# GDP is non-self-averaging

**Let ****Y _{n}**

**denote the output of a model with**

**K**_{n}**cluster discussed below.**

**Suppose that a cluster of size ****n _{i}**

**produces**

**y**_{i}**, and the total output from**

**K**_{n}**clusters is the sum of outputs of all clusters**

**where we assume that**

**We set **1** to 1 without loss of generality.**

**Yamato and Sibuya (2000) show that the coefficient of variation of ****Y _{n}/n^{a}**

**is the same as that of the random variable L.**

**Proposition: GDP normalized by ****n ^{a}**

**is non-self-avaraging in the two- parameter Poisson-Dirichlet process.**

**To see this, define ****aj**** (n) as the number of clusters in which ****j ****innovations have occurred out of ****n**** total innovations. The vector ****a(n) ****consisting of ****aj****(n), ****j =**** 1,2,..., ****n**** is called the ****partition vector,**** where **

Then we can express *K _{n}* as
We approximate y by

and

where *в* = *ln(*7) > 0.

Yamato and Sibuya (2000) have shown that where f is defined by

where the notation [ *j* ] denotes an ascending factorial
Next, normalize the output by *n ^{a}.* Then

Yamato and Sibuya (2000) and Pitman (2002) have shown that

where the convergence is in almost surely sense as well as in distribution, and since 2T and are both constants, that is, not random variables, we conclude that

and