Regenerative process

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Regenerative process

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.

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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]

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.[5] 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.[6] Intuitively this means a regenerative process can be split into i.i.d. cycles.[7]

When T0 = 0, X(t) is called a nondelayed regenerative process. Else, the process is called a delayed regenerative process.[6]

Examples

Properties

where is the length of the first cycle and is the value over the first cycle.

References

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