Bernoulli

  • Bernoulli
  • Volume 25, Number 4A (2019), 2758-2792.

Dickman approximation in simulation, summations and perpetuities

Chinmoy Bhattacharjee and Larry Goldstein

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Abstract

The generalized Dickman distribution $\mathcal{D}_{\theta}$ with parameter $\theta>0$ is the unique solution to the distributional equality $W=_{d}W^{*}$, where \begin{align} W=d U1/θ(W+1),\tag{1} \end{align} with $W$ non-negative with probability one, $U\sim\mathcal{U}[0,1]$ independent of $W$, and $=_{d}$ denoting equality in distribution. These distributions appear in number theory, stochastic geometry, perpetuities and the study of algorithms. We obtain bounds in Wasserstein type distances between $\mathcal{D}_{\theta}$ and the distribution of \begin{eqnarray*}W_{n}=\frac{1}{n}\sum_{i=1}^{n}Y_{k}B_{k},\end{eqnarray*} where $B_{1},\ldots,B_{n},Y_{1},\ldots,Y_{n}$ are independent with $B_{k}$ distributed $\operatorname{Ber}(1/k)$ or $\mathcal{P}(\theta/k)$, $E[Y_{k}]=k$ and $\operatorname{Var}(Y_{k})=\sigma_{k}^{2}$, and provide an application to the minimal directed spanning tree in $\mathbb{R}^{2}$. We also provide bounds with optimal rates for the Dickman convergence of weighted sums, arising in probabilistic number theory, of the form \begin{eqnarray*}S_{n}=\frac{1}{\log(p_{n})}\sum_{k=1}^{n}X_{k}\log(p_{k}),\end{eqnarray*} where $(p_{k})_{k\ge1}$ is an enumeration of the prime numbers in increasing order and $X_{k}$ is geometric with parameter $(1-1/p_{k})$, Bernoulli with success probability $1/(1+p_{k})$ or Poisson with mean $\lambda_{k}$.

Lastly, we broaden the class of generalized Dickman distributions by studying the fixed points of the transformation \begin{eqnarray*}s(W^{*})=_{d}U^{1/\theta}s(W+1)\end{eqnarray*} generalizing (1), that allows the use of non-identity utility functions $s(\cdot)$ in Vervaat perpetuities. We obtain distributional bounds for recursive methods that can be used to simulate from this family.

Article information

Source
Bernoulli, Volume 25, Number 4A (2019), 2758-2792.

Dates
Received: June 2017
Revised: August 2018
First available in Project Euclid: 13 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.bj/1568362042

Digital Object Identifier
doi:10.3150/18-BEJ1070

Mathematical Reviews number (MathSciNet)
MR4003564

Zentralblatt MATH identifier
07110111

Keywords
delay equation distributional approximation primes utility weighted Bernoulli sums

Citation

Bhattacharjee, Chinmoy; Goldstein, Larry. Dickman approximation in simulation, summations and perpetuities. Bernoulli 25 (2019), no. 4A, 2758--2792. doi:10.3150/18-BEJ1070. https://projecteuclid.org/euclid.bj/1568362042


