Large Deviations of Random Vector Fields with Applications to Economics

AbstractLet Zn(p), n=1,2,…, be a sequence of random vector fields on Rl, and let π*n={p∈Rl;Zn(p)=0} denote the random set of zeros of Zn. We study the asymptotics of probabilities of the type P(π*n∩B≠∅) for subsets B⊃Rl. Under suitable regularity conditions these probabilities are of the order e−nI(B) where I(B) is a large deviation rate function obtained as a minimum of an associated entropy function I(p) over B. We study also the large deviations of random graphs of the form {(p,n−1Xn(p));p∈π*n}⊂Rl+d, where Xn is an auxiliary sequence of random maps Xn: Rl→Rd. In economic applications Zn(p) refers to the total excess demand in a random economy of size n and hence π*n becomes the random set of equilibrium prices for the economy. The auxiliary map Xn(p) may denote any price–depending total characteristic like total demand, supply, etc. The conditional laws of large numbers which ensue on the large deviation estimates admit an interpretation in terms of a principle of entropy minimization in analogy with the classical maximum entropy principles in statistics and statistical mechanics