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, ω, 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<Subscript>n</Subscript>, such that q<Subscript>n</Subscript>=1 recovers the Gaussian distribution. Matching low statistical moments of GARCH distribution for returns with a q-Gaussian distribution obtained through maximising the entropy <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$S_{q}=\frac{1-\sum_{i}p_{i}^{q}}{q-1}$</EquationSource> </InlineEquation>, basis of nonextensive statistical mechanics, we obtain a sole analytical connection between q and <InlineEquation ID="Equ2"> <EquationSource Format="TEX">$\left( b,c,q_{n}\right) $</EquationSource> </InlineEquation> 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<Subscript>q</Subscript>, we also analyse the degree of dependence between successive returns, z<Subscript>t</Subscript> and z<Subscript>t+1</Subscript>, of GARCH(1,1) processes. This degree of dependence is quantified by an entropic index, q<Superscript>op</Superscript>. Our analysis points the existence of a unique relation between the three entropic indexes q<Superscript>op</Superscript>, q and q<Subscript>n</Subscript> of the problem, independent of the value of (b,c). Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2005
