Before revision:  Under these assumptions, we are able to prove that , if we consider a sufficiently large number of periods, the expected rate of return and the expected gross return are linear in the initial time value of the log earnings over price ratio , and their variance goes to zero with  rate of convergence equal to minus one .
After revision:  Under these assumptions, we are able to prove that , if we consider a sufficiently large number of periods, the expected rate of return and the expected gross return are linear in the initial logarithmic CAPE , and their variance goes to zero with rate of convergence equal to minus one .