On the connection between ARCH time series and non-extensive statistical mechanics

The ARCH(1) is a generator of stochastic discrete time series, {epsilon(t)}, widely used in finance and characterised by conditional time-varying (and correlated) second-order moment. It involves a parameter, beta and a noise, eta. In this work one presents, through an analytical result, that ARCH(1) stationary distributions are well approached by the distributions that maximise the entropy, S-q = 1-integral[p(x)](q)dx/1-q. Using the generalised Kullback-Leibler relative entropy, I-q, one also quantifies the degree of dependence between variables epsilon(t) and epsilon(t)' and shows that the degree of dependence increases with parameter beta. (C) 2004 Elsevier B.V. All rights reserved.