Optimal Sub-Gamma Scales for Spectrally Positive Infinitely Divisible Laws

Abstract

Let X be a centered spectrally positive infinitely divisible random variable with finite nonzero variance V. We determine the smallest scale c in a right-sided sub-gamma bound whose quadratic proxy is fixed at V. A probability transform of the normalized Kolmogorov canonical measure produces a nonnegative variable R_X for which log E e^{tX} = V t^2 M_{R_X}(t)/2. The scale problem therefore reduces to comparing the MGF of R_X with that of an exponential law, leading to an exact one-dimensional variational formula. If X is non-Gaussian and has a positive exponential moment, the optimal scale is at least kappa_3(X)/(3V), with equality precisely when R_X belongs to the M-class of Klar and Mueller. Equality in the standard sub-gamma envelope occurs only for the Gaussian law or, in the non-Gaussian case, for centered compound Poisson laws with exponential jumps. For a general compound Poisson law, R_X is the second-order equilibrium distribution of the jump size. Exact calculations for gamma and bounded two-point jumps show that the optimum may occur at the origin, at the positive MGF boundary, or at an interior point.