Stationary regimes for inventory processes

The inventory equation, Z(t) = X(t) + L(t), where X = X(t):t ≥ 0 is a given netput process and L(t):t ≥ 0 is the corresponding lost potential process, is explored in the general case when X is a negative drift stochastic process that has asymptotically stationary increments. Our results show that if (as s → ∞) Xs ≜ X(s + t) − X(s):t ≥ 0 converges in some sense to a process X∗ with stationary increments and negative drift, then, regardless of initial conditions, θsZ ≜ Z(s + t):t ≜ 0 converges in the same sense to a stationary version Z∗. We use coupling and shift-coupling methods and cover the cases of convergence in total variation and in total variation in mean, as well as strong convergence in mean. Our approach simplifies and extends the analysis of Borovkov (1976). We remark upon an application in regenerative process theory.