A definition of the coupled-product for multivariate coupled-exponentials

• A generalized product is defined to factor multivariate coupled exponentials.
• The coupling is defined independent of the power and dimensions of the variable.
• The output dimension of the generalized product is the sum of the input dimensions.

The coupled-product and coupled-exponential of the generalized calculus of nonextensive statistical mechanics are defined for multivariate functions. The nonlinear statistical coupling is indexed such that κd=κ/1+dκκd=κ/1+dκ, where d is the dimension of the argument of the multivariate coupled-exponential. The coupled-Gaussian distribution is defined such that the argument of the coupled-exponential depends on the coupled-moments but not the coupling parameter. The multivariate version of the coupled-product is defined such that the output dimensions are the sum of the input dimensions. This enables construction of the multivariate coupled-Gaussian from univariate coupled-Gaussians. The resulting construction forms a model of coupling between distributions, generalizing the product of independent Gaussians.