# Non-self-averaging triangular urn models

In the previous section we considered a simple innovation-driven growth model in which some macroeconomic variables were non-self-averaging. Stochastic events are not confined to innovations, of course. No wonder modern macroeconomicsâ€”rational expectations models, real business cycle theory, labor search theory, and endogenous growth theoryâ€” explicitly takes into account stochastic "shocks" in representative models. Most models can be interpreted as a variety of stochastic processes

Now, many stochastic processes can be interpreted as urn models. For example, the Ewens process, which is a one-parameter Poisson-Dirichlet model in which only one type of innovation occurs, has been implemented as an urn model by Hoppe (1984). More generally, by drawing balls not uniformly but at random times governed by exponential distribution urn models can be reinterpreted as stochastic processes, as shown by Athreya and Karlin (1968).

An important characteristic of urn models is that such processes are *path dependent.* Feller (1968, p. 119) calls path dependence an "aftereffect." he says that "it is conceivable that each accident has an after-effect in that it either increases or decreases the chance of new accidents."

An obvious example would be contagious diseases. In a classic paper by Eggenberger and Polya (1923), an urn model was introduced to describe contagious diseases. In Polya's urn scheme, the drawing of either color increases the probability of the same color at the next drawing, and we are led to such path dependence as seen in contagious diseases. We can easily conceive of path-dependent phenomena in economies. They can be described by urn models. We next show that a class of urn models displays non-self-averaging behavior.

These examples are meant to demonstrate that non-self-averaging is not pathological but generic.