The $GARCH$ algorithm is the most renowned generalisation of Engle's originalproposal for modelising {\it returns}, the $ARCH$ process. Both cases arecharacterised by presenting a time dependent and correlated variance or {\itvolatility}. Besides a memory parameter, $b$, (present in $ARCH$) and anindependent and identically distributed noise, $\omega $, $GARCH$ involvesanother parameter, $c$, such that, for $c=0$, the standard $ARCH$ process isreproduced. In this manuscript we use a generalised noise following adistribution characterised by an index $q_{n}$, such that $q_{n}=1$ recoversthe Gaussian distribution. Matching low statistical moments of $GARCH$distribution for returns with a $q$-Gaussian distribution obtained throughmaximising the entropy $S_{q}=\frac{1-\sum_{i}p_{i}^{q}}{q-1}$, basis ofnonextensive statistical mechanics, we obtain a sole analytical connectionbetween $q$ and $(b,c,q_{n}) $ which turns out to be remarkably good whencompared with computational simulations. With this result we also derive ananalytical 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 isquantified by an entropic index, $q^{op}$. Our analysis points the existence ofa 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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