  Multifractal analysis of stochastic processes deals with the fine scale
properties of the sample paths and seeks for some global scaling property that
would enable extracting the so-called spectrum of singularities. In this paper
we establish bounds on the support of the spectrum of singularities. To do
this, we prove a theorem that complements the famous Kolmogorov's continuity
criterion. The nature of these bounds helps us identify the quantities truly
responsible for the support of the spectrum. We then make several conclusions
from this. First, specifying global scaling in terms of moments is incomplete
due to possible infinite moments, both of positive and negative order. For the
case of ergodic self-similar processes we show that negative order moments and
their divergence do not affect the spectrum. On the other hand, infinite
positive order moments make the spectrum nontrivial. In particular, we show
that the self-similar stationary increments process with the nontrivial
spectrum must be heavy-tailed. This shows that for determining the spectrum it
is crucial to capture the divergence of moments. We show that the partition
function is capable of doing this and also propose a robust variant of this
method for negative order moments.
