Large Deviations of Random Vector Fields with Applications to Economics

✍ Scribed by Esa Nummelin

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 231 KB
• Volume: 24
• Category: Article
• ISSN: 0196-8858
• DOI: 10.1006/aama.1999.0668

✦ Synopsis

Let Z n p n = 1 2, , be a sequence of random vector fields on R l , and let π * n = p ∈ R l Z n p = 0 denote the random set of zeros of Z n . We study the asymptotics of probabilities of the type P π * n ∩ B = for subsets B ⊂ R l . 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 -1 X n p p ∈ π * n ⊂ R l+d , where X n is an auxiliary sequence of random maps X n R l → R d . In economic applications Z n 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 X n 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.