Path-dependent processes and the emergence of macro-structure

Path-dependent systems of the ‘autocatalytic’ or self-reinforcing type typically possess a multiplicity of possible asymptotic outcomes or structures, with early random fluctuations determining which structure is ‘selected’.We explore a wide class of such systems, which we call non-linear Polya systems, where increments to proportions or concentrations occur with probabilities that are non-linear functions of present proportions or concentrations. We show that such systems converge to outcomes (proportions or concentrations) that are represented by the stable fixed points of these functions. These limit theorems are strong laws of large numbers for path-dependent increments, and as such they generalize the standard Borel strong law for independent increments. They are powerful and easy to use.We show applications in chemical kinetics, industrial location theory and in the emergence of technological structure in the economy.