Limit laws in a stochastic model for profit from units sold under pro rata warranty

A stochastic model is developed describing the profit/loss experienced by a firm selling units under a simple pro rata warranty based on the life of the unit. The expected profit and profit variance are computed for this model, and various limit theorems are developed under two scenarios: (1) that expected profit due to the exercise of the warranty (as opposed to the purchase of the initial unit) is zero, and (2) that a constant expected net profit is realized each time the warranty is exercised. It is shown that for (1), scale mixtures of the normal distribution arise as the limiting distributions and for (2), the exponential distribution is the limiting distribution.