FREE EXPONENTIAL FAMILIES AS KERNEL FAMILIES

Free exponential families have been previously introduced as a special case of the ^-exponential family.We show that free exponential families arise also from the approach analogous to the definition of exponential families by using the Cauchy-Stieltjes kernel 1/(1 -Ox) instead of the exponential kernel exp(0x).We use this approach to re-derive some known results and to study further similarities with exponential families and reproductive exponential models.