Let $\displaystyle P $ be an $\displaystyle n \times n $ stochastic matrix. It is well known that if $\displaystyle P$ is aperiodic (or regular), then there exists a $\displaystyle t < \infty$ such that $\displaystyle p_{ij}^{t} > 0$ for $\displaystyle 1 \le i,j \le n$, where $\displaystyle P^{t}$ is the tth power of $\displaystyle P$. However, this result is not very useful for actually checking if $\displaystyle P$ is aperiodic, since one can only be sure that $\displaystyle P$ is periodic after performing an infinite number of multiplications. Is there a practical way to check whether$\displaystyle P $ is aperiodic or not?