Differences between heavy tail and fat tail distributions

I thought heavy tail = fat tail, but some articles I read gave me a sense that they aren't.

One of them says: heavy tail means the distribution have infinite jth moment for some integer j. Additionally all the dfs in the pot-domain of attraction of a Pareto df are heavy-tailed. If the density has a high central peak and long tails, then the kurtosis is typically large. A df with kurtosis larger than 3 is fat-tailed or leptokurtic. I still don't have a concrete distinction between these two (heavy tail vs. fat tail). Any thoughts or pointers to relevant articles would be appreciated.

• Great question. There are a bunch of other tail descriptors that appear to be somewhat interchangeable at first glance. In particular, long-tailed (which is sometimes used interchangably with heavy-, fat-, and right-tail), if you take the first sentence of the wikipedia article on face-value, seems to be a super-set of fat- and heavy- tails (as more rigidly defined on their own pages). Jul 27, 2012 at 3:50
• I encountered a distribution with wild outliers (weekly % change of S&P 500) and got interested in this topic. There are cases where the MGF integral does not converge, but all of the moments exist. For the stock data, a t-distribution with 3 degrees of freedom seems to fit (except for skew). Aug 24, 2014 at 10:42

I would say that the usual definition in applied probability theory is that a right heavy tailed distribution is one with infinite moment generating function on $(0, \infty)$, that is, $X$ has right heavy tail if $$E(e^{tX}) = \infty, \quad t > 0.$$ This is in agreement with Wikipedia, which does mention other used definitions such as the one you have (some moment is infinite). There are also important subclasses such as the long-tailed distributions and the subexponential distributions. The standard example of a heavy-tailed distribution, according to the definition above, with all moments finite is the log-normal distribution.

It may very well be that some authors use fat tailed and heavy tailed interchangeably, and others distinguish between fat tailed and heavy tailed. I would say that fat tailed can be used more vaguely to indicate fatter than normal tails and is sometimes used in the sense of leptokurtic (positive kurtosis) as you indicate. One example of such a distribution, which is not heavy tailed according to the definition above, is the logistic distribution. However, this is not in agreement with e.g. Wikipedia, which is much more restrictive and requires that the (right) tail has a power law decay. The Wikipedia article also suggests that fat tail and heavy tail are equivalent concepts, even though power law decay is much stronger than the definition of heavy tails given above.

To avoid confusions, I would recommend to use the definition of a (right) heavy tail above and forget about fat tails whatever that is. The primary reason behind the definition above is that in the analysis of rare events there is a qualitative difference between distributions with finite moment generating function on a positive interval and those with infinite moment generating function on $(0, \infty)$.

• thanks for your answer. I now have a better understanding. Could you elaborate on your last sentence: "the analysis of extremes there is a qualitative difference between distributions with finite moment generating function on a positive interval and those with infinite moment generating function on (0,∞). " ? May 14, 2011 at 15:29
• @Melon, sure. First, I have edited "extremes" to "rare events", which is more appropriate I believe. What I was referring to, in particular, is that you can use the exponential change of measure technique if you have a light tail (that is, not a heavy tail) and you need other tools, and get different kinds of results, if the tail is heavy. A reference is Chapter XIII in Applied Probability and Queues.
– NRH
May 14, 2011 at 18:25
• Why are you talking about right tails on braces? Aug 5, 2020 at 14:29

NN Taleb, P Cirillo (2019) address this directly in Branching epistemic uncertainty and thickness of tails where they state:

From the point of view of extreme value statistics, both the Gamma and the Lognormal are heavy-tailed distributions, meaning that their right tail goes to zero slower than an exponential function, but not "true" fat-tailed, i.e. their tail decreases faster than a power law [31]. From the point of view of extreme value theory, both distributions are in the maximum domain of attraction of the Gumbel case of the Generalized Extreme Value distribution [9], [14], and not of the Fréchet one, i.e. the proper fat-tailed case. As a consequence, the moments of these distributions will always be finite.

• Since the tail is a property of a distribution, it's hard to determine in the context of this question what you're claiming with "the difference [between heavy and fat tailed distributions] is distribution dependent." I just can't make sense of that.
– whuber
Jul 12, 2020 at 17:07

First there can be left tails and right tails, then long tails and short tails. A short-tailed distribution can be thought of a having a finite range, called its support. And a long tail has semi-infinite support in that direction. For right-tail heaviness one does a comparison of either survival functions (RVs) or complimentary cumulative density functions (1-CDF), typically by examining the logarithm of ratio of two different distributions. In general, heavy tailed means heavier than the exponential distribution, and light tailed means lighter than that. A subset of heavier-tailed distributions are called "fat-tailed." From a historical perspective it is most likely that the concept of a fat tail relates to the type I Pareto distribution, i.e., $$$$\label{eq:PD} \text{PD}(t; \alpha, \beta)= \dfrac{\alpha}{t} \left(\dfrac{\beta}{t}\right) ^{\alpha } \theta(t-\beta)\;,$$$$

where $$\alpha$$ is the shape parameter, $$\beta$$ is a scale parameter and $$\theta(\cdot)$$ is the unit step function such that $$\theta(t-\beta)$$ is the unit step function time-delayed by $$\beta$$, and is used to make a product that is non-zero only when $$t> \beta$$.

From Juran, "The Pareto Principle (Sic, 80-20 rule) gets its name from the Italian-born economist Vilfredo Pareto (1848-1923), who observed that a relative few people held the majority of the wealth (20%) – back in 1895. Pareto developed logarithmic mathematical models to describe this non-uniform distribution of wealth and the mathematician M.O. Lorenz developed graphs to illustrate it."

Next, let us consider the attitude toward wealth at that time. Renzaho quotes Grivetti saying "At the turn of the 20th century in North America obesity was admired; wealthy consumers exhibited their wealth around their waist. Fat cheeks and ample stomachs were visual cues that individuals were healthy, not infected with the dreaded slim tuberculosis. Photographs of American executives taken during the late 19th and early 20th centuries reveal that dietary intakes of wealthy gentlemen regularly exceeded calories expended."

The history of those times lends considerable weight to those words. From the US 1910 census, tuberculosis, A.K.A, "consumption," which literally eats the body from the inside out, had been endemic for decades and was responsible for approximately 15 deaths per 1,000 inhabitants per year, or, if you wish, approximately 30 times the annual death rate from SARS-CoV-2. Thus, the more modern concept of slimness being healthy was not plausibly in vogue.

Next, the term "fat cats" came to describe wealthy political donors in approximately 1920 or earlier and Pareto's work was first translated into English in 1916. Wesolowski et al. summarize the prevailing attitudes of those times in a footnote, "Ironically, the fat-tailed distribution of wealth inspired Karl Marx’s [49], as well as Benito Mussolini’s economic policies as diametrically opposite and extreme reactions to the same statistics [50]." and attribute the meaning of fat tails as follows, "The statistical form for a power law is the Pareto distribution (PD), which like the Cauchy distribution, has tails so heavy they confer unusual statistical properties and have been given the name fat-tailed distributions. Power laws are scale independent and intrinsically fractal."

In that paper, [49] refers to Of Fat Cats and Fat Tails: From the Financial Crisis to the ‘New’ Probabilistic Marxism and [50] refers to Pareto and Fascism Reconsidered by Zanden. That Vilfredo Pareto had Benito Mussolini as his most well known student should be lost on no one.