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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|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 [[#References|[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 [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''2''', Wiley (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> F. Spitzer, "Principles of random walk", Springer (1976)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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</TD></TR></table>

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