Full text: Pareto Distribution

5 
towns grows at the rate of and the number of inhabitants 
of the town grows at the rate of ^ then after a 
sufficiently long time there will be a steady state 
distribution with Pareto coefficient yp • 
Mandelbrot (1960,1961) deals with the problem from the 
point of view of a mathematician and therefore on a 
very general level. He starts from the concept of stable 
a 
laws (compare Feller Vol II ch.VI ). If thre sum of 
independent identically distributed random variables 
is distributed in the same way afjits components, except 
for a scale factor and possibly of a location factor, 
then this distribution is stable. The best known example 
is the normal distribution. It has been shown by P.Levy 
that there is a class of distributions with infinite 
variance which are stable and which converge to the law 
of Pareto when the variable in question ( say income ) 
tends to infinity. The Pareto law in this context is 
confined to the range 1 2. Mandelbrot surmises, 
owing to the reproductive quality, in the above sense, 
of the Pareto law, that its importance empirically must 
be very great. He also considers that this must have 
implications for some statistical meljtods which depend 
on the assumption of normalcy. 
As to income Mandelbrot suggests that it can be regarded 
as composed of a number of independent elements which 
are identically distributed. We can easily imagine 
f 
decomposition into a few parts such as earned income, 
property income and transfer income. Mandelbrot requires^, 
however, in order to assure convergence, a large number of 
components, and these, as he admits, have hardly any
	        

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