If I have a $\sqrt{N}$ asymptotic normal estimator (call it $\boldsymbol{\theta}$, possibly a vector). Say I want to find the asymptotic distribution of $g(\boldsymbol{\theta})$ and suppose $g'(\boldsymbol{\theta})=0$ (assume $g(\cdot)$ real valued).
How would the second order delta method derivation go along in this case? Can easily argue that the quadratic form in the Taylor expansion is asymptotically chi-square? In the univariate case this is trivial, but in the multivariate case there is dependency issues which does not seem obvious. Also, would the estimator converge at $\sqrt{N}$ speed?
Thanks.