  We prove some general results about the asymptotics of the distribution of
the number of cycles of given length of a random permutation whose distribution
is invariant under conjugation. These results were first established to be
applied in a forthcoming paper (Cycles of free words in several random
permutations with restricted cycles lengths), where we prove results about
cycles of random permutations which can be written as free words in several
independent random permutations. However, we also apply them here to prove
asymptotic results about random permutations with restricted cycle lengths.
More specifically, for $A$ a set of positive integers, we consider a random
permutation chosen uniformly among the permutations of $\{1,..., n\}$ which
have all their cycle lengths in $A$, and then let $n$ tend to infinity.
Improving slightly a recent result of Yakymiv (Random A-Permutations:
Convergence to a Poisson Process), we prove that under a general hypothesis on
$A$, the numbers of cycles with fixed lengths of this random permutation are
asymptotically independent and distributed according to Poisson distributions.
In the case where $A$ is finite, we prove that the behavior of these random
variables is completely different: cycles with length $\max A$ are predominant.
