Regenerative process

History

Regenerative processes were first defined by Walter L. Smith in Proceedings of the Royal Society A in 1955.{{Cite journal

Definition

A regenerative process is a stochastic process with time points at which, from a probabilistic point of view, the process restarts itself. 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 1 2 k* 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 k*}

for k ≥ 1. Intuitively this means a regenerative process can be split into i.i.d. cycles.

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

Properties

• By the renewal reward theorem, with probability 1,

::\lim_{t \to \infty} \frac{1}{t}\int_0^t X(s) ds= \frac{\mathbb{E}[R]}{\mathbb{E}[\tau]}.

:where \tau is the length of the first cycle and 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 time

References

References

1. (1967). "An Application of Regenerative Stochastic Processes to a Problem in Inventory Control". Operations Research.
2. (2010). "Introduction to Probability Models".
3. (1955). "Regenerative Stochastic Processes". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.
4. Sheldon M. Ross. (2007). "Introduction to probability models". Academic Press.
5. (2002). "Stochastic Petri Nets".
6. (2003). "Applied Probability and Queues".
7. Sigman, Karl (2009) ''Regenerative Processes'', lecture notes