Zipf-Pareto Law
Pareto observed in 1897 that the number of individuals whose income exceeded the level x was proportional to 1/xa where a is some constant. Pareto observed this phenomenon for data obtained from a number of countries, and concluded that he had found a mathematical law of economics. The phenomenon observed by Zipf illustrated in the figure
was that if the observations (in this case city sizes) are ordered from largest to smallest then their sizes decrease at a rate proportional to 1/nb where b is a constant and this is the size of n th observation from the largest. Zipf's Law and Pareto's observation are intimately related. You can test the hypothesis that data is consistent with Zipf's law.
Convergence Rates for Renewal Sequences
A counter meant to register successes is locked for exactly r-1 epochs following a registration so the occurrence of a success at epoch n would be registered if no registration of one occurred in the preceding r-1 epochs. Upon a registration of a success at epoch n the mechanism would then be locked for epochs n+1 through n+r-1, so that a success could not be registered during any of these. A success would be registered at the n+r th epoch if it occurred since the counter would then be free. Let wn be the probability the counter is free (will register a success if one occurs ) at epoch n. If the probability of success p lies in (0,1) then there is, for each M , 0 < M < 1, greater than the reciprocal of the smallest magnitude of those roots of 1- ( q z + p zr) = 0 outside the unit circle in the complex plane, a finite K such that for all n
while for for every M less than that reciprocal there is no choice of real K for which this holds.
This is but one example of the application of the results on convergence rates of renewal sequences
Minimal-Moment Generating Functions
A characteristic function determines, through an integration formula, the probability distribution
of the associated random variable
so there is a one to one relationship between characteristic functions and probability distributions.
A remarkable result called the continuity theorem says that if the
characteristic functions of a sequence of random variables converge to a function continuous at 0,then that function must be a
characteristic function and furthermore, the sequence of
associated random variables converges weakly to its random variable.
By the time of W. Hoeffding in the early 1950's it was known that in analogy with the
usual moments (expectations of powers) of a random variable,
the maximal moments (expectations of the maxima of independent collections of random variables)
determined the probability distribution. Minimal moments are analogous. Here
a minimal moment
generating function is defined and shown to have properties analogous to those of the characteristic
function.
Refinement Equations
There is a non-trivial integrable solution to a two scale difference equation with non-negative
coefficients involving f if and only if the corresponding random expansion is absolutely continuous. Utilizing this
characterization a simple description in terms of the coefficients is given of all refinement equations
with no more than six coefficients possessing an integrable solution.
The functional equation (D) is variously known as a dilation equation, a refinement
equation,
or a two scale difference equation. Its solutions are important in wavelets and in subdivision schemes.
Our interest is simply in the existence of a solution to (D) without regard to its potential use. We show that there
is a solution to (D) if and only if the measure describing Y, whose cumulative
distribution function F is absolutely continuous with respect to Lebesgue measure, and in that case, f can
be taken as the probability density of Y. The proof of the main theorem involves a
forest of trees related
to points in the complex plane.
