  This is the first of a series of two papers dealing with local limit theorems
in relatively hyperbolic groups. In this first paper, we prove rough estimates
for the Green function. Along the way, we introduce the notion of relative
automaticity which will be useful in both papers and we show that relatively
hyperbolic groups are relatively automatic. We also define the notion of
spectral positive-recurrence for random walks on relatively hy-perbolic groups.
We then use our estimates for the Green function to prove that $p_n \asymp R^n
n^{-3/2}$ for spectrally positive-recurrent random walks, where $p_n$ is the
probability of going back to the origin at time n and where R is the spectral
radius of the random walk.
