# Moving convex functions for classifying the exponential families

Draw the graph of a convex function on $\mathbb{R}^d$, move it, or even do an affine transformation in $\mathbb{R}^{d+1}$ 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 $\mathbb{R}^d,$ you may be relating two very different natural exponential families. For instance for $d=1$ by obtaining $-\sqrt{s}$ from $s^2$ 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 $\mathbb{R}^d$, Abdelhamid Hassairi has found their orbits and Célestin Kokonendji has found the orbits of a rich class in $\mathbb{R}$. 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.