  We provide an upper bound as a random variable for the functions of
estimators in high dimensions. This upper bound may help establish the rate of
convergence of functions in high dimensions. The upper bound random variable
may converge faster, slower, or at the same rate as estimators depending on the
behavior of the partial derivative of the function. We illustrate this via
three examples. The first two examples use the upper bound for testing in high
dimensions, and third example derives the estimated out-of-sample variance of
large portfolios. All our results allow for a larger number of parameters, p,
than the sample size, n.
