Long Memory Properties and Complex Systems
The idea of heterogeneity present in the sum of cross-sectional processes in these original works which results into long memory properties are a consequence of the heterogeneity present in the autoregressive part of each individual agent . These works discuss the rise of long memory properties in the aggregation of stochastic processes of the following form :
where is the autoregressive parameter distributed according to a specific distribution ( so this is the heterogeneity source in the implied process ), are normally distributed disturbances with n , and is the ith individual real-valued stochastic variable that may be aggregated . In an adaptive learning framework , this can be thought as the way individuals make their respective forecast about the system next state .
On the other hand , in the underlying process here discussed , is a binary stochastic process ( which has a completely different nature ); all autoregressive parts are kept constant across the different agents , and a bias factor
is introduced ( each agent with its own bias ). Nonetheless , are distributed according to a Bernoulli distribution , and there is a linear feedback relationship between the collective behavior of the agents and its individual behavior , which is fed into their individual process according to the same parameter weights .
When all these factors are put together and compared against the traditional literature , an important conclusion can be made . If all agents forecast exactly in the same way ( which is an explicit assumption in this model ), the way they perceive these same forecasts generate such long memory behavior , which is something not yet discussed in the existing literature .
In addition to the mentioned points above , it is worth mentioning that the original authors have chosen a Bernoulli distributed variable for the external information , given the fact that it is very easy to obtain a map within the range that models the behavior of the mean field of the proposed Boolean Network .
It is straightforward to derive :
And according to the rules implied in each agent ( node ), it follows that ( for a generic probability distribution of the variable B ):
As can be seen in the original work , it is possible to derive the following map , when B follows a Bernoulli process :
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