  A permutation sequence is said to be convergent if the density of occurrences
of every fixed permutation in the elements of the sequence converges. We prove
that such a convergent sequence has a natural limit object, namely a Lebesgue
measurable function $Z:[0,1]^2 \to [0,1]$ with the additional properties that,
for every fixed $x \in [0,1]$, the restriction $Z(x,\cdot)$ is a cumulative
distribution function and, for every $y \in [0,1]$, the restriction
$Z(\cdot,y)$ satisfies a "mass" condition. This limit process is well-behaved:
every function in the class of limit objects is a limit of some permutation
sequence, and two of these functions are limits of the same sequence if and
only if they are equal almost everywhere. An ingredient in the proofs is a new
model of random permutations, which generalizes previous models and might be
interesting for its own sake.
