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 Incoming links: Integrable singularity

Endpoint concentration in the arcsine distribution 2026-10-18

An unbounded probability density function does not imply an atom. Here

P (X = 0) = P (X = 1) = 0, P (0 < X < ε) = \frac{2}{π} arcsin ε \sim \frac{2}{π} ε (ε ↓ 0) .

The last equivalence follows from

arcsin u / u \to 1

. The endpoint interval has much more probability than the same interval under a uniform distribution, whose value is

ε

, but its probability still tends to zero. This is an integrable singularity.

For example, the first 1% of the unit interval contains about

6.377%

of the probability; the last 1% contains the same amount because the arcsine distribution is a symmetric probability distribution. Each exact endpoint still has zero probability. Density and interval probability answer different questions.

Figure 1.
Endpoint density diverges, but every finite bin has finite probability
. Original exact curves and bin probabilities. Left: the arcsine cumulative distribution function compared with the uniform one. Right: ten equal-width bins have probabilities $F ((j + 1) /10) - F (j /10)$ , not heights sampled from the divergent density. Their total is one. Details.

A numerical integration routine must respect these endpoint integrable singularities. The substitution

x = sin^{2} θ

gives

f (x) d x = \frac{2}{π} d θ, 0 < θ < π /2.

It turns a difficult-looking density integral into a constant measure in angle. Integrating finite bins is also a faithful way to draw the distribution near its endpoints; sampling

f (0)

to make a bar height is not.

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Integrable singularity 2026-10-18

A function has an integrable singularity at a point if it becomes unbounded there but its absolute value has a finite improper integral over a sufficiently small surrounding interval, one-sided at an endpoint. The endpoint behavior of the arcsine distribution's probability density function is an example.

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