  Multifractal formalisms provide an apt framework to study random cascades in
which multifractal spectrum width $\Delta\alpha$ fluctuates depending on the
number of estimable power-law relationships. Then again, multifractality
without surrogate comparison can be ambiguous: the original measurement series'
multifractal spectrum width $\Delta\alpha_\mathrm{Orig}$ can be sensitive to
the series length, ergodicity-breaking linear temporal correlations (e.g.,
fractional Gaussian noise, $fGn$), or additive cascade dynamics. To test these
threats, we built a suite of random cascades that differ by the length, type of
noise (i.e., additive white Gaussian noise, $awGn$, or $fGn$), and mixtures of
$awGn$ or $fGn$ across generations (progressively more $awGn$, progressively
more $fGn$, and a random sampling by generation), and operations applying noise
(i.e., addition vs. multiplication). The so-called ``multifractal
nonlinearity'' $t_\mathrm{MF}$ (i.e., a $t$-statistic comparing
$\Delta\alpha_\mathrm{Orig}$ and multifractal spectra width for
phase-randomized linear surrogates $\Delta\alpha_\mathrm{Surr}$) is a robust
indicator of random multiplicative rather than random additive cascade
processes irrespective of the series length or type of noise. $t_\mathrm{MF}$
is more sensitive to the number of generations than the series length.
Furthermore, the random additive cascades exhibited much stronger ergodicity
breaking than all multiplicative analogs. Instead, ergodicity breaking in
random multiplicative cascades more closely followed the ergodicity-breaking of
the constituent noise types -- breaking ergodicity much less when arising from
ergodic $awGn$ and more so for noise incorporating relatively more correlated
$fGn$. Hence, $t_\mathrm{MF}$ is a robust multifractal indicator of
multiplicative cascade processes and not spuriously sensitive to ergodicity
breaking.
