Regenerative process

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Regenerative processes have been used to model problems in inventory control. The inventory in a warehouse such as this one decreases via a stochastic process due to sales until it gets replenished by a new order.[1]

In applied probability, a regenerative process is a class of stochastic process with the property that certain portions of the process can be treated as being statistically independent of each other.[2] This property can be used in the derivation of theoretical properties of such processes.

History

Regenerative processes were first defined by Walter L. Smith in Proceedings of the Royal Society A in 1955.[3][4]

Definition

A regenerative process is a stochastic process with time points at which, from a probabilistic point of view, the process restarts itself.[2] These time point may themselves be determined by the evolution of the process. That is to say, the process {X(t), t ≥ 0} is a regenerative process if there exist time points 0 ≤ T0 < T1 < T2 < ... such that the post-Tk process {X(Tk + t) : t ≥ 0}

  • has the same distribution as the post-T0 process {X(T0 + t) : t ≥ 0}
  • is independent of the pre-Tk process {X(t) : 0 ≤ t < Tk}

for k ≥ 1.[5] Intuitively this means a regenerative process can be split into i.i.d. cycles.[6]

When T0 = 0, X(t) is called a nondelayed regenerative process. Else, the process is called a delayed regenerative process.[5]

Examples

Properties

\lim_{t \to \infty} \frac{1}{t}\int_0^t X(s) ds= \frac{\mathbb{E}[R]}{\mathbb{E}[\tau]}.
where \tau is the length of the first cycle and R=\int_0^\tau X(s) ds is the value over the first cycle.
  • A measurable function of a regenerative process is a regenerative process with the same regeneration time[7]

References

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  7. 7.0 7.1 Sigman, Karl (2009) Regenerative Processes, lecture notes