Summary: We consider a random walk on the affine group of the real line, we denote by \(P\) the corresponding Markov operator on \(\mathbb R\), and we study the Birkhoff sums associated with its trajectories. We show that, depending on the parameters of the random walk, the normalized Birkhoff sums converge in law to a stable law of exponent \(\alpha \in ]0,2[\) or to a normal law. The corresponding analysis is based on the spectral properties of two families of associated transfer operators \(P_{t},T_{t}\). The operator \(P_{t}\) is a Fourier operator and is considered here as a perturbation of the Markov operator \(P=P_{0}\) of the random walk. The operator \(T_{t}\) is related to \(P_{t}\) by a symmetry of Heisenberg type and is also considered as a perturbation of the Markov operator \(T_{0}=T\). We prove that these operators have an isolated dominant eigenvalue which has an asymptotic expansion involving fractional powers of \(t\). The parameters of this expansion have simple expressions in terms of tails and moments of the stationary measures of \(P\) and \(T\).