Fluctuations in the Curie-Weiss version of the random field Ising model

Discrete Choice with Social Interactions

“…The reason for this is that there does not currently exist any results on laws of large numbers for the Curie-Weiss model where the individual random variables are not identically distributed. Results in Amaro de Matos and Perez (1991) suggest such a generalization should be possible.…”

“…Existence of an equilibrium choice level will follow from the same argument given for Proposition 1 above; similarly, the equivalence of the proportional spillovers and conformity cases (Proposition 7) is unaffected. So long as dF h,I (·) converges weakly to some probability measure dF h (·), Proposition 6 can be generalized accordingly using techniques developed in Amaro de Matos and Perez (1991). Analysis of the other propositions will require the imposition of some restrictions on dF h,I (·).…”

Alternative maps of the world of collective behaviors

“…One pattern that is consistent with behavior along the far eastern edge of the map is the breakdown of the Law of Large Numbers and the Central Limit Theorem. The idea is this: As the product of the intensity of choice and the strength of social interactions, b t J t , grows larger than some threshold, one can show (Amaro de Matos & Perez 1991;Brock & Durlauf 2001b;Repetto 2006) that the Central Limit Theorem underlying the Gaussian distribution breaks down, and more-complicated distributionsmixtures of Gaussian distributionsappear. This behavior is consistent with the east side of the map because it can't happen unless there is positive social influence.…”

Identification of Social Interactions

“…Another approach to the identification of social interactions via qualitative features of sample moments was suggested in the context of financial applications in an early paper by Brock (1993) and uses bifurcations around certain parameter values of a type where the Law of Large Numbers and the Central Limit Theorem break down as in, for example, the statistical mechanics models of Amaro de Matos and Perez (1991) and Ellis (1985). The basic idea of this second approach is to explore how strong dependence between choices can lead to qualitative changes in the properties of the joint stochastic process for a set of choices.…”