References & Further Reading

References

1. J. A. Gubner, Probability and Random Processes for Electrical and Computer Engineers, Cambridge University Press, 2006

   The primary textbook for Caire's FSP course. Chapters 5-6 cover transforms and limit theorems in the same unified framework used here.

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

   The definitive graduate-level treatment of characteristic functions and the continuity theorem. Chapters 26-27 provide the measure-theoretic foundations for the results in this chapter.

3. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, Wiley, 2nd ed., 1971

   The classic reference for characteristic functions and their applications. Chapter XV on characteristic functions remains unsurpassed in depth and elegance.

4. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer, 2nd ed., 1998

   The standard reference for large deviations theory. Chapter 2 develops Cramer's theorem rigorously, including the multidimensional case.

5. G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, Oxford University Press, 3rd ed., 2001

   Excellent treatment of generating functions, branching processes, and random walks. Chapters 5 and 11 complement the material here.

6. K. B. Athreya and P. E. Ney, Branching Processes, Springer, 1972

   The definitive monograph on branching processes. Goes far beyond the Galton-Watson process covered here, into multi-type and continuous-time branching.

7. P. Levy, Calcul des Probabilites, Gauthier-Villars, 1925

   Levy's original work establishing the continuity theorem for characteristic functions, the key tool for proving the CLT.

8. H. Cramer, Sur un nouveau theoreme-limite de la theorie des probabilites, 1938

   Cramer's original paper on large deviations for sums of i.i.d. random variables, establishing the exponential decay rate.

9. F. Galton and H. W. Watson, On the Probability of the Extinction of Families, 1874

   The founding paper of branching process theory. Galton posed the problem of family name extinction; Watson provided the generating function analysis.

10. A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, 4th ed., 2002

    A widely used engineering textbook with clear treatment of characteristic functions and their role in proving the CLT.

11. R. Durrett, Probability: Theory and Examples, Cambridge University Press, 5th ed., 2019

    Modern graduate-level probability. Chapter 3 provides a rigorous treatment of characteristic functions and weak convergence.

12. T. E. Harris, The Theory of Branching Processes, Springer, 1963

    The first comprehensive treatment of branching processes. Accessible and still relevant for the classical Galton-Watson theory.

Further Reading

• Large deviations beyond Cramer's theorem

  Dembo and Zeitouni, Large Deviations Techniques and Applications (1998)

  Cramer's theorem is just the beginning. The general theory handles non-i.i.d. sequences, Markov chains, and empirical measures — essential for the error exponents in information theory (Book ITA).

• Stein's method and rates of convergence in CLT

  Chen, Goldstein, and Shao, Normal Approximation by Stein's Method (2011)

  The Berry-Esseen theorem gives $O(1/\sqrt{n})$ convergence rate; Stein's method provides sharper, distribution-free bounds with explicit constants.

• Multitype and continuous-time branching processes

  Athreya and Ney, Branching Processes (1972)

  The Galton-Watson process is the simplest branching model. Real applications (epidemic models, population genetics) require multi-type and age-dependent generalizations.

• Moment problems and determinacy

  Stoyanov, Counterexamples in Probability (2014)

  When does a distribution's moment sequence uniquely determine it? This is the question the MGF answers (when it exists), but the answer is subtle for distributions with heavy tails.