Cascade models

1.1 Borel-Tanner distribution

A delta Lagrangian distribution, the Borel distribution is the distribution of a cascade size starting from a population size of $k=1$. We can generalize it to $k>$, in which case it is the Borel-Tanner distribution.

Spelled

$\mathrm{Borel-Tanner}(k,\eta )$

Pmf

$\mathbb{P}(X=x;k,\eta )=\frac{k}{x}\frac{e^{-\eta x}(\eta x{)}^{x-k}}{(x-k)}$

Mean

$\frac{k}{1-\eta }$

Variance

$\frac{k\eta }{(1-\eta {)}^3}$

Note to self: Wikipedia suggests an intriguing correspondence with random walks, which I should follow up (Pitman 1998; Dwass 1969).

The only R implementation I could find for this variate is in VGAM, although it is not so complicated.

1.2 Poisson-Poisson Lagrangian

See (P. C. Consul and Famoye 2006 Ch 9.3). Also known as the Generalised Poisson, although there are many things called that.

Spelled

$\mathrm{GPD}(\mu ,\eta )$

Pmf

$\mathbb{P}(X=x;\mu ,\eta )=\frac{\mu (\mu +\eta x{)}^{x-1}}{x!e^{\mu +x\eta }}$

Mean

$\frac{\mu }{1-\eta }$

Variance

$\frac{\mu }{(1-\eta {)}^3}$

Returning to the cascade interpretation: Suppose we have

• an initial population is distributed $\mathrm{Poisson}(\mu$)
• and everyone in the population has a number of offspring distributed $\mathrm{Poisson}(\eta$).

Then the total population is distributed as $\mathrm{GPD}(\mu ,\eta )$.

Notice that the Poisson-Poisson can produce long tails, in the sense that it can have a large variance with finite mean, but not heavy tails, in the sense of the variance becoming infinite while retaining a finite mean; both variance and expectation go to infinity together.

Here, I implemented the GPD for you in python. There are versions for R, presumably. A quick search turned up RMKDiscrete and LaplacesDemon.

1.3 General Lagrangian distribution

A larger family of Lagrangian distributions (the largest?) family is summarised in (P. Consul and Shenton 1972), in an unintuitive (for me) way.

One “parameter”: a differentiable (infinitely differentiable?) function, not necessarily a pgf, $g:[0,1]\to \mathbb{R}$ such that $g(0)\ne 0\text{ and }g(1)=1$. Now we define a pgf $\psi (s)$ implicitly as the smallest root of the Lagrange transformation $z=sg(z)$. The paradigmatic example of such a function is $g:z\mapsto 1-p+pz$;

How does this fall out as an actual distribution?

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Pmf

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Mean

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Variance

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