A.7 Poisson random variables and the Ewens sampling formula

A.7.1 Approtximations by Poisson random variables

A large body of probability and statistics literature exists on approximating sums of nonnegative integer-valued random variables as Poisson variables if the summands have sufficiently high probability of taking 0 value and sufficiently weak mutual dependence: Arratia and Tavare (1992), Arratia, Barbour and Tavare (1992,1999), Galambos (1987), Steele (1994), and Chen (1975), among others.

In economic modeling, there exist many situations where approximations by Poisson random variables are appropriate. We summarize here some relevant facts that are deemed useful for economic applications. In particular, random combinatorial structures such as Poisson-Dirichlet limits seem to be applicable to problems in industrial organization, such as market shares and entry and exit phenomena, to name just two. Kaplan (1977) discusses Poisson approximation for urn schemes.

In the area of industrial organization, suppose that X_i = 1 when firm i is in the market and X_i = 0 otherwise. Then, S_n = X₁ +--------------------------------------------------- + X_n is the total

number of firms in the market. Define T_n = Z₁ +--- + Z_n, where

where Q_λ is the Poisson distribution with mean λ.

Next replace X_i with an independent Bernoulli random variable X* with parameter p', and define Z' using X'. Do likewise with T_n. Then, we have

and

Then,

Let π be a permutation of n symbols, {1, 2,..., n}, say.

A.7.2 Conditional Poisson random variables

We have observed that Poisson random variables play important roles in de­scribing random combinatorial structures in patterns formed by agents. Fol­lowing Shepp and Lloyd (1966) and Arratia and Tavare (1994), we assume that

The probability of the sum, T_n, is obtained simply by calculating the proba­bility generating function for it,

one, respectively. Note that this conditional probability is independent of the means of the Poisson distributions.

As an example of unequal assignments of probabilities to combinatorial structures, we consider permutations. Instead of treating each permutation as equally probable, we can twist the distribution, as is done in large-deviation theory. See Shwartz and Weiss (1995)or Arratia and Tavare (1994) for examples. Here we explain the simplest case, where we posit mutually independent Poisson random variables Z_i having mean θx^l / i, where θ is nonnegative. We show that this one parameter generates the Ewens sampling formula.

First, note that the joint distribution is

where

and

The weighted sum of Zs has the probability

This follows as before from the Cauchy formula.

Next, introduce a new measure P_θ under which the Zs are mutually inde­pendent:

We have

where the joint probability of the Zs is equal to

On the other hand, we have

and hence

These two expressions are both proportional to the same expression P (Z = a), and both are probability densities on Zⁿ₊. Hence, the two normalizing constants are equal to each other:

With parameter θ, we weight each permutation unequally The probability is

as explained above.

Equate the coefficient of xⁿ of both sides in the identity (Kelly 1979, Kendall 1975)

to denote ascending factorials. The right-hand side yields

Recalling the Cauchy formula, we obtain

Arratia et al. (1992) have established that by assigning probabilities, not equally to all permutations of n symbols, but according to

where ∖π | is the total number of cycles in the product representation of the permutation π, the Ewens distribution can be expressed by

Iterating this relation, we derive

This leads to the density of the limit T_n/n, denoted by T. It is given by

for 0 < x < 1.