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Tossing half-coins and other partial coins: signed probabilities and Sibuya distribution

Nikolai Leonenko, Igor Podlubny

TL;DR

This work addresses the challenge of simulating signed probability distributions that arise in partial coins and fractional-order processes by leveraging the Sibuya distribution and a generating-function decomposition. The authors provide explicit constructions for nonnegative distributions $g$ and $h$ such that $f g = h$ for the $1/n$-coin, establishing positivity of the corresponding coefficients and enabling Monte Carlo simulation via cumulated-mass methods. They develop a concrete numerical framework, derive recurrence-based coefficient evaluations, and implement a MATLAB toolbox to perform partial-coin and multi-coin simulations, including biased cases. The results extend signed-distribution simulation to infinite-support cases and have potential implications for fractional differentiation, uncertainty quantification, and related stochastic modeling tasks.

Abstract

A method for the numerical simulation of signed probability distributions for the case of tossing $1/n$-th of a coin is presented and illustrated by examples.

Tossing half-coins and other partial coins: signed probabilities and Sibuya distribution

TL;DR

This work addresses the challenge of simulating signed probability distributions that arise in partial coins and fractional-order processes by leveraging the Sibuya distribution and a generating-function decomposition. The authors provide explicit constructions for nonnegative distributions and such that for the -coin, establishing positivity of the corresponding coefficients and enabling Monte Carlo simulation via cumulated-mass methods. They develop a concrete numerical framework, derive recurrence-based coefficient evaluations, and implement a MATLAB toolbox to perform partial-coin and multi-coin simulations, including biased cases. The results extend signed-distribution simulation to infinite-support cases and have potential implications for fractional differentiation, uncertainty quantification, and related stochastic modeling tasks.

Abstract

A method for the numerical simulation of signed probability distributions for the case of tossing -th of a coin is presented and illustrated by examples.

Paper Structure

This paper contains 14 sections, 1 theorem, 34 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Half-coins and other partial coins
  3. Sibuya probability distribution
  4. Simulation of the signed distribution
  5. Coefficients of the generating functions $f$, $g$, and $h$
  6. Numerical simulations
  7. Evaluation of the coefficients of $f(x)$ and $g(x)$
  8. Evaluation of the coefficients of $h(x)$
  9. Method of simulation
  10. Examples of numerical simulations of partial coins
  11. Examples of numerical simulation of tossing two partial coins
  12. Tossing biased partial coins
  13. Conclusion
  14. Acknowledgements

Key Result

Theorem 1

For every generalized generating function $f$ of a signed probability distribution there exist two generating functions $g$ and $h$ of ordinary non-negative probability distributions such that $fg =h$.

Figures (14)

  • Figure 1: Coefficients of $g(x)$ and $f(x)$
  • Figure 2: A quarter-coin. Flips: 10000; ones: 1183; zeros: 8817; expectation: 0.1183
  • Figure 3: A one-third-coin. Flips: 10000; ones: 1650; zeros: 8350; expectation: 0.1650
  • Figure 4: A half-coin. Flips: 10000; ones: 2519; zeros: 7481; expectation: 0.2519
  • Figure 5: A three-quarters-coin. Flips: 10000; ones: 3716; zeros: 6284; expectation: 0.3716
  • ...and 9 more figures

Theorems & Definitions (1)

  • Theorem 1: G. Székely Szekely-2005