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)=∑n=1∞e-t(1-e-t)n-1Qn+(F)(v). Here, Qn+(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 ∥Qn+(F)-M∥L1(R) tends to zero. In the case of the Kac model, we prove that for every ε>0, if F has moments of every order and finite Fisher information, there is a constant C so that for all n, ∥Qn+(F)-M∥L1(R)⩽CnΛ+ε where Λ is the least negative eigenvalue for the linearized collision operator. We show that Λ is the best possible exponent by relating this estimate to a sharp estimate for the rate of relaxation of f(·,t) to M. A key role in the analysis is played by a decomposition of Qn+(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.