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Bellow and Calderón proved that the sequence of convolution powers ${\mu }_{n}f\left(x\right)={\sum }_{k\in \mathbb{Z}}{\mu }^n\left(k\right)f\left(T^{k}x\right)$ converges a.e, when $\mu$ is a strictly aperiodic probability measure on $\mathbb{Z}$ such that the expectation is zero, $E\left(\mu \right)=0$, and the second moment is finite, $m_2\left(\mu \right)\<\infty$. In this paper we extend this result to cases where $m_2\left(\mu \right)=\infty$.