  Given two weighted automata, we consider the problem of whether one is big-O
of the other, i.e., if the weight of every finite word in the first is not
greater than some constant multiple of the weight in the second.
  We show that the problem is undecidable, even for the instantiation of
weighted automata as labelled Markov chains. Moreover, even when it is known
that one weighted automaton is big-O of another, the problem of finding or
approximating the associated constant is also undecidable.
  Our positive results show that the big-O problem is polynomial-time solvable
for unambiguous automata, coNP-complete for unlabelled weighted automata (i.e.,
when the alphabet is a single character) and decidable, subject to Schanuel's
conjecture, when the language is bounded (i.e., a subset of $w_1^*\dots w_m^*$
for some finite words $w_1,\dots,w_m$) or when the automaton has finite
ambiguity.
  On labelled Markov chains, the problem can be restated as a ratio total
variation distance, which, instead of finding the maximum difference between
the probabilities of any two events, finds the maximum ratio between the
probabilities of any two events. The problem is related to
$\varepsilon$-differential privacy, for which the optimal constant of the big-O
notation is exactly $\exp(\varepsilon)$.
