# Balkema-de Haan-Pickands theorem, generalized Pareto and lognormal

On the wikipedia page on the Balkema-de Haan-Pickands theorem, en.wikipedia.org/wiki/Pickands-Balkema-de_Haan_theorem, it is said the "for a large class of underlying distribution functions", their far-right values can be approximated by a generalized Pareto.

What is and what is not that "large class"?

• Doesn't the comment thread in the referenced post answer your question?
– whuber
Sep 9, 2019 at 16:56
• Not that I can see or understand. That discussion is more focussed on the fact that a lognormal does not have asymptotic properties like a GP. But that is an observation after the fact of the BHP theorem. I'm still trying to understand the general properties of the BHP theorem, and which of those general properties disqualify the lognormal. Help is useful. Sep 9, 2019 at 17:16
• So, for example, in this paper: arxiv.org/pdf/1708.01686.pdf, it is said "The heavy tail phenomenon and the GPD are linked through the famous Pickands-Balkema-de Hann theorem (Balkema and De Haan (1974) and Pickands III (1975)) which states that, for an arbitrary distribution of which the sample maximum tends to a non-degenerate distribution after suitable standardization, the distribution function of its exceedances over a large threshold converges to the GPD." I suppose that buried in there is some quality that disqualifies the lognormal. Sep 9, 2019 at 17:18
• The wikipedia page is not very useful: en.wikipedia.org/wiki/Pickands-Balkema-de_Haan_theorem , where it is said "for a large class of underlying distribution functions". Well, what is and what is not that "large class"? Sep 9, 2019 at 17:26
• The comment thread is far more general than your characterization of it: it concerns the sense in which the BHP theorem means that a tail can be "approximated by" a Pareto.
– whuber
Sep 9, 2019 at 18:36

The standard requirement is that $$F$$ is in the domain of attraction of the GEV (generalized extreme value distribution). If F is in said domain of attraction, then the distribution of excesses over a high threshold (conditional on clearing that threshold) is well approximated by a distribution in the GP family.

"In the domain of attraction of the GEV" means that block maxima $$M_n = \max_iX_i$$, for some sequence of constants $$a_n$$, $$b_n$$, where $$n$$ refers to block size, the distribution of the standardized sample maximum doesn't degenerate.

$$\lim\limits_{n\to\infty} \text{P}\left(\frac{M_n - b_n}{a_n} \leq x\right) = G(x)$$

If F is in the domain of attraction of a GEV, then $$\lim\limits_{u\to\infty}\text{P}\left(X \geq u + y \mid x \geq u\right) = \lim\limits_{u\to\infty}\frac{1 - F(u + y)}{1 - F(u)} = H(y)$$

Where $$H$$ is generalized Pareto.

• There seems to be some confusion. I am interested in knowing which distributions are not in the domain of attraction of the GEV. Jun 23, 2021 at 23:22
• Then you might want to look at Fisher et al, 1928--that describes the three forms of the limiting distributions of extrema of a sample. Later authors reformulate those three forms as a single form, the GEV. But it's those distributions that fall into the domain of attraction of the GEV for which excesses over a threshold can be modeled with GP. That's the "large class of distributions" they're referring to. Jun 25, 2021 at 1:09
• Incidentally, with regard to "lognormal" in the title: lognormal falls into domain of attraction of Gumbel, which is special case of GEV. Therefore, excesses over a threshold in lognormal can be represented as GP. Jun 25, 2021 at 1:14
• Can you please give me an example of a distribution that is not in the GEV domain? I had thought the lognormal was an example, but I now know that I was wrong. A specific example or two is what I'd be interested in. Jun 26, 2021 at 4:46
• Well, as a trivial answer, Bernoulli isn't in the DOA of the GEV. Nor will any ordinal distribution, or any distribution established using point masses (for that matter). You're probably unsatisfied with that answer. I'd start looking at the convergence condition for heavy tailed distributions, cauchy in particular (as it doesn't meet the convergence criteria for CLT, I don't know if it would for Fisher-Tippet.) Beyond that, I imagine for mixtures you would only need consider the right-most mixture component (and that all components fall into some DOA). Jul 11, 2021 at 1:43

There are some helpful comments on this in a draft textbook by Smith and Weissman, available at https://rls.sites.oasis.unc.edu/s834-2020/s834.html.

