A.14 Random growth processes
Consider a discrete-time stochastic process
where {λt} is a sequence of i.i.d.
positive-valued random variables. On considering λt as a growth factor, and S as the size or share of some economic variable (such as share of markets, size of firms, etc.), this equation describes a random growth process in which the product of random variables rather than the more common sum of random variables is involved.10
where λ has the same distribution as λ1, and the equality is in distribution. Proceeding informally, suppose f (∙) is the density function for the λ's. Suppose that
i.e.,
Let
Then,
In steady state, we arrive at
10 Differences between the product and the sum of random variables are sometimes striking.
