Based on some experiments, I conjecture that the average power of an inharmonic "PM" (in the sense that's used in music, i.e. Phase Modulated) signal tends to be the same as that of the carrier, "in the long run". Can this be proved though?
Let
$$s(t)=\sin(t + a\sin(t\sqrt{2}))$$ be our signal where the modulator has a frequency that's not harmonic to the carrier's, and $a$ is some arbitrary "modulation index" in engineering/music speak. Then let the square of the [RMS] power up to time $t$ be:
$$ P(t) = \frac{1}{t}\int_0^t(s(x))^2 dx$$
Can it be shown that $$L = \lim_{t\to \infty} P(t) = \frac{1}{2}$$?
(This [trivially] doesn't hold in some harmonic cases though, e.g. for $\sin(t + a\sin(t))$, where the modulator frequency is harmonic to the carrier's.)
I suppose I'm overcomplicating this (in the comments). It looks like a simple argument is that since no two frequencies coincide in the Bessel expansion in the inharmonic case, at the limit the (square of) integral/sum of power is half the sum of squares of the (Bessel) coefficients, i.e. $$L = \frac{1}{2}\sum_{n=-\infty}^\infty (J_n(a))^2= \frac{1}{2}$$
due to the Neuman's addition theorem (10.23.3 in DMLF). Somewhere in the background we're also probably using that the sum and integral commute since we're only summing/integrating positive values there.
