The GARCH algorithm is the most renowned generalisation of Engle's original proposal for modelising returns, the ARCH process. Both cases are characterised by presenting a time dependent and correlated variance or volatility. Besides a memory parameter, b, (present in ARCH) and an independent and identically distributed noise, omega, GARCH involves another parameter, c, such that, for c=0, the standard ARCH process is reproduced. In this manuscript we use a generalised noise following a distribution characterised by an index q(n), such that q(n)=1 recovers the Gaussian distribution. Matching low statistical moments of GARCH distribution for returns with a q-Gaussian distribution obtained through maximising the entropy S-q = 1-Sigma(i) p(i)(q) / q-1 basis of nonextensive statistical mechanics, we obtain a sole analytical connection between q and {b, c, q(n)} which turns out to be remarkably good when compared with computational simulations. With this result we also derive an analytical approximation for the stationary distribution for the (squared) volatility. Using a generalised Kullback-Leibler relative entropy form based on S-q, we also analyse the degree of dependence between successive returns, z(t) and z(t+1), of GARCH(1,1) processes. This degree of dependence is quantified by an entropic index, q(op). Our analysis points the existence of a unique relation between the three entropic indexes q(op), q and q(n) of the problem, independent of the value of (b,c).
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