It is known that, in general, an affine or Gabor AP-frame is an \(L^2(\mathbb {R})\) -frame and conversely. In part as a consequence of the Ergodic Theorem, we prove a necessary and sufficient condition for an affine (wavelet) system \(\mathcal {A}=\{a^{j/2} \psi _{j,k}(t):=a^{-j/2} \psi (a^{-j} t -k):j\in \mathbb {Z}, k\in \mathbb {K}:=b\mathbb {Z}\}\) to be an affine AP-Frame in terms of Gaussian stationary random processes expanding in this way what we have done recently for Gabor systems. Likewise, we study a connection between the decay of the associated stationary sequences \(\{\langle {X,\psi _{j,k}}\rangle : k\in \mathbb {K}\}\) for each \(j\in \mathbb {Z}\) , and a smoothness condition on a Gaussian stationary random process \(X=(X(t))_{t\in \mathbb {R}}\) .
