Fisher Information Matrix of General Stable Distributions Close to the Normal Distribution

Abstract

We investigate behavior of the Fisher information matrix of general stable distributions. DuMouchel (1975, 1983) proved that the Fisher information of characteristic exponent \alpha diverges to infinity as \alpha approaches 2. Nagaev and Shkol'nik (1988) made more detailed analysis and derived asymptotic behavior of the Fisher information matrix of \alpha diverging to infinity as \alpha approaches 2 in the symmetric case. Extending their work in this paper we have obtained behavior of the Fisher information matrix of general stable distributions as \alpha approaches 2 by detailed study of behavior of the corresponding density and its score functions. We clarify the limiting values of the 4*4 Fisher Information matrix with respect to the location \mu, the scale \sigma, the characteristic exponent \alpha and the skewness parameter \beta.