Counter-examples to an Assertion concerning the Normal Distribution and a New Stochastic Price Fluctuation Model

Consider the following assertion: If X1, ..., X, are random variables with finite second moments and if all non-trivial linear combinations a1X, +... + a,X, have the same distribution except for location and scale, then that distribution must be normal. True or false? The assertion would be true if any pair of the variables were assumed to be stochastically independent, but it is false in general, even for variables assumed to be uncorrelated. In Section I, we demonstrate that the assertion is false via counter-examples. A recent commentary in this Review by Borch [1], Feldstein [4], and Tobin [13] indicates that the truth of the above assertion has gone unquestioned by some eminent economic theorists. Nevertheless, the generality of Tobin's mean-variance approach to risk analysis is not much enhanced by our revelation; the numerous theoretical criticisms of the mean-variance approach (e.g. Borch [1] and Feldstein [4]) still obtain. As a practical matter, our counter-examples merely extend the class of approximating distributions available to portfolio analysts. A more intriguing ramification of our counter-examples is the possibility of a viable alternative to the stable Paretian model of stochastic price fluctuation [7], [8], [2], [3], [10]. The stable Paretian model follows from the assumption of independent stable increments in the log-price process. In Section II, we construct an alternative model with uncorrelated (but dependent) stable increments.