# Concluding remarks

The analysis of this chapter and the related numerical studies cited in this chapter suggest that the phenomena of non-self-averaging abound in economics, and cannot be ignored. In particular, this chapter suggests that economies may contain both self-averaging and non-self-averaging sectors interacting with each other.

Given this realization, we should not persist in using only self-averaging models in our studies of economic phenomena but amend our attitude and examine in our policy analysis the possibilities of sectors of two or more different types of agent mutually interacting and with non-self-averaging behavior (see also Aoki, 1996, 2000a, 2000b).

We have also established that policy becomes less effective as the value of the CVs falls towards zero.

# Appendix: Mittag-Leffler function and its distribution

A short description of Mittag-Leffler distribution is available in Kotz, Read, Balakrishnan, Vidakovic, eds, 2006. See also Erdelyi (1955).

The Mittag-Leffler function was introduced by G. Mittag-Leffler in (1905) as a generalization of the exponential function. For *t *non-negative, and 0 < *a <* 1, it is given by

This function is generalized to a two-parameter version. H. Pollard (1948), Feller (1970, vol. 2, XIII.1 XIII.8), Blumenfeld and Mandelbrot (1997), and others have contributed to clarifying the properties of this function.

For example, it is shown that *Fa* is the Laplace transform of a probability measure on R_{+}.

It is also known that the Mittag-Leffler density *F _{a}* has the momentgenerating function

Its density function is given by

See also Blumenfeld and Mandelbrot (1997) and Pollard (1948).

The moment condition

uniquely determines the distribution, which is completely monotone. From this condition we see that the integral of *f _{a}* from 0 to infinity is 1, as it should be. Moments uniquely determine the distribution that is completely monotone. The moment-generating function recovers the function

*Fa(t).*