Let A be a nonnegative $n \times n$ matrix. In this paper we study the growth of the powers $A^m, m = 1,2,3, \cdots $ when $\rho ( A ) = 1$. These powers occur naturally in the iteration process \[x^{( m + 1 )} = Ax^{( m )} ,\quad x^{( 0 )} \geqq 0,\] which is important in applications and numerical techniques. Roughly speaking, we analyze the asymptotic behavior of each entry of $A^m $. We apply our main result to determine necessary and sufficient conditions for the convergence to the spectral radius of A of certain ratios naturally associated with the iteration above.