Almost everywhere convergence of convolution powers without finite second moment

We generalize a theorem of Bellow and Calderón concerning the a.e. convergence of the convolution powers \(\ds \mu^nf(x)=\sum_{k}\mu^n(k)f(T^k x)\) where \(T\) is a measure preserving transformation of a probability space and \(\mu\) is a probability measure on the integers.