From author’s inroduction. Let T be an invertible positive isometry on \(L_ 1\) of a \(\sigma\)-finite measure space. It is proved that if f and p are in \(L_ 1\), and p is nonnegative, then the ratios \((\sum^{n}_{i=m}T^ if)/(\sum^{n}_{i=m}T^ ip)\) converge almost everywhere on the set \(\{\sum^{+\infty}_{-\infty}T^ ip>0\}\) as \(m\to -\infty\) and \(n\to +\infty\), independently; and the identification of the limit is obtained.