| Popis: |
Let X be a real random variable with cumulative distribution function F on R. Assume that the moment generating function Φ(t) := ∫∞−∞etxdF(x) is finite in a right neighborhood of 0. As a first topic treated in this paper, we consider distribution functions such that −log F(x−) = sup (xy − log Φ(y), y ≻ 0)(1 + o(1)) as x → ∞, where F(x −) := (1 − F)(x −). The classical Chernoff Inequality is shown to be extended into this tail equivalent statement in many cases, including many infinitely divisible distributions and most of the distribution functions used in statistics. We explore various Tauberian Theorems in this direction with a special attention to the case when −log F is a function rapidly varying at infinity; this latter case yields to an extension of Stirling′s formula to a large class of functions. Set Sn = X1 + · · · + Xn, where the Xi′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 observations which determine the weak behaviour of Sn for fixed n, stemming from the case when F is a subexponential c.d.f. up to the case when −log F is a function rapidly varying at infinity. These results provide new insights on the the problem of closure under convolution operations for large classes of distributions. |
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