Magnitude of the analytic signal
By definition, the magnitude of the analytic signal $s_a(t)$ can always be considered the envelope of the signal in a purely mathematical sense, simply because the real-valued signal is given by
$$s(t)=\textrm{Re}\big\{s_a(t)\big\}=|s_a(t)|\cos\big(\arg\{s_a(t)\}\big)\tag{1}$$
However, this does not mean that $|s_a(t)|$ has necessarily a physical significance as an actual envelope of $s(t)$. It does have physical significance if the term $\cos(\arg\{s_a(t)\})$ in $(1)$ has the function of a carrier. This carrier doesn't need to be sinusoidal but it must be a narrowband signal, i.e., its center frequency must be large compared to its bandwidth.
If you want a strict mathematical definition of the conditions under which $|s_a(t)|$ is an envelope in a physically significant way, then you must define what you mean by 'envelope' and by 'physical significance'. This is not trivial and probably also not very useful.
If your question is just about the feasibility of envelope detection, then that's a very different question, and one that's much easier to answer. If the modulated signal is given by
$$s(t)=m(t)\cos(\omega_ct)\tag{2}$$
then the magnitude of the corresponding analytic signal equals $m(t)$ if

$m(t)$ has a maximum frequency smaller than the carrier frequency $\omega_c$
$m(t)\ge 0$

These conditions are met for standard AM, but not for DSB-SC.

EDIT: For the signal $(2)$ it's straightforward to show that $|s_a(t)|=m(t)$ if above conditions are met. If $m(t)$ is a low pass signal with a maximum frequency smaller than the carrier frequency $\omega_c$, then the analytic signal is given by
$$s_a(t)=m(t)e^{j\omega_ct}\tag{3}$$
with magnitude
$$|s_a(t)|=|m(t)|\tag{4}$$
Clearly, if $m(t)\ge 0$ then $|s_a(t)|=m(t)$.