Sibuya distribution

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The Sibuya distribution is a heavy-tailed distribution that arises from the waiting time for the first success in Bernoulli trials [1]. These trials have probabilities of success that are inversely proportional to the number of trials. It is a member of the (a, b, 1) class of distributions and is derived from the extended truncated negative binomial (ETNB) distribution [2].

The distribution is named after Mitsuru Sibuya, a Japanese mathematician known for his contributions to probability theory. In his 1979 paper, Discrete analogues of stable distributions [3], Sibuya introduced this distribution. At the time, he held a professorship at the University of Tokyo and had expertise in various areas including branching processes and the theory of extreme values.

Properties of the Sibuya distribution

The Sibuya distribution, denoted as Sib(α), can be described using its Laplace transform (which converts integral and differential equations into algebraic equations):

sibuya distribution laplace transform

where t ∈ [0,∞).

Its cumulative distribution function (CDF) is defined as

sibuya distribution cdf

where k ∈ ℕ.

Alternatively, the probability mass function (PMF) is given by

sibuya distribution pmf

where k ∈ ℕ.

In both cases, α ∈ (0, 1] and t ∈ [0, ∞).

In the case of the Sibuya distribution, the mean and variance are infinite. This means that there isn’t a single average value of the distribution, and the distribution is spread out over an infinite number of values.

The absence of finite moments in the Sibuya distribution is due to its power-law tail, which is characterized by a gradual decrease in the probability of observing larger values. There is always a non-zero probability of observing extremely large values, which results in infinite mean and variance. As a result of lacking finite moments, the Sibuya distribution is not suitable for modeling data with a restricted range of values. However, it can be suitable for modeling data with a long tail, such as the number of earthquakes of a specific magnitude or word counts in text.


[1] Kozubowski, T. & Podgotski, K. A generalized Sibuya distribution. Tokyo Vol. 70, Iss. 4,  (2018): 855-887. DOI:10.1007/s10463-017-0611-3

[2] Ma, D. (2019). The (a,b,1) class.

[3] M. Sibuya. Generalized hypergeometric, digamma and trigamma distributions. Annals of the Institute of Statistical Mathematics, 31(1):373390,
Dec. 1979

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