The Accuracy of the Gaussian Tail and Dual Dirac Model in Jitter Histogram and Probability Density Functions

Jitter can be generally modeled as a superposition of an unbounded random component that follows a Gaussian distribution and a bounded deterministic component. It is usually assumed that the probability distribution of total jitter for very large values of jitter tends to a purely Gaussian distribution. The standard deviation of this Gaussian distribution is identical to the standard deviation of the random component and the mean value of the distribution is related to the deterministic component. A mathematical justification for this assumption is, however, lacking in the literature. In this work, a general asymptotic expression is derived for the tail of the total jitter distribution. It is shown that, to first order, the tail of the distribution can be expressed as a Gaussian function divided by a power function of jitter. Asymptotic expressions for the cumulative distribution function and the Q-scale are also derived. The implications of these results for the accuracy of broadly accepted tail fitting routines are discussed.