  The purpose of this paper is to establish the convergence in law of the
sequence of "midpoint" Riemann sums for a stochastic process of the form f'(W),
where W is a Gaussian process whose covariance function satisfies some
technical conditions. As a consequence we derive a change-of-variable formula
in law with a second order correction term which is an It\^{o} integral of
f''(W) with respect to a Gaussian martingale independent of W. The proof of the
convergence in law is based on the techniques of Malliavin calculus and uses a
central limit theorem for q-fold Skorohod integrals, which is a
multi-dimensional extension of a result proved by Nourdin and Nualart [J.
Theoret. Probab. 23 (2010) 39-64]. The results proved in this paper are
generalizations of previous work by Swanson [Ann. Probab. 35 (2007) 2122-2159]
and Nourdin and R\'{e}veillac [Ann. Probab. 37 (2009) 2200-2230], who found a
similar formula for two particular types of bifractional Brownian motion. We
provide three examples of Gaussian processes W that meet the necessary
covariance bounds. The first one is the bifractional Brownian motion with
parameters $H\le1/2$, HK=1/4. The others are Gaussian processes recently
studied by Swanson [Probab. Theory Related Fields 138 (2007) 269-304], [Ann.
Probab. 35 (2007) 2122-2159] in connection with the fluctuation of empirical
quantiles of independent Brownian motion. In the first example the Gaussian
martingale is a Brownian motion, and expressions are given for the other
examples.
