Asymptotic Normality

The exact distribution of a statistic is usually highly complicated and difficult to work with. Hence the need to approximate the exact distribution by a distribution of a simpler form whose properties are more transparent. The limit theorems* of probability theory provide an important tool for such approximations. In particular, the classical central limit theorems* state that the sum of a large number of independent random variables is approximately normally distributed under general conditions (see the section “Central Limit Theorems for Sums of Independent Random Variables”). In fact, the normal distribution* plays a dominating role among the possible limit distributions. To quote from Gnedenko and Kolmogorov [18, Chap. 5]: “Whereas for the convergence of distribution functions of sums of independent variables to the normal law only restrictions of a very general kind, apart from that of being infinitesimal (or asymptotically constant), have to be imposed on the sum mands, for the convergence to another limit law some very special properties are required of the summands.” Moreover, many statistics behave asymptotically like sums of independent random variables (see the fifth, sixth, and seventh sections). All of this helps to explain the importance of the normal distribution* as an asvmototic distribution.