"Square roots" of probability distributions

Think about probability distributions supported on the positive integers. Not all of them have a "square root" -- that is, given a random variable X supported on the positive integer, there do not exist independent, identically distributed variables Y₁, Y₂ such that X has the same distribution as Y₁ + Y₂.

You might be wondering why it's natural to refer to this as a "square root". Well, let p_k be the probability of the event X = k. Then the probability generating function for X is

.

Similarly, let q_j be the probability of the event Y = j, and define the probability generating function

.

Let Y, Y₁, Y₂ be equidistributed. Then X is equidistributed with Y₁ + Y₂ if and only if f(z) = g(z)², since addition of independent random variables corresponds to convolution of distributions and multiplication of their generating functions.

Conversely, the random variable X has a square root in this sense if and only if its generating function f(z) has a square root which has a Taylor series at z = 0 with all coefficients positive.

The simplest case is when X has finite support. Then X has a square root if and only if its generating function f(z) is the square of a polynomial. For example, the random variable which takes values 0, 1, 2 with probabilities 1/4, 1/2, 1/4 has a square root; its generating function is

.

But the random variable taking values 0, 1, 2 with probability 1/3 each is not, since 1/3 + z/3 + z²/3 is not a square.

But what about distributions with infinite support? Some of these are easy -- the square root of the Poisson distribution with mean λ is Poisson with mean λ/2. This can easily be seen since the probability generating function of a Poisson(λ) random variable is

.

(In fact, the Poisson distribution is infinitely divisible; in the nomenclature used in this post one might say it has roots of all orders.)

Now consider a random variable X, with geometric distribution with parameter 1/2 supported on {0, 1, 2, 3, ...}; this has P(X = k) = 2^-(k+1). This is a special case of the negative binomial distribution, which is infinitely divisible. We have f(z) = 1/2 + z/2² + z²/2³ + ... = 1/(2-z). So the square root distribution has generating function

and in general the coefficient of zⁿ is, by the binomial theorem,

That binomial coefficient is ${1 \over 2} {2n \choose n}$; we have ${2n \choose n} \sim 4^n/\sqrt{\pi n}$, so the coefficient of zⁿ in our probability generating function, which we'll call q_k, is asymptotic to

In particular, the "square root" distribution decays just a bit faster than the distribution that it's the square root of, the main difference being the additional factor of n^-1/2. This is reasonable, if you think about the process of convolution. We have

p_n = q₀ q_n + q₁ q_n-1 + q₂ q_n-2 + ... + q_n-1 q₁ + q_n q₀

and each of the n terms is roughly 1/(n2ⁿ). This is just another negative binomial distribution. (I actually didn't realize that until I started writing this post; the post was originally titled "what's the name of this distribution?" and then I did some research.)