## Domains of attraction

Section 2 discusses domains of attraction and says:

In practice nearly all continuous distributions are in the domain of attraction of some extreme value limit.

Page 12 presents some counterexamples:

Continuous distributions not in any domain of attraction These are hard to construct but they do exist! Here are two examples: (a) Any distribution function with slowly varying tail, for instance $$F(x) = 1 − 1/ \log x$$ for $$x > e$$. (b) Consider $$F_\delta(x) = 1-x^{-1}\{1 + \delta \sin(\log x)\}$$ valid for $$x \geq$$ some $$x_0$$, with $$|\delta|$$ small enough to make it a valid distribution function.

The second example is credited to Resnick (1971) "Tail equivalence and its applications".

## Log-normal

The question title mentions the log-normal distribution. Page 12 of the book has another example showing the log-normal is in the GPD domain of attraction. It might be worth pointing out that this involves a more general version of the Pickands theorem than is given on the wikipedia page currently, which says $$\Pr(X-u \leq y | X>u)$$ converges to a GPD (with constant scale $$\sigma$$) as $$u \to \infty$$. A fuller vesion is that $$\Pr(\frac{X-u }{h(u)} \leq y | X>u)$$ converges to a GPD for some scaling function $$h(u)$$. (The first pages of Tawn and Papastathopolous https://arxiv.org/abs/1111.6899 have a nice summary of this.) The example in the book requires $$h(u)$$ to be non-constant. So I suspect the log-normal only lies in the GPD domain of attraction for the general theorem, and not for the simplified wikipedia form.

Speaking informally, if centered, normalized maximum of $$n \to \infty$$ i.i.d. random variables sampled from your distribution is in the domain of attraction of Gumbel Type I EVD, corresponding residual life time $$p(\frac{ \xi - u }{ a(u) } > x)$$ for $$u \to \infty$$ converges to Generalized Pareto $$(1 + \gamma x)^{1/\gamma} \xrightarrow{\gamma \to 0} (1 - e^{-x})$$. E.g. if you have a sample of wagons aged $$u=20$$ (with their survival function being exponential), the chance that they serve $$x=3$$ more years is still exponential. Here $$a(u)$$ is a normalising function, called auxiliary function. All auxiliary functions are asymptotically equivalent as $$u \to \infty$$ by Khinchin's theorem. For instance, one can use $$a(u) = \frac{1}{r(u)}$$, the inverse of hazard rate $$r(u) = \frac{F'(u)}{1 - F(u)}$$ as auxiliary function.

If your distribution is in the domain of attraction of Inverse Weibull Type III EVD, corresponding residual life time converges to "specialized" Generalized Pareto $$(1 + x)^{1/\gamma}$$. E.g. if you have a sample of horses, aged 20, the chance that they survive $$x=3$$ more years is specialized Generalized Pareto-distributed. "Specialized", because it is not $$(1 + \gamma x)^{\frac{1}{\gamma}}$$, it is just $$(1 + x)^{1/\gamma}$$.

The difference is that survival function of horses has long fat tails (decaying as a polynomial hyperbola $$1/x^\alpha$$) and has a finite upper end point (all horses are dead by 40, chance of surviving past some point is exactly 0), while the survival function of wagons has a light sub-exponential tail and might have or have not an upper end point.

Centered, normalized maximum of most distributions converges to one of 3 types of EVD, those are the only max-stable distributions, i.e. such distributions that maximum of $$n$$ i.i.d. random variables, sampled from them converges to themselves, this is Fisher-Tippett-Gnedenko theorem. The specific set of distributions is determined by application of necessary and sufficient conditions of convergence to EVD.

Some notable exceptions are geometric and Poisson distribution, they do not converge to any max-stable distribution due to discontinuity of their pdf at integer points (there is a criterion that requires that $$\frac{S(x)}{S(x-)} \xrightarrow{x \to \infty} 1$$ for a distribution to be in a domain of attraction of max-stable distribution), which is clearly not satisfied for these two, as survival function at integer point $$S(x)$$ is different from survival function at its left proximity $$S(x-)$$.