Regenerative process

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 < 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. 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,

${\displaystyle \underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}{\int }_0^{t}X(s)ds=\frac{\mathbb{E}[R]}{\mathbb{E}[\tau ]}.}$

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