  We provide a new version of delta theorem, that takes into account of high
dimensional parameter estimation. We show that depending on the structure of
the function, the limits of functions of estimators have faster or slower rate
of convergence than the limits of estimators. We illustrate this via two
examples. First, we use it for testing in high dimensions, and second in
estimating large portfolio risk. Our theorem works in the case of larger number
of parameters, $p$, than the sample size, $n$: $p>n$.
