An n-variate Farlie-Gumbel-Morgenstern (FGM) copula consists of 2n - n - 1 parameters that express multivariate dependence among random variables. Motivated by the dependence structure of the n-variate FGM copula, we derive the exact range of the n-variate FGM copula's parameter. The exact range of the parameter is given by a closed-form expression under the condition that all parameters take the same value. Moreover, under the same condition, we reveal that the n-variate FGM copula becomes the independence copula for n → ∞. This result contributes to the dependence modeling such as reliability analysis considering dependent failure occurrence.

In this paper, I investigate a property of self-organizing feature map (SOFM) in terms of reference vector density q(x) when probability density function of input signal fed into SOFM is p(x). Difficulty of general analysis on this property is briefly discussed. Then, I employ an assumption (conformal map assumption) to evaluate this property, and it is shown that for equilibrium state, q(x)p(x)s holds. By giving Lyapunov functioin for time evolution of reference vector density q(x) in SOFM, the equilibrium state is proved to be stable in terms of distribution. Comparison of the result with one which is based on different assumption reveals that there is no unique result of a simple form, such as conjectured by Kohonen. However, as there are cases in which these assumptions hold, these results suggest that we can consider a range of the property of SOFM. On the basis of it, we make comparison on this property between SOFM and fundamental adaptive vector quantization algorithm, in terms of the exponent s of the relation q(x)p(x)s. Difference on this property between SOFM and fundamental adaptive vector quantization algorithm, and propriety of mean squared quantization error for a performance measure of SOFM, are discussed.