Regenerative process
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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.[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
- Renewal processes are regenerative processes, with T1 being the first renewal.[5]
- Alternating renewal processes, where a system alternates between an 'on' state and an 'off' state.[5]
- A recurrent Markov chain is a regenerative process, with T1 being the time of first recurrence.[5] This includes Harris chains.
- Reflected Brownian motion is a regenerative process (where one measures the time it takes particles to leave and come back).[7]
Properties
- By the renewal reward theorem, with probability 1,[8]
- where is the length of the first cycle and is the value over the first cycle.
- A measurable function of a regenerative process is a regenerative process with the same regeneration time[8]
References
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