Definition of heavy-tailed distribution:

The distribution of a random variable \(X\) with distribution function \(F\) is said to have a heavy right tail if the moment generating function of \(X\), \(M_X(t)\) is infinite for all \(t > 0\).

This implies that:

\begin{equation} \forall t > 0, \mathbb{E}[e^{tX}] = \int_{-\infty}^{\infty} e^{tx} dF(x) = \infty \end{equation}

and given that \(dF(x) = f(x)dx\) this is reducible to:

\begin{equation} \int_{-\infty}^{\infty} e^{tx} f(x)dx = \infty \end{equation}

The distribution of prime numbers is heavy-tailed:

Due to the Prime Number Theorem, given the set of primes \(\mathbb{P} \subset \mathbb{N}\) for large \(N\) we have:

\begin{equation} \frac{\pi(N)}{N} \sim \frac{1}{\ln(N)} \end{equation}

\begin{equation} \forall x \in [1,N], P(N \in \mathbb{P}) \leq P(x \in \mathbb{P}) \end{equation}

and since there is at most one prime number in each interval \((n,n+1]\) we have:

\begin{equation} \frac{1}{N \cdot \ln(N)} \leq P(N \in \mathbb{P}) \leq \frac{1}{\ln(N)} \end{equation}

so if we define:

\begin{equation} f(x) := P(x \in \mathbb{P}) \end{equation}

we find:

\begin{equation} \forall t > 0, \int_{2}^{\infty} e^{tx} f(x)dx \geq \lim_{N \to \infty} \int_{2}^{N} e^{tx} \cdot \frac{1}{x \cdot \ln(N)} dx = \infty \end{equation}

so the prime numbers have a heavy-tailed distribution.

References:

  1. Doron Zagier. Newman’s short proof of the Prime Number Theorem. The American Mathematical Monthly, Vol. 104, No. 8 (Oct., 1997), pp. 705-708

  2. Embrechts P.; Klueppelberg C.; Mikosch T. (1997). Modelling extremal events for insurance and finance. Stochastic Modelling and Applied Probability. 33. Berlin: Springer