I’m trying to get an intuition for why the superposition of independent renewal processes in equilibrium converges – locally, on small timescales – to a Poisson process. (I mean a general renewal process; that the superposition of Poisson processes yields a Poisson process is a special case of this.)
Let $(N_i(t))_{i=1\ldots n}$ be $n$ independent renewal processes with i.i.d. interarrival times $X$, with CDF $F$ and density $f$, and set $μ := \mathbb{E}(X)$. Suppose the interarrival time $X$ is not arithmetic. I’m interested in the interarrival time of the “merged” process $N(t) = \sum_{i=1}^n N_i(t)$.
Here is what I know:
A key result in renewal theory is that in equilibrium, the excess life of the process $N_i$ has CDF:
$$\mathbb{P}(E(t) \le y) \to \frac{1}{μ}\int_0^y [1-F(x)]\,dx$$
Therefore, in the superposition $N(t)$ in equilibrium, by taking the minimum over $N$ realizations of $X$, the interarrival time $Y$ has CDF:
$$\begin{aligned} \mathbb{P}(Y \le y) &= 1 - \left(1-\frac{1}{μ}\int_0^y[1-F(x)]\,dx\right)^n \\ &= 1-\left(\frac{1}{μ}\int_y^\infty[1-F(x)]\,dx\right)^n \\ \end{aligned}$$
Based on the references below, $N(t)$ converges to a Poisson process for $n\to\infty$, so for large $n$ it should be “approximately Poisson”, so that the interarrival time $Y$ should be “approximately exponential”.
In other words, I think the following holds, but I’d appreciate your help explaining why this seems to hold or how fast the convergence happens:
$$\mathbb{P}(Y \le y) \approx 1-e^{-λy}$$
Based on a few numeric examples I tried out, it seems $λ \approx \frac{n}{μ}$.
References
- Cox, D. R., & Smith, W. L. (1954). On the Superposition of Renewal Processes. Biometrika, 41(1/2), 91. doi:10.2307/2333008 – Original article studying this problem for the first time, and deriving the result. Specifically equations (30)+(31), and section 5 claims: In particular, the interval between successive events tends to be distributed in an exponential distribution, as can also be proved directly from (31). I don’t understand how it can be proved directly from (31) and the paper omits the details.
- Kallenberg (1997), Thm. 14.18 (p. 266) has a proof, it’s short and not giving me a great intuition.
- There is also the Palm–Khintchine theorem for which I’ve not been able to find a proof.
I think I acquired some intuition now, sharing what I learned. They key is to realize that in general, the superposition of many independent renewal processes does not converge to a Poisson process globally, just locally.
Feller Vol II 1968, XI.4 example (a), covers this topic and puts it thus:
The paper Superposition of many independent spike trains is generally not a Poisson process has a good explanation why one cannot in general hope to obtain a Poisson process globally, only on short time scales:
Intuition: For a distribution function $F$ of the interarrival time, in many well-behaved cases, $F(ε)\approx 0$ for $ε \ll μ$. Then $1-F(ε) \approx 1$, and therefore:
$$\begin{aligned} \mathbb{P}(Y>ε) &= \left(1-\frac{1}{μ}\int_0^ε [1-F(x)]\,dx\right)^n \\ &\approx \left(1-\frac{ε}{μ}\right)^n \\ &\approx e^{-εnμ^{-1}} \end{aligned}$$
As an example, consider renewals with a Gamma density and $α=100, β=200$, so that $μ=\frac{1}{2}$: