Transition-rate specifications in a partition vector
When an agent leaves a group of j agents, the number of clusters of size j is reduced by one and that of size j — 1 increases by one, that is, aj is reduced by one and aj—1 increases by one.
Unlike the transition-rate expressions in terms of n, an entry of one agent into a cluster of size i in terms of the partition vector is denoted by w(a, a + ui), with ui := ei — ei—1, where e0 is vacuous. Exitfrom a cluster of size j by one agent is expressed by w(a, a — u j). The next example illustrates this.In Section 10.5.1, we describe the transition rate of a type change which we specify as a composite event of
and
See the Ewens sampling formula description in Section 10.5.2.
10.4 Logarithmic series distribution
In this section, we connect the (ordered) market-share distributions with ZipLs distributions and the like used in the economics literature on size distributions. Kelly (1979) used a birth-and-death process with immigration to study family sizes and clustering of agents in social environments such as at cocktail parties. His models may be reinterpreted in terms of agents of different types in the spirit of this chapter. The state space of a continuous-time Markov process is a = (a1, a2,...), where only a finite number of components are nonzero.
A birth in a collection of categories with j agents may be interpreted as one agent changing his type, i.e., moving from a category that contains j agents to form a new category with j + 1 agents, i.e., a changes into
A death in a category with j agents means that a changes into
The boundary condition is a changing into (α1 + 1, a2,...) when an agent of a new type enters and when a single agent departs (α1 - 1, a2,...).
In a simple case of constant birth rate λ and death rate μ with immigration rate ν, Kelly has established that the equilibrium distribution is given by
where
and where x is assumed to be less than one, is a stationary distribution for a = (a1, a2,...), where aj is either the number of types containing exactly j agents, or the number of agents of type j, depending on the interpretation. Note that
The random variable aj is Poisson with a mean that is inversely related to j, and is called a logarithmic series distribution in Watterson (1976).
To extend this model to market-share models, we follow Watterson in part, and assume that the components ai of the partition vector are independent Poisson random variables with mean 
From the assumption, we have
subject to the constraints
and
Using the dummy variable Sj, the joint probability generating function of these random variables is
Set
Then
since
for the second exponential expres
sion, since the powers of s match up to the nth in our generating-function calculations.
The coefficient of sn yields
where we use the relation between the negative binomial and binomial coefficients
We thus recover the Ewens formula by dividing the joint probability by the above:
By setting Sj to s, we obtain 
After replacing Sj by sφj and proceeding analogously, we use
to obtain
By taking the derivative with respect to s, and setting it equal to one, we can extract the expression for E (K = k) in terms of the unsigned Stirling number of the first kind. See Hoppe (1987).
10.4.1 Frequency spectrum of the logarithmic series distribution
A special case is obtained by setting kj = 1, all other ks being zero:
When we evaluate the right-hand side by approximating the factorials by the Stirling’s formula, it is seen to be approximately equal to
The second and the third factors converge to 1 as n become large. The last factor converges to
Therefore we recover an approximate
expression for the frequency spectrum, which is defined to be the expected value of the clusters of size j,
where x is the relative frequency j/n and dx is approximated by 1/n. See further discussions in Section 10.8.
10.5