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History

Regenerative processes were first defined by Walter L. Smith in Proceedings of the Royal Society A in 1955.²³

Definition

A regenerative process is a stochastic process with time points at which, from a probabilistic point of view, the process restarts itself.⁴ 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.⁷

Examples

• Renewal processes are regenerative processes, with T1 being the first renewal.⁸
• Alternating renewal processes, where a system alternates between an 'on' state and an 'off' state.⁹
• A recurrent Markov chain is a regenerative process, with T1 being the time of first recurrence.¹⁰ This includes Harris chains.
• Reflected Brownian motion is a regenerative process (where one measures the time it takes particles to leave and come back).¹¹

Properties

• By the renewal reward theorem, with probability 1,¹²

lim t → ∞ 1 t ∫ 0 t X ( s ) d s = E [ R ] E [ τ ] . {\displaystyle \lim _{t\to \infty }{\frac {1}{t}}\int _{0}^{t}X(s)ds={\frac {\mathbb {E} [R]}{\mathbb {E} [\tau ]}}.} where τ {\displaystyle \tau } is the length of the first cycle and R = ∫ 0 τ X ( s ) d s {\displaystyle 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 time¹³

References

1. Ross, S. M. (2010). "Renewal Theory and Its Applications". Introduction to Probability Models. pp. 421–641. doi:10.1016/B978-0-12-375686-2.00003-0. ISBN 9780123756862. 9780123756862 ↩

2. Schellhaas, Helmut (1979). "Semi-Regenerative Processes with Unbounded Rewards". Mathematics of Operations Research. 4: 70–78. doi:10.1287/moor.4.1.70. JSTOR 3689240. /wiki/Mathematics_of_Operations_Research ↩

3. Smith, W. L. (1955). "Regenerative Stochastic Processes". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 232 (1188): 6–31. Bibcode:1955RSPSA.232....6S. doi:10.1098/rspa.1955.0198. /wiki/Wally_Smith_(mathematician) ↩

4. Sheldon M. Ross (2007). Introduction to probability models. Academic Press. p. 442. ISBN 0-12-598062-0. 0-12-598062-0 ↩

5. Haas, Peter J. (2002). "Regenerative Simulation". Stochastic Petri Nets. Springer Series in Operations Research and Financial Engineering. pp. 189–273. doi:10.1007/0-387-21552-2_6. ISBN 0-387-95445-7. 0-387-95445-7 ↩

6. Asmussen, Søren (2003). "Regenerative Processes". Applied Probability and Queues. Stochastic Modelling and Applied Probability. Vol. 51. pp. 168–185. doi:10.1007/0-387-21525-5_6. ISBN 978-0-387-00211-8. 978-0-387-00211-8 ↩

7. Haas, Peter J. (2002). "Regenerative Simulation". Stochastic Petri Nets. Springer Series in Operations Research and Financial Engineering. pp. 189–273. doi:10.1007/0-387-21552-2_6. ISBN 0-387-95445-7. 0-387-95445-7 ↩

8. Sheldon M. Ross (2007). Introduction to probability models. Academic Press. p. 442. ISBN 0-12-598062-0. 0-12-598062-0 ↩

9. Sheldon M. Ross (2007). Introduction to probability models. Academic Press. p. 442. ISBN 0-12-598062-0. 0-12-598062-0 ↩

10. Sheldon M. Ross (2007). Introduction to probability models. Academic Press. p. 442. ISBN 0-12-598062-0. 0-12-598062-0 ↩

11. Asmussen, Søren (2003). "Regenerative Processes". Applied Probability and Queues. Stochastic Modelling and Applied Probability. Vol. 51. pp. 168–185. doi:10.1007/0-387-21525-5_6. ISBN 978-0-387-00211-8. 978-0-387-00211-8 ↩

12. Sigman, Karl (2009) Regenerative Processes, lecture notes ↩

13. Sigman, Karl (2009) Regenerative Processes, lecture notes ↩