View source for Invariance, principle of

<!--
i0521901.png
$#A+1 = 27 n = 0
$#C+1 = 27 : ~/encyclopedia/old_files/data/I052/I.0502190 Invariance, principle of
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

Let  $  X _ {1} , X _ {2} \dots $
be independent identically-distributed real-valued random variables with zero expectation and variance  $  \sigma  ^ {2} $; 
consider the random polygonal line

$$ 
Y _ {n} ( t )  =  
\frac{1}{\sigma \sqrt n }

\{ S _ {[ nt ] }  + ( nt - [ nt ] ) X _ {[ nt ] + 1 }  \} ,\ \ 
0 \leq  t \leq  1 ,
$$

where  $  S _ {m} = \sum_{i=1}^ {m} X _ {i} $. 
If  $  f $
is a real-valued continuous function on the space  $  C [ 0 , 1] $
of continuous functions on  $  [ 0 , 1 ] $
with the supremum norm (or only continuous everywhere except on a set of Wiener measure zero), then  $  f ( Y _ {n} ) $
converges in distribution to  $  f ( W ) $, 
where  $  W $
is a Wiener random function. Thus, the limiting distribution for the  $  f ( Y _ {n} ) $
does not depend on any special properties of the  $  X _ {1} , X _ {2} , .  .  . $.

A typical scheme for the use of the invariance principle consists in finding the limiting distribution for the  $  f ( Y _ {n} ) $
by finding the limiting distribution for  $  f ( Y _ {n}  ^  \prime  ) $, 
where  $  Y _ {n}  ^  \prime  $
is a random polygonal line constructed in the same way as  $  Y _ {n} $
from some specially chosen sequence  $  X _ {1}  ^  \prime  , X _ {2}  ^  \prime  , .  .  . $. 
For example, if

$$ 
f ( x )  =  \sup _ {0 \leq  t \leq  1 }  x ( t ) ,
$$

then  $  f $
is continuous on  $  C $, 
and, since

$$ 
f ( Y _ {n} )  =  
\frac{1}{\sigma \sqrt n }
  \max _ {1 \leq  m \leq 
n }  S _ {m} ,
$$

one has that

$$ 

\frac{1}{\sigma \sqrt n }
  \max _ {1 \leq  m \leq  n }  S _ {m}  $$

converges in distribution to  $  \sup _ {t}  W ( t ) $. 
To find the distribution of  $  \sup _ {t}  W ( t ) $, 
the sequence  $  \{ X _ {n}  ^  \prime  \} : $
$  {\mathsf P} \{ X _ {n}  ^  \prime  = 1 \} = {\mathsf P} \{ X _ {n}  ^  \prime  = - 1 \} = 1 / 2 $
is used, and as a result of the calculations one obtains

$$ 
{\mathsf P} \left \{ \sup _ { t }  W ( t ) \leq  a \right \}  = \ 
\sqrt {
\frac{2} \pi 
 } \int\limits _ { 0 } ^ { a }  e ^ {- u  ^ {2} / 2 }  du ,\  a
\geq  0 .
$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Donsker,   "An invariance principle for certain probability limit theorems"  ''Memoirs Amer. Math. Soc.'' , '''6'''  (1951)  pp. 1–12</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.V. Prokhorov,   "Convergence of random processes and limit theorems in probability theory"  ''Theor. Probab. Appl.'' , '''1'''  (1956)  pp. 157–214  ''Teor. Veroyatnost. Prilozhen.'' , '''1''' :  2  (1956)  pp. 177–238</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Billingsley,   "Convergence of probability measures" , Wiley  (1968)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.P. Breiman,   "Probability" , Addison-Wesley  (1968)</TD></TR></table>