TAUBERIAN-THEOREMS, CHERNOFF INEQUALITY, AND THE TAIL BEHAVIOR OF FINITE CONVOLUTIONS OF DISTRIBUTION-FUNCTIONS

Let X be a real random variable with cumulative distribution function F on R. Assume that the moment generating function Phi(t): = integral( - infinity)(infinity) dF(x) is finite in a right neighborhood of 0. As a first topic treated in this paper, we consider distribution functio ns such that - log (F) over bar(x -) = sup (xy - log Phi(y), y > 0)(1 + o(1)) as s --> infinity, where (F) over bar(x -) : = (1 - F)(x -). T he classical Chernoff Inequality is shown to be extended into this tai l equivalent statement in many cases, including many infinitely divisi ble distributions and most of the distribution functions used in stati stics. We explore various Tauberian Theorems in this direction with a special attention to the case when - log F is a function rapidly varyi ng at infinity; this latter case yields to an extension of Stirling's formula to a large class of functions. Set S-n = X(1) + ... + X(n), wh ere the X(i)'s are i.i.d, with common c.d.f. F. As an extension of our Tauberian results we consider the proportion of the sample observatio ns which determine the weak behaviour of S-n for fixed n, stemming fro m the case when Fis a subexponential c.d.f. up to the case when - log (F) over bar is a function rapidly varying at infinity. These results provide new insights on the the problem of closure under convolution o perations for large classes of distributions. (C) 1995 Academic Press, Inc.