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Stationary regimes for inventory processes

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  • Bardhan, Indrajit
  • Sigman, Karl

Abstract

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 --> [infinity]) Xs [triangle, equals] 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, [theta]sZ [triangle, equals] Z(s + t):t [triangle, equals] 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.

Suggested Citation

  • Bardhan, Indrajit & Sigman, Karl, 1995. "Stationary regimes for inventory processes," Stochastic Processes and their Applications, Elsevier, vol. 56(1), pages 77-86, March.
    • Handle: RePEc:eee:spapps:v:56:y:1995:i:1:p:77-86
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    References listed on IDEAS

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    1. Karl Sigman, 1990. "One-Dependent Regenerative Processes and Queues in Continuous Time," Mathematics of Operations Research, INFORMS, vol. 15(1), pages 175-189, February.
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    Cited by:

    1. Guillemin, Fabrice M. & Mazumdar, Ravi R., 1997. "On pathwise analysis and existence of empirical distributions for G/G/1 queues," Stochastic Processes and their Applications, Elsevier, vol. 67(1), pages 55-67, April.

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