  We prove a Bernstein-von Mises result for generalized fiducial distributions
following the approach based on quadratic mean differentiability in Le Cam
(1986); van der Vaart (1998). Building on their approach, we introduce only two
additional conditions for the generalized fiducial setting. While asymptotic
normality of generalized fiducial distributions has been studied before,
particularly in Hannig (2009) and Sonderegger and Hannig (2014), this work
significantly extends the usefulness of such a result by the much more general
condition of quadratic mean differentiability. We demonstrate the applicability
of our result with two examples that necessitate these more general
assumptions: the triangular distributions and free-knot spline models.
