Assume we have two random variables $G_1$ and $G_2$ that both follow a geometric distribution of type B (so RVs can assume 0) with success probability $p = p_1 = p_2$ that describe the interarrival times of independent arrival processes.
From Kingman, we know that the superposition of poisson point processes (independent, memoryless, negative exponentially distributed interarrival times) again results in a poisson point process.
Since in the scenario above, the processes generated by $G_1$ and $G_2$ are both memoryless and independent and the geometric distribution is the discrete analogon to the continuous exponential distribution, why does the superposition of these processes not result in a process whose interarrival times follow the geometric distribution?
What property of the geometric distribution destroys this behavior?
Below is a quick R snippet that simulates the superposition of two geometrically distributed RVs.
nSamples <- 5e6 s1 <- cumsum(rgeom(nSamples, prob = 1/100)) s2 <- cumsum(rgeom(nSamples, prob = 1/100)) minMax <- min(max(s1), max(s2)) s1 <- s1[s1 < minMax] s2 <- s2[s2 < minMax] s3 <- diff(sort(c(s1, s2))) data.table(x = s3)[, .(n = .N), by = x][order(x)][, pct := n / sum(n)][1:10]
The resulting distribution is not strictly monotonically decreasing.
x n pct 1: 0 149315 0.01493716 2: 1 195471 0.01955451 3: 2 191880 0.01919528 4: 3 187642 0.01877131 5: 4 183838 0.01839077 6: 5 180202 0.01802703 7: 6 177592 0.01776593 8: 7 173753 0.01738189 9: 8 170504 0.01705686 10: 9 166771 0.01668342 ...