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Motivated by Yoshida's martingale expansion theory, we study the limit of thejoint distribution of a multidimensional Generalized Tempered Stable (GTS)process and its quadratic covariation process when the stable index tends totwo. Under a proper scaling, the GTS processes converge to a Brownian motionthat is a stable process with stable index two. We renormalize their quadraticcovariation processes so that they have a nondegenerate limit distribution. Weshow that the limit is a stable process with stable index one and isindependent of the limit Brownian motion of the GTS processes.

A limit theorem for generalized tempered stable processes and their quadratic variations with stable index tending to two