  We revisit the periodic complexity function $h_{\bf w}(n)$ introduced by
Mignosi and Restivo. This function gives the average of the first $n$ local
periods of a recurrent infinite word ${\bf w}$. We give a different method than
that of Mignosi and Restivo for computing the asymptotics of the periodic
complexity function of the Thue-Morse word and show how to apply the method to
other automatic sequences, like the Rudin-Shapiro word and the period-doubling
word.
