Correlation in DSBSC modulation
The random phase $\phi$ makes the carrier signal $c(t)=A\cos(2\pi f_0+\phi)$ wide-sense stationary with the autocorrelation function

$$R_c(\tau)=\frac{A^2}{2}\cos(2\pi f_0\tau)\tag{1}$$

The Fourier transform of $(1)$ is the power spectrum of the carrier:

$$S_c(f)=\frac{A^2}{4}\left[\delta(f-f_0)+\delta(f+f_0)\right]\tag{2}$$

where $\delta(f)$ is the Dirac impulse.

Since the carrier signal $c(t)$ and the message signal $m(t)$ are assumed to be uncorrelated, the autocorrelation function of the modulated signal $s(t)=c(t)m(t)$ equals the product of the autocorrelation functions of $c(t)$ and $m(t)$:

$$R_s(\tau)=R_c(\tau)R_m(\tau)\tag{3}$$

From $(3)$ it follows that the power spectrum $S_s(f)$ of $s(t)$ is the convolution of the power spectra of $c(t)$ and $m(t)$:

$$S_s(f)=S_c(f)\star S_m(f)=\frac{A^2}{4}\left[S_m(f-f_0)+S_m(f+f_0)\right]\tag{4}$$