Double Pareto

There is a fundamental link between the lognormal cascade distribution and the Double Pareto distribution. I have not been able to make full sense of it.

Diffusion and confinement

Double Pareto is not only a stable solution of the Kolmogorov diffusion equation. It is also an extreme case of the ground state of a quantum well when the width of the well is zero. A Double Pareto distribution, or a lognormal cascade, therefore implies that the underlying population is in a confined state. That thought leads directly to financial cycles.

Random growth and inequality

An interesting book is The Distribution of Income between Persons by D. G. Champernowne. It pushed me to think outside quantitative finance. The world operates on random growth. Such an economic system is fundamentally volatile and will produce inequality no matter what. Billionaires and political strongmen are products of that random growth system.

This helps explain how the Dow could rise 10 times, from 1,000 to 10,000, in 20 years and then go nowhere for another 20 years. That is part of life. Treasury yields will do the same. Inflation, gold and commodities can behave in the same way.

The fitted evidence is in the data atlas. The same distribution family can be explored in the simulator, and the market level scaling argument is in the N^3/2 law note.