Do we assume modulating and carrier signal uncorrelated in AM modulation?
The total power of an amplitude modulated signal is
$$\begin{align}\overline{s^2_{AM}(t)}&amp;=\overline{\big(A+m(t)\big)^2\cos^2(2\pi f_ct)}\\&amp;=\frac12\overline{\big(A+m(t)\big)^2}+\frac12\overline{\big(A+m(t)\big)^2\cos(4\pi f_ct)}\tag{1}\end{align}$$
The second term on the right-hand side of $(1)$ is zero if $m(t)$ is a lowpass signal, and if $f_c$ is sufficiently large (i.e., greater than the highest frequency in $m(t)$).
Consequently, we have
$$\begin{align}\overline{s^2_{AM}(t)}&amp;=\frac12\overline{\big(A+m(t)\big)^2}\\&amp;=\frac12\big(A^2+2A\cdot\overline{m(t)}+\overline{m^2(t)}\big)\tag{2}\end{align}$$
The term $\overline{m(t)}$ is the DC value of the message signal, which is assumed to be zero. With this assumption we finally arrive at
$$\overline{s^2_{AM}(t)}=\frac12\big(A^2+\overline{m^2(t)}\big)\tag{3}$$
In sum, the assumptions for which $(3)$ is valid are that $m(t)$ is a lowpass signal, that the carrier frequency is sufficiently high, and that $m(t)$ has a DC value equal to zero. It is usually straightforward to satisfy all of these assumptions to a very high degree of accuracy.