Necessary and sufficient conditions for local second-order identifiability

This note presents a new formulation and proof of the result the Hessian of the likelihood function of an observed process at the point
\theta
in a parameter space, computed under the assumption that the process is i.i.d. Gaussian, is asymptotically nonsingular if and only if
\theta
is locally second-order identifiable. That is to say, if and only if the parameters in a neighborhood of
\theta
are in one-to-one correspondence with the second-order statistics of the observed process.