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Complexity of Anticipated Rejection Algorithms and the Darling–Mandelbrot Distribution

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Abstract

We study in limit law the complexity of some anticipated rejection random sampling algorithms. We express this complexity in terms of a probabilistic process, the threshold sum process. We show that, under the right conditions, the complexity is linear and admits as a limit law a so-called Darling–Mandelbrot distribution, studied by Darling (Trans Am Math Soc 73:95–107, 1952) and Lew (Constr Approx 10(1):15–30, 1994). We also give an explicit form to the density of the Darling–Mandelbrot distribution and derive some of its analytic properties.

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Notes

  1. Given the definition of the Gamma function \(\varGamma (y) = \int _0^{\infty } x^{y-1} e^{-x} {\mathrm {d}}x\), the upper and lower incomplete versions are defined through the corresponding integrals on modified domains \(\varGamma (y,z) = \int _z^{\infty } \cdot \;\) and \(\gamma (y,z) = \varGamma (y)-\varGamma (y,z) = \int _0^z \cdot \;\), respectively. Non-positive real values of z are reached by analytic continuation.

  2. The sum in this expression can be seen as a special case of the Lauricella function \(F_B^{(k)}\) where all variables are specialized to \(-x\).

  3. To sample these, a better (in fact, optimal) algorithm consists in using the algorithm of [2] to sample a pointed binary plane tree and using classical bijections to get a Dyck prefix.

  4. The condition on the form of the left-hand side can be relaxed to some extent, we treat here a simplified situation in order to lighten the notation.

  5. This question has been posed also by the anonymous referee.

  6. The positivity property still holds for the homogeneous, more refined polynomials associated to the equation

    $$\begin{aligned}&(2x+y+z)F_n = F_{n+1} + x(y+z)(2x+y+z) F_{n-2}\\&F_0=0\,;\quad F_1=1\,;\quad F_2=x + y\,. \end{aligned}$$

  7. We thank M. Bousquet-Mélou and K. Rashel or pointing out this reference.

  8. This is not discussed in the synthetic presentation of [19], but it could be derived on identical basis, and it is implicit in the large generality of the results in [9].

  9. It is worth noting that, still on \({\mathbb {N}} \times {\mathbb {Z}}\), and at generic k, in the variant in which the endpoints are prescribed, exact enumeration formulas allow for an efficient algorithm, involving no anticipated rejection (see [6, Chapt. 4]). We thank the anonymous referee for pointing us towards this reference.

  10. Incidentally, note that also in the directed case the formula for \(\alpha (k)\) matches with the trivial value \(\alpha =0\) for problem (P1).

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Authors and Affiliations

  1. RISC, Johannes Kepler University, Altenbergerstraße 69, 4040, Linz, Austria

    Axel Bacher

  2. LIPN, University Paris Nord, 99 av. J.-B. Clément, 93430, Villetaneuse, France

    Andrea Sportiello

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  1. Axel Bacher
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  2. Andrea Sportiello
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Correspondence to Andrea Sportiello.

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Axel Bacher: Supported by FWF project F050-04 (Austria).

Andrea Sportiello: Supported by ANR Magnum project BLANC 0204 (France).

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Bacher, A., Sportiello, A. Complexity of Anticipated Rejection Algorithms and the Darling–Mandelbrot Distribution. Algorithmica 75, 812–831 (2016). https://doi.org/10.1007/s00453-015-0040-8

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