Falling and thinning
Recap
This post is a follow-up to my earlier post on the falling moment generating function
I argued that the FMGF is a natural discrete equivalent to the moment generating function (MGF)
After that blogpost, Oliver Johnson pointed out an even better argument for the FMGF, based on the idea of “thinning”. (Olly is the author of the Substack “Logging the World” and the book “Numbercrunch”, and was my PhD supervisor back in the day.)
Scaling and thinning
All the “GFs” work well with adding up independent random variables – to find the GF of the independent sum
The MGF works particularly well with scaling random variables. By scaling, I just mean multiplying by a constant, so going from
But, while scaling is a natural operation for continuous random variables, it doesn’t really make sense for discrete random variables. (From now on, when I say “discrete”, I’ll specifically mean random variables that take values in the non-negative integers
Instead, for discrete random variables, a more natural operation is thinning. To thin a discrete random variable
where the
Like scaling, thinning reduces the expectation
So, what happens to the FMGF if we thin a random variable. Well, it’s not to difficult to check that we get
where we used the fact that, conditional on
So we see that the way the FMGF behaves under thinning,
Comparing coefficients of
Large numbers
The way that the MGF works so well with adding independent random variables and scaling random variables allows us to prove a very important result called the law of large numbers.
Let
The law of large numbers says that
The law of large numbers can be proved using the MGF. Note that by the properties of the MGF we’ve discussed in this post, we have
of the MGF of
But
Thin numbers
But we can now follow exactly the same argument for discrete random variables, with thinning instead of scaling, and the FMGF instead of the MGF.
Let
The “law of thin numbers” (as Harremoës, Johnson and Kontoyiannis call it) says that
The law of thin numbers can be proved using the FMGF. Note that by the properties of the FMGF we’ve discussed in this post, we have
of the FMGF of
But