In applied probability

Applied probability

Much research involving probability is done under the auspices of applied probability, the application of probability theory to other scientific and engineering domains...

, a regenerative process is a special type of stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

 that is defined by having a property whereby certain portions of the process can be treated as being statistically independent of each other. This property can be used in the derivation of theoretical properties of such processes.

Definition

A regenerative process is a stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic 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 in to i.i.d. cycles.

Regenerative processes were first defined by W. L. Smith in Proceedings of the Royal Society A in 1955.

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