References & Further Reading

References

1. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I, Wiley, 3rd ed., 1968

   The classic reference for discrete probability. Chapter II covers combinations, the hypergeometric distribution, and occupancy problems with exceptional clarity. Chapter VI gives the Poisson approximation and numerous applications.

2. S. M. Ross, A First Course in Probability, Pearson, 10th ed., 2019

   An accessible introduction with many worked examples. Chapter 4 covers the hypergeometric, binomial, and Poisson distributions with practical applications.

3. G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, Oxford University Press, 4th ed., 2020

   A comprehensive reference for both discrete and continuous probability. The treatment of generating functions and the Poisson approximation is particularly thorough.

4. R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 2nd ed., 2012

   The definitive reference for Stirling numbers, Bell numbers, integer partitions, and generating functions. Chapter 1 covers the foundations; Chapter 5 treats exponential generating functions in depth.

5. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 5th ed., 1979

   Chapter 19 covers the theory of partitions, including the generating function, Euler's pentagonal theorem, and the Hardy-Ramanujan asymptotic formula.

6. P. Billingsley, Probability and Measure, Wiley, 3rd ed., 1995

   A rigorous measure-theoretic treatment. The Poisson limit theorem is proved as a special case of convergence in distribution.

7. L. Le Cam, An Approximation Theorem for the Poisson Binomial Distribution, 1960

   The original paper proving the total variation bound for the Poisson approximation to sums of independent (non-identical) Bernoullis.

8. Y. Polyanskiy, A Perspective on Massive Random Access, 2017

   Introduces the unsourced random access framework where the number of active users is modeled as Poisson. A foundational paper for massive IoT research.

9. A. D. Barbour, L. Holst, and S. Janson, Poisson Approximation, Oxford University Press, 1992

   The definitive monograph on Poisson approximation methods, including the Stein-Chen method for dependent indicators. Goes far beyond Le Cam's inequality.

10. D. E. Knuth, The Art of Computer Programming, Vol. 1: Fundamental Algorithms, Addison-Wesley, 3rd ed., 1973

    Sections 1.2.6 and 1.2.9 cover Stirling numbers, their notation, recurrences, and applications in the analysis of algorithms.

11. L. von Bortkiewicz, Das Gesetz der kleinen Zahlen, B.G. Teubner, 1898

    The original "Law of Small Numbers" monograph that empirically validated the Poisson distribution on data including Prussian cavalry deaths and child suicides.

Further Reading

• Analytic combinatorics and generating functions

  P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009

  A comprehensive treatment of generating functions as a systematic method for solving counting problems. Goes far beyond the preview in this chapter.

• Stein-Chen method for Poisson approximation

  A. D. Barbour, L. Holst, and S. Janson, Poisson Approximation, Oxford, 1992

  Extends Le Cam's inequality to dependent random variables using Stein's method — essential for birthday-type problems and random graph applications.

• Partition theory and $q$-series

  G. E. Andrews, The Theory of Partitions, Cambridge University Press, 1998

  The standard reference on integer partitions, including Ramanujan's congruences, the Rogers-Ramanujan identities, and connections to modular forms.

• Random access and massive IoT

  Y. Polyanskiy, A Perspective on Massive Random Access, ISIT 2017

  Shows how the Poisson user model leads to fundamental limits on random access — directly connecting this chapter to modern wireless communications research.