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.

Regenerative processes were first defined by Walter L. Smith in Proceedings of the Royal Society A in 1955.^[3][4]

A regenerative process is a stochastic process with time points at which, from a probabilistic point of view, the process restarts itself.^[5] 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.^[6] Intuitively this means a regenerative process can be split into i.i.d. cycles.^[7]

When T₀ = 0, X(t) is called a nondelayed regenerative process. Else, the process is called a delayed regenerative process.^[6]

• Renewal processes are regenerative processes, with T₁ being the first renewal.^[5]
• Alternating renewal processes, where a system alternates between an 'on' state and an 'off' state.^[5]
• A recurrent Markov chain is a regenerative process, with T₁ being the time of first recurrence.^[5] This includes Harris chains.
• Reflected Brownian motion is a regenerative process (where one measures the time it takes particles to leave and come back).^[7]

• By the renewal reward theorem, with probability 1,^[8]

${\displaystyle \lim _{t\to \infty }{\frac {1}{t}}\int _{0}^{t}X(s)ds={\frac {\mathbb {E} [R]}{\mathbb {E} [\tau ]}}.}$

where ${\displaystyle \tau }$ is the length of the first cycle and ${\displaystyle R=\int _{0}^{\tau }X(s)ds}$ is the value over the first cycle.

• A measurable function of a regenerative process is a regenerative process with the same regeneration time^[8]