A Quantitative Weak Law of Large Numbers and Its Application to the Delta Method

Let T-n be a statistic of the formT(n) = f((X) over bar (n)), where (X) over bar (n) is the samplemean of a sequence of independent random variables and f denotes a prescribed function taking values in a separable Banach space. In order to establish asymptotic expansions for bias and variance of T-n conventional theorems typically impose restrictive boundedness conditions upon f or its derivatives; moreover, these conditions are sufficient but not necessary. It is shown that a quantitative version of the weak law of large numbers is both sufficient and necessary for the accuracy of Taylor expansions of T-n. In particular, boundedness conditions may be replaced by mild requirements upon the global growth of f.