Moving convex functions for classifying exponential families
Draw the graph of a convex function on Rd, move it, or even do an affine transformation in Rd+1 of such a surface. What you obtain is probably not the graph of a function anymore, but a small proportion of it could be the graph of a new convex function. When this process is applied to the convex function which is the log of the Laplace transform of a positive measure in Rd, you may be relating two very different natural exponential families. For instance for d=1 by obtaining −√s from s2 you link the Gaussian laws N(m,1) with the inverse Gaussian distributions. In many cases a probabilistic interpretation of the link can be provided in terms of stopping times of Lévy processes or of random walks on the integers. This is a genuine equivalence relation on the set E of exponential families, and an equivalence class is called an orbit on E. In general, the members of the orbit of a familiar exponential family are quite interesting objects. Through a tool called the variance functions the above affine transformation can be read as a transformation of the variance function: the four orbits generated by the quadratic real variance functions are due to Marianne Mora, Muriel Casalis has described the simple quadratic variance functions in Rd, Abdelhamid Hassairi has found their orbits and Célestin Kokonendji has found the orbits of a rich class in R. This lecture is a tribute to the work made around 1990 by these four graduate students. Some challenging questions in the classification of the exponential families will be mentioned.