In probability theory, the term subexponential distribution has historically been used for a distribution whose CDF $F(x)$ satisfies the relation $$ n(1-F(x)) \sim 1 - F^{*n}(x) $$ for any $n \ge 1$ as $x \to \pm \infty$ (depending on the context), where $F^{*n}(x)$ is the CDF of the $n$-fold additive convolution. The interpretation is that for large $x$, $F(x) \approx 1-\epsilon$ and so $$ n(1-(1-\epsilon)) = n\epsilon \approx 1 - (1-\epsilon)^n $$ and the latter represents the maximum of $n$ independent copies of the distribution, so we can say that $$ \mathbb{P}[X_1 + X_2 + \ldots + X_n > x] \sim \mathbb{P}[\max\{X_1,X_2,\ldots,X_n\}>x] $$ for large $x$, or specifically that it is a distribution such that if the sum of $n$ independent copies is large, it is likely due to the contribution of one particularly large jump. So subexponential is a type of large-tailed distribution, subject to extreme events.

On the other hand, a subgaussian distribution is one whose moment generating function does not grow faster than the Gaussian-like function, so $$ \mathbb{E}[e^{\lambda(X-\mathbb{E}[X])}] \le e^{\lambda^2\sigma^2/2} $$ for all $\lambda$ and some $\sigma^2 >0$. This guarantees tails that are no larger than Gaussian. If we relax this so that it only holds in a neighborhood of zero, specifically that $$ \mathbb{E}[e^{\lambda(X-\mathbb{E}[X])}] \le e^{\lambda^2\sigma^2/2}, \ \ \ \ \ |\lambda| < 1/\alpha $$ then the tail bounds become $$ \mathbb{P}[|X-\mathbb{E}[X]|>t] \le 2e^{-t/2\alpha}, \ \ \ \ \ t> \sigma^2/\alpha $$ It seems that in some circles, these distributions are now being referred to as subexponential. See, for instance, https://www.stat.berkeley.edu/~bartlett/courses/2013spring-stat210b/notes/4notes.pdf, and other top hits on "subexponential distribution".

Frustratingly, these conditions are quite opposite of one another: one implies heavy tails, the other light tails. The latter seems more appropriate, meaning "tails no heavier than exponential". My question is, what is the historical background that led to calling the former subexponential?


In the known survey Subexponential distributions by Goldie and Klüppelberg, we find this:

"The name arises from one of their properties, that their tails decrease more slowly than any exponential tail".

So, the logic behind this term seems to be that the rate of decrease of the tails of such distributions is less than the rate of decrease of the tail of any exponential distribution.

I am not sure if this logic is compelling. I guess people did not think much about the choice of the term.

So far, there seems to be no written record like this: "Let us refer to the distributions satisfying this condition as subexponential." One of the earliest papers in the field, by Teugels, says just this: "The distribution $F$ is said to belong to the subexponential class $\mathscr S$ if [...]". I guess the term first appeared during a rather casual presentation or conversation, and then took root.


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