GDP is non-self-averaging
Let Yn denote the output of a model with Kn cluster discussed below.
Suppose that a cluster of size ni produces yi , and the total output from Kn 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 Yn/na is the same as that of the random variable L.
Proposition: GDP normalized by na 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 Kn as We approximate y by
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