Renewal Processes

The Poisson process models events (order arrivals, defaults, trades) whose gaps between events are exponentially distributed — a specific, memoryless shape. A renewal process asks the more general question: what if the gaps between events follow some distribution, not necessarily exponential, as long as each gap is drawn independently from the same distribution as the others? Order interarrival times that cluster around a typical spacing (rather than the exponential's characteristic mix of very short and occasionally long gaps) are a renewal process but not a Poisson process — the Poisson process is really just the special case of a renewal process with exponential gaps.

The structure

A renewal process is a sequence of event times $T_1, T_2, T_3, \dots$ where the gaps $X_i = T_i - T_{i-1}$ are independent and identically distributed random variables (with $T_0 = 0$). Because every gap resets the process afresh — the future depends only on "a new gap starts now," not on how long the process has already been running — the process "renews" itself at every event, which is where the name comes from. The count of events by time $t$, written $N(t)$, is the quantity of practical interest, and a key long-run result (the elementary renewal theorem) says that the average rate of events settles down to $1/\mathbb{E}[X]$ — the reciprocal of the average gap length — no matter what specific shape the gap distribution has, as long as it has a finite mean.

A quick numeric illustration: if trade executions arrive with gaps averaging 5 seconds (whatever their specific distribution — could be tightly clustered around 5 seconds, or a mix of short and long gaps that averages to 5), the long-run rate of executions settles to $1/5 = 0.2$ per second, or 12 per minute — a fact that depends only on the average gap, not its full shape.

What this means in practice

Renewal processes are the natural model whenever events repeat with a stable but not-necessarily-exponential gap structure: maintenance and replacement cycles, order reposting in an execution algorithm, or any repeating cycle where the classic memoryless assumption of the Poisson process is too restrictive to fit the data. The renewal reward theorem (see The Renewal Reward Theorem) builds directly on this framework to compute long-run average rates of cost or profit, not just event counts, using exactly the same "average per cycle divided by average cycle length" logic.