\mathsf P} \left \{ \frac{\tau _ {t} }{t} < x \right \} = \ F _ {1/2} (x) = \ \frac{2} \pi \mathop{\rm arcsin} \ \sqrt {x } ,\ 0 \leq x \leq 1,\ t > 0.$$

It was subsequently noted  that a random walk with discrete time obeys the following arcsine law: Let $S _ {0} = 0 \dots S _ {n} \dots$ be the successive locations in the random walk,

$$S _ {n} = \sum _ {u = 1 } ^ { n } \xi _ {u} ,$$

where $\xi _ {1} , \xi _ {2} \dots$ are independent and identically distributed, let $T _ {n}$ be equal to the number of indices $k$ among $0 \dots n$ for which $S _ {k} > 0$, and let

$$K _ {n} = \mathop{\rm min} \left \{ {k } : {S _ {k} = \max _ {0 \leq m \leq n } S _ {m} } \right \} ,$$

then the relationships

$$\lim\limits _ {n \rightarrow \infty } {\mathsf P} \left \{ \frac{T _ {n} }{n} < x \right \} = \ \lim\limits _ {n \rightarrow \infty } {\mathsf P} \left \{ \frac{K _ {n} }{n} < x \right \} = F _ {a} (x) ,$$

$$\lim\limits _ {n \rightarrow \infty } \frac{ {\mathsf P} \{ S _ {1} < 0 \} + \dots + {\mathsf P} \{ S _ {n} < 0 \} }{n} = \alpha$$

are all satisfied or not satisfied at the same time; here, ${F _ \alpha } (x)$ for $0 < \alpha < 1$ is the generalized arcsine distribution,

$$F _ {1} (x) = {\mathsf E} (x) \ \textrm{ and } \ F _ {0} (x) = \ {\mathsf E} ( x - 1 ) ,$$

where ${\mathsf E} (x) = 0$ if $x \leq 0$ and ${\mathsf E} (x) = 1$ if $x > 0$.

The arcsine law in renewal theory states that for $0 < \alpha < 1$ the following equalities are valid:

$${\mathsf P} \{ \xi _ {1} \geq 0 \} = 1$$

and for

$$y _ {t} = t - S _ {\eta _ {t} } ,$$

where $\eta _ {t}$ is defined by the relation ${S _ {\eta _ {t} } } < t \leq S _ {\eta _ {t + 1 } }$,

$$\lim\limits {\mathsf P} \left \{ \frac{y _ {t} }{t} < x \right \} = \ F _ \alpha (x)$$

if and only if

$${\mathsf P} \{ \xi _ {1} > x \} = x ^ {- \alpha } L (x)$$

for $x > 0$, where $L(x)$ is a function which is defined for $x > 0$ and which has the property

$$\lim\limits _ {x \rightarrow \infty } \frac{L (xy) }{L (x) } = 1 \ \ \textrm{ for } \ 0 < y < \infty .$$

There exists a close connection between the arcsine law in renewal theory and the arcsine law governing a random walk .