Regenerative process explained

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.^[1] 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.^{[2] [3]}

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 is a regenerative process if there exist time points 0 ≤ T₀ < T₁ < T₂ < ... such that the post-T_k process

• has the same distribution as the post-T₀ process
• is independent of the pre-T_k process

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.

Examples

• Renewal processes are regenerative processes, with T₁ being the first renewal.^[4]
• Alternating renewal processes, where a system alternates between an 'on' state and an 'off' state.^[4]
• A recurrent Markov chain is a regenerative process, with T₁ being the time of first recurrence.^[4] 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]

\lim _{t} (1)/(t) \int ^{t} _{0} X(s) ds = (E [R])/(E [\tau]) .

where

\tau is the length of the first cycle and R = \int ^{\tau} _{0} 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]

Notes and References

1. Book: Ross . S. M. . Renewal Theory and Its Applications . 10.1016/B978-0-12-375686-2.00003-0 . Introduction to Probability Models . 421–641 . 2010 . 9780123756862 .
2. Schellhaas . Helmut. 10.1287/moor.4.1.70. Semi-Regenerative Processes with Unbounded Rewards. Mathematics of Operations Research. 4. 70–78. 1979. 3689240.
3. Smith . W. L. . Wally Smith (mathematician). Regenerative Stochastic Processes . 10.1098/rspa.1955.0198 . Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences . 232 . 1188 . 6–31. 1955 . 1955RSPSA.232....6S .
4. Book: Introduction to probability models. Sheldon M. Ross. 0-12-598062-0. 442. 2007. Academic Press.
5. Book: Peter J. . Haas. Peter J. Haas (computer scientist). 10.1007/0-387-21552-2_6 . Regenerative Simulation . Stochastic Petri Nets . Springer Series in Operations Research and Financial Engineering . 189–273 . 2002 . 0-387-95445-7 .
6. Book: Søren . Asmussen. 10.1007/0-387-21525-5_6 . Regenerative Processes . Applied Probability and Queues . Stochastic Modelling and Applied Probability . 51 . 168–185 . 2003 . 978-0-387-00211-8 .
7. Sigman, Karl (2009) Regenerative Processes, lecture notes