The Cantor distribution is defined as a random serieswhere J is a parameter and the X_{i} are random variables that take the values 0 and 1 with probability 1/2. The moments and order statistics are discussed, as well as a ``Fibonacci'' variation. Connections to certain trees and splitting processes are also mentioned.
1J
J
å i³1 X_{i}J^{i},
X= 


X_{i} J^{i}, 
value(w_{1} w_{2} ···)= 


w_{i} J^{i}. 
a_{n}= 




^{nk} J^{k} a_{k}, a_{0}=1. 
A(z)= 


. 
E[X^{n}]=a_{n}= F(log 

n)n 

æ ç ç è 
1+ O 
æ ç ç è 

ö ÷ ÷ ø 
ö ÷ ÷ ø 
. 
 

ó õ 



e 

x 

dx. 
(2^{n}2J)a_{n}= 

+ J 


J a_{k}. 

(z)= 

A(z)= 


_{n} 

, 

(2z)= J 

(z)+ 

. 
a_{n}= 





, 
a_{n} ~ n 


(  G(log_{2} J) z(log_{2} J) + d(log_{2} n)  )  , 
value(000)=0, value(001)= 

, value(010)= 

, value(100)= 

, value(101)= 

F(z)= 

= 

F_{m+2}z^{m}. 
F_{n}= 

(  a^{n}b^{n}  )  with a= 

and b= 

. 
G_{n}(z):= 

(  value (w)  ) 

z^{w}, 
[z^{m}]G_{n}(z)  
[z^{m}]F(z) 
value(0w)  =J · value(w)  
value(10w) 

G_{n}(z)= 

é ê ê ë 

^{n} z+ z^{2} 



^{ni} J^{2i}G_{i}(z) 
ù ú ú û 
. 
M_{n}= 


= 


. 
M_{n}= 




^{ni} J^{2i}M_{i}. 

(z)= 


(J z)+ 


(J^{2}z). 
M_{n}= 
æ è 
1+ 

ö ø 

F(log 

n) n 

æ ç ç è 
1+O 
æ ç ç è 

ö ÷ ÷ ø 
ö ÷ ÷ ø 
, 
 

ó õ 



(J z) z 

dz. 
This document was translated from L^{A}T_{E}X by H^{E}V^{E}A.