Light tails: all summands are large when the empirical mean is large

It is well known that for a fixed number of independent identically distributed summands with light tail, large values of the sample mean are obtained only when all the summands take large values. This paper explores this property as the number of summands tends to infinity. It provides the order of magnitude of the sample mean for which all summands are in some interval containing this value and it also explores the width of this interval with respect to the distribution of the summands in their upper tail. These results are proved for summands with log-concave or nearly log concave densities. Making use of some extension of the Erdös-Rényi law of large numbers it also explores the forming of aggregates in a sequence of i.i.d. random variables. As a by product the connection is established between large exceedances of the local slope of a random walk on growing bins and the theory of extreme order statistics.