Regeneration Times and Regenerative Cycles

What Regeneration Means

A regenerative model divides a process into successive cycles so that, at the end of each cycle, its future starts afresh. This is stronger than saying that cycles have similar shapes: the future must have the same law as a fresh process and must be independent of the past. When this structure holds, questions about a long path can often be organized around a sequence of cycle records.

Let \((\Omega,\mathcal{F},\mathbb{P})\) be a probability space, let \(S\) be a finite or countable state space, and let \((X_t)_{t\ge0}\) be an \(S\)-valued process in continuous time. Let \((\mathcal{F}_t)_{t\ge0}\) be a filtration to which the process is adapted. A random time \(T\) is a stopping time if \(\{T\le t\}\in\mathcal{F}_t\) for every \(t\ge0\). Write \(\mathcal{F}_T\) for the information available up to \(T\). We assume the path-valued random variables below are measurable and that the times considered are finite almost surely.

The equality of laws refers to the entire future path, not just the state at one time. Independence means that for every event \(A\in\mathcal{F}_{T_k}\) and every measurable set \(H\) of future paths, the probability that \(A\) occurs and the shifted path belongs to \(H\) is \(\mathbb{P}(A)\mathbb{P}((X_t)_{t\ge0}\in H)\). In particular, the future is not merely independent of a summary of the past; it is independent of all information in \(\mathcal{F}_{T_k}\).

Define the length of cycle \(k\) by \(L_k=T_k-T_{k-1}\), for \(k\ge1\). A cycle record may contain \(L_k\) and the path observed within the cycle, measured relative to its start. A convenient convention is to record the path on \(0\le u<L_k\), assigning the endpoint \(T_k\) to the next cycle. If the endpoint itself carries information needed to describe a cycle, include that information explicitly and use the same convention for every cycle.

Regeneration and the Cycle Records

Regeneration gives a precise reason that cycle records can be treated as repeated independent observations. To state the implication carefully, assume that the rule identifying the next regeneration time is the same after every regeneration: applying it to the shifted future identifies the next cycle in that shifted path. Let \(C_{k+1}\) be the resulting record of the cycle from \(T_k\) to \(T_{k+1}\). Suppose the cycle record is a measurable function of the shifted path up to its next regeneration time, and the law of that record in a fresh process is a fixed probability distribution \(\nu\).

Proof. The shifted future at \(T_k\) has the fresh-process law and is independent of \(\mathcal{F}_{T_k}\). Applying the same measurable cycle-extraction rule to that future shows that \(C_{k+1}\) has law \(\nu\) and is independent of \(\mathcal{F}_{T_k}\). Each earlier record \(C_1,\ldots,C_k\) is determined by the path up to \(T_k\), so it is \(\mathcal{F}_{T_k}\)-measurable. For measurable sets \(A_1,\ldots,A_{k+1}\) of cycle records, independence therefore gives \(\mathbb{P}(C_1\in A_1,\ldots,C_k\in A_k,C_{k+1}\in A_{k+1})=\mathbb{P}(C_1\in A_1,\ldots,C_k\in A_k)\nu(A_{k+1})\). Starting with \(k=0\) and applying this identity successively factors the joint probability as \(\prod_{j=1}^{k+1}\nu(A_j)\). This is the definition of independence, and the same law \(\nu\) for each record gives identical distribution. \(\square\)

The proposition uses both parts of regeneration. Independence from the past gives independence of the records; the common shifted-future law and common extraction rule give identical distribution. The converse is also useful when building models.

Proof. Fix a cycle boundary \(T_k\). The shifted future from \(T_k\) is formed by concatenating records \(C_{k+1},C_{k+2},\ldots\). Because the records are independent and identically distributed, this sequence has the same law as \(C_1,C_2,\ldots\). The shifted path, being formed from that sequence by the same concatenation rule, consequently has the same law as the process started at time \(0\). The information up to \(T_k\) is determined by the first \(k\) records. Independence of the records implies that the sequence from \(C_{k+1}\) onward is independent of those first \(k\) records, hence of \(\mathcal{F}_{T_k}\) when that filtration contains no additional information about future records. Thus the shifted path is independent of the past. The boundary times are stopping times by assumption, so they satisfy the definition of regeneration times. \(\square\)

What to Verify in a Regenerative Model

When proposing regeneration times, check the definition rather than relying on the visual appearance of repeated cycles. In particular, identify the filtration, verify that the times are stopping times, and specify exactly what is meant by the future process after a boundary. If the state accumulates a count or another quantity that cannot restart, determine whether the intended property is instead regeneration of recentered increments or of a cycle-relative path.

A common mistake is to call renewal epochs regeneration times for the absolute renewal count because the interarrival times after each epoch are fresh. The calculation \(N(T_n+0)=N(T_n)=n\), while \(N(0)=0\), shows why that claim is false under the shifted-process definition: even at time zero, the shifted absolute path does not have the fresh-process value. The correction is to use \(N(T_n+t)-N(T_n)\), or to define cycle records using within-cycle counts. The distinction matters whenever the process retains accumulated history in its state.

Finally, regeneration is not the same as stationarity. Regeneration asserts independence and a repeated law after selected times; it does not assert that the distribution of \(X_t\) is constant in \(t\). Nor does the definition by itself give a long-run average formula. Cycle lengths and cycle rewards provide the structure for such calculations, but the corresponding limit theorems require their own hypotheses and arguments.