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'' $  \omega  ^ {2} $-
distribution''

The [[Probability distribution|probability distribution]] of the random variable

$$ 
\omega  ^ {2}  =  \int\limits _ { 0 } ^ { 1 }  Z  ^ {2} ( t)  dt,
$$

where  $  Z( t) $
is a conditional [[Wiener process|Wiener process]] (conditioned on  $  Z ( 1) = 0 $). 
The characteristic function of the  "omega-squared"  distribution is expressed by the formula

$$ 
{\mathsf E} e ^ {it \omega  ^ {2} }  =  \prod _ { k= } 1 ^  \infty  
\left ( 1 - 
\frac{2it }{\pi  ^ {2} k  ^ {2} }
 \right )  ^ {-} 1/2 .
$$

In mathematical statistics, the  "omega-squared"  distribution is often found in the following circumstances. Let  $  X _ {1} \dots X _ {n} $
be independent random variables, uniformly distributed on  $  [ 0, 1] $, 
according to which an empirical distribution function  $  F _ {n} ( \cdot ) $
is constructed. In this case, the process

$$ 
Z _ {n} ( t)  =  \sqrt n ( F _ {n} ( t) - t)
$$

converges weakly to a conditional Wiener process, from which it follows that

$$ 
\lim\limits _ {n \rightarrow \infty }  {\mathsf P} \left \{ \int\limits _ { 0 } ^ { 1 }  Z _ {n}  ^ {2} ( t)  dt &lt; \lambda \right \}  = \ 
{\mathsf P} \{ \omega  ^ {2} &lt; \lambda \} =
$$

$$ 
= \ 
1 - 
\frac{2} \pi 
 \sum _ { k= } 1 ^  \infty   (- 1)  ^ {k-} 1 \int\limits _ {( 2k- 1) \pi } ^ { {2k }  \pi } 
\frac{e ^ {- t  ^ {2} \lambda / 2 }
 }{\sqrt {- t  \sin  t } }
  dt,\  \lambda &gt; 0 .
$$

See also [[Cramér–von Mises test|Cramér–von Mises test]].

====References====
<table><tr><td valign="top">[1]</td> <td valign="top">  N.V. Smirnov,   "On the $\omega ^ { 2 }$-distribution"  ''Mat. Sb.'' , '''2'''  (1937)  pp. 973–993  (In Russian)</td></tr><tr><td valign="top">[2]</td> <td valign="top">  T.W. Anderson,   D.A. Darling,   "Asymptotic theory of certain  "goodness of fit"  criteria based on stochastic processes"  ''Ann. Math. Stat.'' , '''23'''  (1952)  pp. 193–212</td></tr></table>

====Comments====
The  "conditional Wiener process"   $  Z $
is usually referred to in the Western literature as tied-down Brownian motion, pinned Brownian motion or as the Brownian bridge.

The pioneering paper is [[#References|[a1]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  D.A. Darling,   "The Cramér–Smirnov test in the parametric case"  ''Ann. Math. Stat.'' , '''26'''  (1955)  pp. 1–20</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J. Durbin,   "Distribution theory for tests based on the sample distribution function" , SIAM  (1973)</td></tr></table>