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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

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

 
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