# 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.

## 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 ≤ *T*_{0} < *T*_{1} < *T*_{2} < ... such that the post-*T _{k}* process {

*X*(

*T*+

_{k}*t*) :

*t*≥ 0}

- has the same distribution as the post-
*T*_{0}process {*X*(*T*_{0}+*t*) :*t*≥ 0} - is independent of the pre-
*T*process {_{k}*X*(*t*) : 0 ≤*t*<*T*}_{k}

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

When *T*_{0} = 0, *X*(*t*) is called a **nondelayed regenerative process**. Else, the process is called a **delayed regenerative process**.^{[5]}

## Examples

- Renewal processes are regenerative processes, with
*T*_{1}being the first renewal.^{[2]} - Alternating renewal processes, where a system alternates between an 'on' state and an 'off' state.
^{[2]} - A recurrent Markov chain is a regenerative process, with
*T*_{1}being the time of first recurrence.^{[2]}This includes Harris chains. - Reflected Brownian motion is a regenerative process (where one measures the time it takes particles to leave and come back).
^{[6]}

## Properties

- By the renewal reward theorem, with probability 1,
^{[7]}

- 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
^{[7]}

## References

- ↑ Hurter, A. P.; Kaminsky, F. C. (1967). "An Application of Regenerative Stochastic Processes to a Problem in Inventory Control".
*Operations Research*.**15**(3): 467. JSTOR 168455. doi:10.1287/opre.15.3.467. - ↑
^{2.0}^{2.1}^{2.2}^{2.3}^{2.4}Ross, S. M. (2010). "Renewal Theory and Its Applications".*Introduction to Probability Models*. pp. 421–641. ISBN 9780123756862. doi:10.1016/B978-0-12-375686-2.00003-0. Cite error: Invalid`<ref>`

tag; name "Ross" defined multiple times with different content - ↑ Schellhaas, Helmut (1979). "Semi-Regenerative Processes with Unbounded Rewards".
*Mathematics of Operations Research*.**4**: 70–78. JSTOR 3689240. doi:10.1287/moor.4.1.70. - ↑ Smith, W. L. (1955). "Regenerative Stochastic Processes".
*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*.**232**(1188): 6–4. Bibcode:1955RSPSA.232....6S. doi:10.1098/rspa.1955.0198. - ↑
^{5.0}^{5.1}Haas, Peter J. (2002). "Regenerative Simulation".*Stochastic Petri Nets*. Springer Series in Operations Research and Financial Engineering. pp. 189–273. ISBN 0-387-95445-7. doi:10.1007/0-387-21552-2_6. - ↑
^{6.0}^{6.1}Asmussen, Søren (2003). "Regenerative Processes".*Applied Probability and Queues*. Stochastic Modelling and Applied Probability.**51**. pp. 168–185. ISBN 978-0-387-00211-8. doi:10.1007/0-387-21525-5_6. - ↑
^{7.0}^{7.1}Sigman, Karl (2009)*Regenerative Processes*, lecture notes