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Let $J_n = \sum_{i=1}^n S_i$ be a renewal process with gamma distributed interval times $S_i \sim Gamma(\alpha,\beta)$.

Consider the value $J_k$ of the first event when the process passes a certain point $t_{boundary}$.

What is the distribution of $t = J_k - t_{boundary}$? More specifically what is the distribution in the limit $t_{boundary} \to \infty$?

Example computation/simulation of the process and the distribution for $S_i \sim Gamma(\alpha = 5,\beta = 20)$ and $t_{boundary} = 10$

example

alpha = 5
beta = 20

sim = function(x = 10) {
  t = 0
  ### keep adding gamma distributions
  ### untill t is above some boundary point
  while(t<x) {
    dt = rgamma(1,alpha,beta)
    t = t+dt
  }
  return(t-x) ### return the difference with the boundary point
}

set.seed(1)
time = replicate(10^4,sim())  
hist(time, freq = 0, breaks = seq(0,max(time)+0.2,0.05))
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1 Answer 1

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We can argue that the interval length of the last interval $S_k$ is distributed as $f(t) \sim Gamma(\alpha+1,\beta)$. That is, the $\alpha$ parameter is one more than for the unconditional distribution of interval times $S$. See for a motivation: Intuitive explanation of paradoxical interval times distribution

If the interval had size $S_k$, we can imagine the point $t_{boundary}$ to be uniform distributed within that interval. Then the distribution of $t$ is like a compound distribution

$$t \sim U(0,S_k) \\ S_k \sim Gamma(\alpha+1,\beta)$$

This is related to the integral

$$g(t) = \int_t^\infty \frac{1}{s} f(s,\alpha+1,\beta) ds$$ and we can express this in terms of the cumulative distribution of a Gamma distribution

$$g(t) = \frac{\beta}{\alpha} (1-F(t,\alpha,\beta))$$

Here is a code that adds this distribution to the histogram plot

ts = seq(0,max(time),0.01)
lines(ts,beta/alpha*(1-pgamma(ts,alpha,beta)))

histogram with distribution added


More generally the solution is

$$g(t) = \frac{(1-F(t,\alpha,\beta))}{E[S]}$$

as mentioned in How to calculate average length of adherence to vegetarianism when we only have survey data about current vegetarians?

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