A quantitative weak law of large numbers and its application to the delta method

Let T _n be a statistic of the form T _n = f(), where 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.