Summary: For a critical branching process with immigration, \(\{X_ n\}\), \((\log X_ n)/\log n\) is shown to converge almost surely to 1 when \(\{X_ n\}\) is transient. The rates of growth for \(\sum^ n_{i = 0} X_ i\) and \(\sum^ n_{i = 0} (1 + X_ i)^{-1}\) are then derived and used to obtain convergence rates for the conditional weighted least squares estimators of the generation and immigration means.