Size distribution for old and new goods

We have described distributions of cluster sizes based on Dirichlet distributions. These are stationary distributions from the viewpoint of random combinatorial analysis. Here, we examine the same subject from a different perspective.

8.9.1 Diffusion-equation approximation

We calculate the probability that there are k clusters formed by n agents at time t starting from n singletons, that is, n individuals initially. We follow the analyses by Derrida and Peliti (1991).

Suppose that there are n agents of either type 1 or 2. Suppose that a Markov chain X (t) is defined by

where X(t) is the number of agents of type 1. This process is known to be approximated by a diffusion process with the forward Kolmogorov equation for the density with mean zero and variance σ (x)²:

with σ (x)² = x(1 - x)/n. This is the equation for the density of the fraction x of the diffusion approximation with a system of n agents. We define τ = t/n and rewrite the time derivative in terms of τ to remove the 1 /n in the variance equation. See Ewens (1979, p. 140).

This diffusion equation was first solved by Kimura (1955). We discuss the solution of this equation with the initial condition δ(x — p).

We posit f (x, τ; p) = T(τ)X(x; p) to try the separation of the variables to solve the equation. The equation separates into

where κ is a constant.

The functionis immediate, and we change variable from x

to z = 1 — 2x in X.It satisfies

where' denotes differentiation with respect to z now.

is known as the Gegenbauer function, a type of hypergeometric function. See Morse and Feshbach (1953, pp. 547, 731) on the Gegenbauer functions. We see that the case with β = 1 and α = i - 1 is for our function X with κ = -i(i + 1)/2. These values are the eigenvalues, and the corresponding ψ are eigenfunctions. To be explicit, we have

The class of Gegenbauer polynomials is known to be a system of complete orthogonal polynomials with weight 1 — z² on the interval [-1, 1]: For any positive integers m and n,

Hence the solution of the diffusion equation can be expressed as

The recursion relations are given by

8.9.2 Lines of product developments and inventions

We derive joint distributions for shares of old and new goods in a sense we now explain. Pick some past time instant t. Some of the goods or products currently available on the markets were in existence at least t time units or periods ago, that is, when we go back in time t periods, these products were already invented or being produced. Call these goods or products old goods or products. The remainder of goods or products that are currently available but were not available t periods in the past have been either invented or improved upon since that time.

In this section, we use the theory of coalescents, which was invented by Kingman (1982) and has been applied extensively in the genetics literature, to explain shares of old and new goods (species). See, among others, Watterson (1984), Donnelly and Tavare (1987), Ewens (1990), and Hoppe (1987) on coa­lescents and related topics. We describe the distributions of the numbers of old goods and new goods, and the probability density of the shares or fractions of these goods.

At present time, we take a sample of n products or goods, and we examine their histories of developments, going back in time. Some goods can trace their developmental history to an invention or innovation that took place some time ago. Others may have branched out from common prototypes some time in the past.

Pick a time t units in the past, and fix a sample of size n goods out of all goods that are available now in the markets. When the histories of development or improvements, or mere existence in the markets, are traced back in time for these n goods, they can be put in equivalence relations using the notion of defining events in the terminology of Ewens (1990, Sec. 7). A defining event is either the emergence of two products from a common prototype, or the invention of a new good or product some time ago. It is assumed that the overall rate at which the former takes place in an interval of length h is [k(k - 1)∕2]h + o(h) when there are k goods, and the overall rate of invention is specified by [kθ∕2]h + o(h). The rate of arrival of defining events is therefore k(k + θ — 1)/2 when there are k products. The mean time of arrival is 2∕{k(k + θ — 1)}.

Using the notation in Watterson (1984), at time t ago there were D_t old goods, and their equivalence classes are denoted by ξ_i, i = 1, 2,..., D_t. Theirsizesare λ_i = |ξ_i |. New goods are also put into equivalence classes, η j, j = 1, 2,..., F_t (where F_t is the number of the equivalence classes), having the sizes μ j = | η j |.

Watterson (1984, (2.9)) has shown

where k is the number of old goods and l the number of new goods. As t recedes into remote past, k will become zero, because all goods coalesce to a single prototype good, and the above simplifies to

and

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i = 1, 2,..., k, where u₁,..., u_k are uniformly distributed on [0, 1] and sum to one, and x_k+1 = (1 - v_k)z₁, x_k+2 = (1 - v_k)z₂(1 - z₁), and so on, where the z's are i.i.d. with the density of Beta(1, θ),

0 ≤ z ≤ 1. The factor 1 - v_k is residually allocated to the shares of new goods, in the order of appearance, that is, in the order of ages of the new goods. See also Ewens (1990).