Distinguishing Stable Probability Measures - Part I: Discrete Time

The problem of classifying a series of observations as coming from one of two or more possible classes or hypotheses has received a great deal of attention in the statistical and engineering literature. In many physical situations, a variety of disturbances corrupt the observations; rather than model each disturbance separately, it is often argued on physical grounds that the disturbances add and are independent, and the central limit theorem is invoked to model this sum using a gaussian distribution. This approach is adequate as long as the sum is not dominated by one or a few of the summands; if one or a few of the summands does dominate the sum, the disturbances can possibly be modeled as a stable distribution, one member of a family of probability distributions which includes the gaussian, by invoking a frequently overlooked generalization of the central limit theorem. 1125 The gaussian distribution has enjoyed great popularity in hypothesis testing because it is analytically tractable and because it is the only stable distribution with finite variance. Although it may be argued that mathematical models with infinite variance are physically inappropriate, this view conveniently overlooks the fact that the gaussian distribution is unbounded, which is also a physically inappropriate mathematical model. The gaussian model may adequately model disturbances over a narrow range of amplitudes; an infinite-variance, stable-distribution model may adequately model disturbances over a larger range of amplitudes.