Gamma Random Variable Confusion

Question

I know that the gamma random variable can be thought of as an extension to the exponential random variable. The exponential random variable measures the time it takes for one event to appear, while the gamma random variable measures the time it takes for k events to appear.

We can then think of gamma as $$\Gamma \sim X_1 + X_2 + \cdots + X_k $$ where each of the $X_i$'s is a independent exponential random variable.

Here's is where my confusion begins: If $X_1$ is the time for let's say the first earthquake to occur in country A and $X_2$ is the time for the second earthquake to occur in the same country, then $X_1+X_2$ is obviously gamma because the 2nd earthquake cannot occur before the first. We are waiting for two separate, independent events to happen. We start the "clock" for $X_2$ right after the first earthquake.

However, let's consider this scenario. Let $X_1$ be the time for first earthquake to occur in country A and $X_2$ is the time for the second earthquake to occur in the country B. Because these are two different places, we can don't necessarily start the "clock" after one earthquake happens. Is $X_1+X_2$ still measured by a gamma random variable?

So which interpretation of the gamma random variable is correct?

Answer 1

Its more about your model than about gamma random variables. If you set $\Gamma_A$ to be the time of the second earthquake to occur in country A, then you are not capturing the whole system, since an earthquake in country B also affects country A. The best interpretation would be to look at $\Gamma_W$, the hole world instead of only one country.

But if you want to understand when an earthquake will happen and also where then you need to introduce a counting process. Indeed, it was done by Alan G. Hawkes and this type of process is called Hawkes Processes. It is an active research area. The main idea is that the occurence of an earthquake in country A increases the probability (with some proportionality constant) that an earthquake will happens in country B. Then, we construct one timeline for each country, but with an underlying process that relates the history of earthquakes of the whole world.