  Gaussian differential privacy (GDP) is a single-parameter family of privacy
notions that provides coherent guarantees to avoid the exposure of sensitive
individual information. Despite the extra interpretability and tighter bounds
under composition GDP provides, many widely used mechanisms (e.g., the Laplace
mechanism) inherently provide GDP guarantees but often fail to take advantage
of this new framework because their privacy guarantees were derived under a
different background. In this paper, we study the asymptotic properties of
privacy profiles and develop a simple criterion to identify algorithms with GDP
properties. We propose an efficient method for GDP algorithms to narrow down
possible values of an optimal privacy measurement, $\mu$ with an arbitrarily
small and quantifiable margin of error. For non GDP algorithms, we provide a
post-processing procedure that can amplify existing privacy guarantees to meet
the GDP condition. As applications, we compare two single-parameter families of
privacy notions, $\epsilon$-DP, and $\mu$-GDP, and show that all $\epsilon$-DP
algorithms are intrinsically also GDP. Lastly, we show that the combination of
our measurement process and the composition theorem of GDP is a powerful and
convenient tool to handle compositions compared to the traditional standard and
advanced composition theorems.
