THE LINDSAY TRANSFORM OF NATURAL EXPONENTIAL-FAMILIES

Let mu be an infinitely divisible positive measure on R. If the measure rho(mu) is such that x-2[rho(mu(dx)-rho(mu({0})delta0(dx)] is the Levy measure associated with mu and is infinitely divisible, we consider for all positive reals alpha and beta the measure T(alpha,beta)(mu) which is the convolution of mu(alpha) and rho(mu)beta. For example, if mu is the inverse Gaussian law, then rho(mu) is a gamma law with parameter 3/2. Then T(alpha,beta)(mu) is an extension of the Lindsay transform of the first order, restricted to the distributions which are infinitely divisible. The main aim of this paper is to point out that it is possible to apply this transformation to all natural exponential families (NEF) with strictly cubic variance functions P. We then obtain NEF with variance functions of the form square-root DELTAP(square-root DELTA), where DELTA is an affine function of the mean of the NEF. Some of these latter types appear scattered in the literature.