A short exposition of the use of the generalized Jirina sequential tolerance limits procedure, and the necessary assumptions, is given. A method of obtaining the confidence level of the coverage for various values of the parameters from existing tables is included. Formulae for the expectation and variance of the random sample size are derived and usable approximations obtained. That the random sample size, appropriately scaled by one of the parameters, has an asymptotic distribution as the parameter increases is proved and the Laplace transform of this distribution is found. Also formulae for the asymptotic mean and variance are found and methods for their calculation exhibited. Some applications are given in the following paper.