Stable Law and Self-similarity

整理一下兩者之間的關係：

Def. Self-similar: A process X(t) is said to be self-similar if
 
從 chf 來看， self-similarity 可以表示為：

Def. Stable under addition (denoted by +stable): A random variable X has distribution that is stable under addition if

where {X_i} are iid copies of X.

Note. +stable implies infinitely divisible because
Note2. In chf, +stable is equivalent to for all n

Def. Stable: A random variable X is stable if

 

Note. Self-similar implies stable without the shift term g. It is called strickly stable. We can also define strickly +stable analogously.

Thm.
H. X is +stable
C.   

Note. This is to say +stable = stable, because stable is +stable by definition. Therefore.

 

Hence, for Levy processes, we define self-similarity as

By the theorem above, we know immediately that Levy process is self-similar iff it is stable.