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 [2] 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 [3].

References

[1]	W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971)
[2]	F. Spitzer, "Principles of random walk", Springer (1976)
[3]	B.A. Rogozin, "The distribution of the first ladder moment and height and fluctuation of a random walk" Theory Probab. Appl., 16 : 4 (1971) pp. 575–595 Teor. Veroyatnost. i Primenen., 16 : 4 (1971) pp. 593–613

Comments

The function $ L $ in the article above is called a slowly varying function, cf. [1], p. 269.