regenerative process

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

| last1 = Schellhaas | first1 = Helmut

| doi = 10.1287/moor.4.1.70

| title = Semi-Regenerative Processes with Unbounded Rewards

| journal = Mathematics of Operations Research

| volume = 4

| pages = 70–78

| year = 1979

| jstor = 3689240}}{{Cite journal | last1 = Smith | first1 = W. L. | author-link1 = Wally Smith (mathematician)| title = Regenerative Stochastic Processes | doi = 10.1098/rspa.1955.0198 | journal = Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | volume = 232 | issue = 1188 | pages = 6–31| year = 1955 |bibcode = 1955RSPSA.232....6S }}

A regenerative process is a stochastic process with time points at which, from a probabilistic point of view, the process restarts itself.{{cite book|title=Introduction to probability models|author=Sheldon M. Ross|isbn=0-12-598062-0|page=442|year=2007|publisher=Academic Press}} 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-T_k process {X(T_k + t) : t ≥ 0}

• has the same distribution as the post-T₀ process {X(T₀ + t) : t ≥ 0}
• is independent of the pre-T_k process {X(t) : 0 ≤ t < T_k}

for k ≥ 1.{{Cite book | first1 = Peter J. | last1 = Haas|author-link= Peter J. Haas (computer scientist)| doi = 10.1007/0-387-21552-2_6 | chapter = Regenerative Simulation | title = Stochastic Petri Nets | series = Springer Series in Operations Research and Financial Engineering | pages = 189–273 | year = 2002 | isbn = 0-387-95445-7 }} Intuitively this means a regenerative process can be split into i.i.d. cycles.{{Cite book | first1 = Søren | last1 = Asmussen| doi = 10.1007/0-387-21525-5_6 | chapter = Regenerative Processes | title = Applied Probability and Queues | series = Stochastic Modelling and Applied Probability | volume = 51 | pages = 168–185 | year = 2003 | isbn = 978-0-387-00211-8 }}

When T₀ = 0, X(t) is called a nondelayed regenerative process. Else, the process is called a delayed regenerative process.

• 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).

• By the renewal reward theorem, with probability 1,Sigman, Karl (2009) Regenerative Processes, lecture notes

::$\backslash lim\_\{t\; \backslash to\; \backslash infty\}\; \backslash frac\{1\}\{t\}\backslash int\_0^t\; X(s)\; ds=\; \backslash frac\{\backslash mathbb\{E\}[R]\}\{\backslash mathbb\{E\}[\backslash tau]\}.$

:where $\backslash tau$ is the length of the first cycle and $R=\backslash int\_0^\backslash tau\; X(s)\; ds$ is the value over the first cycle.

• A measurable function of a regenerative process is a regenerative process with the same regeneration time

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{{Stochastic processes}}

Category:Stochastic processes