Before revision:  In the classical Wiener setting, this limit is an inverse Gaussian \quad  L\'evy process. By continuous mapping, analogous results follow for any functional, such as a derivative payoff or price, of our volatility paths. As an example, the classical Heston log-price limit is shown to be very closely related, but not equivalent, to the normal-inverse Gaussian L\'evy process. This exposes the origin of the marginal connection between these two processes revealed in Mechkov (2015) .
After revision:  In the classical Wiener setting, this limit is an inverse Gaussian \quad L\'evy subordinator .