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References

  • [1] Arras, B. and Houdré, C. (2017). On Stein’s method for infinitely divisible laws with finite first moment. Available at https://arxiv.org/abs/1712.10051.
  • [2] Arras, B., Mijoule, G., Poly, G. and Swan, Y. (2016). Distances between probability distributions via characteristic functions and biasing. Available at https://arxiv.org/abs/1605.06819v1.
  • [3] Arras, B., Mijoule, G., Poly, G. and Swan, Y. (2017). A new approach to the Stein–Tikhomirov method: With applications to the second Wiener chaos and Dickman convergence. Available at https://arxiv.org/abs/1605.06819.
  • [4] Arratia, R. (2002). On the amount of dependence in the prime factorization of a uniform random integer. In Contemporary Combinatorics. Bolyai Soc. Math. Stud. 10 29–91. Budapest: János Bolyai Math. Soc.
  • [5] Arratia, R. (2017). Personal communication.
  • [6] Arratia, R., Barbour, A.D. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach. Zürich: European Mathematical Society (EMS).
  • [7] Barbour, A.D. and Nietlispach, B. (2011). Approximation by the Dickman distribution and quasi-logarithmic combinatorial structures. Electron. J. Probab. 16 880–902.
  • [8] Baxendale, P. (2017). Personal communication.
  • [9] Bernoulli, D. (1738). Specimen theoriae novae de mensura sortis. Comentarii Academiae Scientiarum Imperiales Petropolitanae 5 175–192.
  • [10] Bernoulli, D. (1954). Exposition of a new theory on the measurement of risk. Econometrica 22 23–36.
  • [11] Bhatt, A.G. and Roy, R. (2004). On a random directed spanning tree. Adv. in Appl. Probab. 36 19–42.
  • [12] Bhattacharjee, C. and Goldstein, L. (2018). Supplement to “Dickman approximation in simulation, summations and perpetuities.” DOI:10.3150/18-BEJ1070SUPP.
  • [13] Cellarosi, F. and Sinai, Y.G. (2013). Non-standard limit theorems in number theory. In Prokhorov and Contemporary Probability Theory. Springer Proc. Math. Stat. 33 197–213. Heidelberg: Springer.
  • [14] Chen, L.H.Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein’s Method. Heidelberg: Springer.
  • [15] Devroye, L. and Fawzi, O. (2010). Simulating the Dickman distribution. Statist. Probab. Lett. 80 242–247.
  • [16] Dickman, K. (1930). On the frequency of numbers containing prime factors of a certain relative magnitude. Ark. Mat. Astron. Fys. 22 1–14.
  • [17] Eeckhoudt, L., Gollier, C. and Schlesinger, H. (2005). Economic and Financial Decisions Under Risk. Princeton: Princeton Univ. Press.
  • [18] Fill, J.A. and Huber, M.L. (2010). Perfect simulation of Vervaat perpetuities. Electron. J. Probab. 15 96–109.
  • [19] Goldstein, L. (2017). Non asymptotic distributional bounds for the Dickman approximation of the running time of the Quickselect algorithm. Available at https://arxiv.org/abs/1703.00505v1.
  • [20] Goldstein, L. (2018). Non asymptotic distributional bounds for the Dickman approximation of the running time of the quickselect algorithm. Electron. J. Probab. 23 1–13.
  • [21] Goldstein, L. and Reinert, G. (1997). Stein’s method and the zero bias transformation with application to simple random sampling. Ann. Appl. Probab. 7 935–952.
  • [22] Hwang, H.-K. and Tsai, T.-H. (2002). Quickselect and the Dickman function. Combin. Probab. Comput. 11 353–371.
  • [23] Peköz, E.A. and Röllin, A. (2011). New rates for exponential approximation and the theorems of Rényi and Yaglom. Ann. Probab. 39 587–608.
  • [24] Peköz, E.A., Röllin, A. and Ross, N. (2013). Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Probab. 23 1188–1218.
  • [25] Penrose, M.D. and Wade, A.R. (2004). Random minimal directed spanning trees and Dickman-type distributions. Adv. in Appl. Probab. 36 691–714.
  • [26] Pinsky, R. (2016). A natural probabilistic model on the integers and its relation to Dickman-type distributions and Buchstab’s function. Available at https://arxiv.org/abs/1606.02965.
  • [27] Pinsky, R. (2016). On the strange domain of attraction to generalized Dickman distributions for sums of independent random variables. Available at https://arxiv.org/abs/1611.07207.
  • [28] Rachev, S.T. (1991). Probability Metrics and the Stability of Stochastic Models. Chichester: Wiley.
  • [29] Ramanujan, S. (1919). A proof of Bertrand’s postulate. J. Indian Math. Soc. (N.S.) 11 27.
  • [30] Rényi, A. (1962). Théorie des éléments saillants d’une suite d’observations. Ann. Fac. Sci. Univ. Clermont-Ferrand No. 8 7–13.
  • [31] Ross, N. (2011). Fundamentals of Stein’s method. Probab. Surv. 8 210–293.
  • [32] Royden, H.L. and Fitzpatrick, P. (1988). Real Analysis, 3rd ed. New York: Macmillan Publishing Company.
  • [33] Rudin, W. (1964). Principles of Mathematical Analysis, Vol. 3. New York: McGraw-Hill.
  • [34] Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. 583–602.
  • [35] Stein, C. (1986). Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes – Monograph Series 7. Hayward, CA: IMS.
  • [36] Stolz, O. (1885). Vorlesungen über Allgemeine Arithmetik: Nach Den Neueren Ansichten (Vol. 1). Leipzig: BG Teubner.
  • [37] Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. Probab. 11 750–783.

Supplemental materials

  • Supplement to: Dickman approximation in simulation, summations and perpetuities. This self contained article (also available at https://arxiv.org/abs/1706.08192) gives detailed proofs of results that were required, but not provided, in the current manuscript.