  The field of analytic combinatorics is dedicated to the creation of effective
techniques to study the large-scale behaviour of combinatorial objects.
Although classical results in analytic combinatorics are mainly concerned with
univariate generating functions, over the last two decades a theory of analytic
combinatorics in several variables (ACSV) has been developed to study the
asymptotic behaviour of multivariate sequences. In this work we survey ACSV
from a probabilistic perspective, illustrating how its most advanced methods
provide efficient algorithms to derive limit theorems, and comparing the
results to past work deriving combinatorial limit theorems. Using the results
of ACSV, we provide a SageMath package that can automatically compute (and
rigorously verify) limit theorems for a large variety of combinatorial
generating functions. To illustrate the techniques involved, we also establish
explicit local central limit theorems for a family of combinatorial classes
whose generating functions are linear in the variables tracking each parameter.
Applications covered by this result include the distribution of cycles in
certain restricted permutations (proving a limit theorem stated as a conjecture
in recent work of Chung et al.), integer compositions, and $n$-colour
compositions with varying restrictions and values tracked. Key to establishing
these explicit results in arbitrary dimension is an interesting symbolic
determinant, which we compute by conjecturing and then proving an appropriate
$LU$-factorization. It is our hope that this work provides readers a blueprint
to apply the powerful tools of ACSV to prove central limit theorems in their
own work, making them more accessible to combinatorialists, probabilists, and
those in adjacent fields.
