In the case of Maxwellian molecules, the Wild summation formula gives an expression for the solution of the spatially homogeneous Boltzmann equation in terms of its initial data \(F\) as a sum \(f(v,t)= \sum_{n=1}^\infty e^{-t} (1-e^{-t})^{n-1} Q_n^+(F)(v)\). Here, \(Q_n^+(F)\) is an average over \(n\)-fold iterated Wild convolutions of \(F\). If \(M\) denotes the Maxwellian equilibrium corresponding to \(F\), then it is of interest to determine bounds on the rate at which \(\|Q_n^+(F)- M\|_{L^1(\mathbb R)}\) tends to zero. In the case of the Kac model, we prove that for every \(\varepsilon>0\), if \(F\) has moments of every order and finite Fisher information, there is a constant \(C\) so that for all \(n\), \(\|Q_n^+(F)- M\|_{L^1(\mathbb R)}\leq Cn^{\Lambda+ \varepsilon}\) where \(\Lambda\) is the least negative eigenvalue for the linearized collision operator. We show that \(\Lambda\) is the best possible exponent by relating this estimate to a sharp estimate for the rate of relaxation of \(f(\cdot,t)\) to \(M\). A key role in the analysis is played by a decomposition of \(Q_n^+(F)\) into a smooth part and a small part. This depends in an essential way on a probabilistic construction of McKean. It allows us to circumvent difficulties stemming from the fact that the evolution does not improve the qualitative regularity of the initial data.