Pre-publicaciones Del Seminario Matematico 2005 the Simplest Example of a Normal Asymptotic Expansion the Simplest Example of a Normal Asymptotic Expansion

1. INTRODUCTION. In his 1974 book entitled " The Life and Times of the Central Limit Theorem, " Adams [1] describes this theorem as " one of the most remarkable results in all of mathematics " and " a dominating personality in the world of probability and statistics. " More than three decades later, his description is not only still pertinent but has also been corroborated and reinforced by developments in different branches of knowledge. In fact, the central limit theorem owes much of its importance to its proven application well beyond the field of probability. Diverse problems arising in economics, engineering, the social sciences, medicine, physics, chemistry, and other areas can be modelled in such a way that the central limit theorem comes into play. Empirically, one observes that a great many natural phenomena, such as the heights of individuals in a given population, obey an approximately normal distribution, that is, a symmetric bell-shaped distribution with scores more concentrated in the middle than in the tails (see Figure 1). One explanation suggested for this is that these phenomena are sums of a large number of independent random effects, none of which is predominant. Actually, the classical version of the central limit theorem asserts that the sum of many independent random variables is asymptotically normally distributed provided that each summand is small with high probability.