# Feller Vol 1 & 2 — Extractor candidates

## VOLUME 1 (discrete)

1. Discrete Sample Space — axiom — finite/countable set of outcomes.
2. Event — axiom — subset of a discrete sample space.
3. Probability Measure on a Discrete Sample Space — axiom — nonneg weights summing to 1.
4. Axioms of Probability (countable additivity) — axiom.
5. Equally Likely Outcomes (classical probability) — axiom — P(A)=|A|/|S|.
6. Multiplication (Fundamental Counting) Principle — technique.
7. Ordered Samples with Replacement — state — n^r ways.
8. Permutations (ordered samples without replacement) — state — (n)_r.
9. Factorial — axiom — n!.
10. Binomial Coefficient (combinations) — axiom — C(n,k).
11. Binomial Theorem — theorem.
12. Multinomial Coefficient — axiom.
13. Multinomial Theorem — theorem.
14. Occupancy (balls in cells) Model — axiom.
15. Maxwell-Boltzmann Statistics — state — distinguishable balls, n^r.
16. Bose-Einstein Statistics — state — indistinguishable, C(n+r-1,r).
17. Fermi-Dirac Statistics — state — ≤1 per cell, C(n,r).
18. Combinations with Repetition (stars and bars) — technique.
19. Hypergeometric Distribution — state.
20. Stirling's Formula — theorem.
21. Combination of Events (set ops) — axiom.
22. Addition Rule / Boole's Inequality — theorem.
23. Inclusion-Exclusion Principle (Poincaré) — theorem.
24. Probability of Realizing Exactly m Events — theorem.
25. Bonferroni Inequalities — theorem.
26. Matching (Rencontres) Problem / derangements — state.
27. Conditional Probability — axiom.
28. Multiplication Rule for Probabilities — theorem.
29. Law of Total Probability — theorem.
30. Bayes' Formula — theorem.
31. Stochastic Independence of Events — axiom.
32. Independent Trials / Product Space — axiom.
33. Bernoulli Trials — axiom.
34. Binomial Distribution — state.
35. Most Probable Value of the Binomial — theorem (central term).
36. Geometric Distribution — state.
37. Negative Binomial (Pascal) Distribution — state.
38. Multinomial Distribution — state.
39. Poisson Distribution — state.
40. Poisson Approximation to Binomial (Law of Rare Events) — theorem.
41. Poisson Process (discrete development) — state.
42. Random Walk (simple, on integers) — axiom.
43. Path Representation of Coin Tossing — technique.
44. Reflection Principle — technique.
45. Ballot Problem (Bertrand) — theorem.
46. Returns to the Origin — state — u_{2n}.
47. First Return to Origin Distribution — theorem — f_{2n}.
48. Probability of No Return / Sign Persistence — theorem.
49. Arcsine Law for Last Visit (Leads) — theorem.
50. Arcsine Law for Time Spent Positive (Sojourn) — theorem.
51. First Passage Times (random walk) — state.
52. Changes of Sign — theorem.
53. Maximum of a Random Walk — theorem.
54. Gambler's Ruin Problem — state.
55. Ruin Probability Formula — theorem.
56. Expected Duration of the Game — theorem.
57. Difference Equation Method (boundary conditions) — technique.
58. Generating Function for Duration of Play — technique.
59. Integer-Valued Random Variable — axiom.
60. Expectation (mean) of a Discrete Variable — axiom.
61. Variance and Standard Deviation — axiom.
62. Linearity of Expectation — theorem.
63. Additivity of Variance for Independent Variables — theorem.
64. Covariance and Correlation Coefficient — axiom.
65. Chebyshev's Inequality — theorem.
66. Probability Generating Function — technique.
67. Moments from the Generating Function — theorem.
68. Convolution of Distributions — theorem.
69. Compound Distribution (random sum) — state.
70. Continuity Theorem for Generating Functions — theorem.
71. Tauberian/Abelian Argument for Generating Functions — technique.
72. Branching (Galton-Watson) Process — state.
73. Iterated Generating Functions for Branching — theorem.
74. Extinction Probability Theorem — theorem.
75. Recurrent Event — axiom.
76. Recurrence-Time (renewal) Distribution — axiom.
77. Renewal Equation (u_n, f_n) — theorem — discrete.
78. Persistent vs Transient Recurrent Event — axiom.
79. Mean Recurrence Time and Periodicity — axiom.
80. Discrete Renewal Theorem — theorem — u_n → 1/μ.
81. Delayed Recurrent Event — state.
82. Number of Occurrences in n Trials — theorem.
83. Markov Chain — axiom.
84. Transition Probability Matrix — axiom.
85. n-Step Transition / Chapman-Kolmogorov — theorem.
86. Higher Transition Probabilities via Matrix Powers — technique.
87. Classification of States (recurrent/transient) — axiom.
88. Periodicity of a State — axiom.
89. Closed Sets and Irreducibility — axiom.
90. Communication / Equivalence Classes of States — state.
91. First-Passage Probabilities in a Chain — state.
92. Recurrence Criterion via Σ p_ii^(n) — theorem.
93. Positive vs Null Recurrence — axiom.
94. Stationary (Invariant) Distribution — state.
95. Existence/Uniqueness of Stationary Distribution — theorem.
96. Ergodic Theorem for Markov Chains — theorem.
97. Cesàro/Average Limit for Periodic Chains — theorem.
98. Absorbing States and Absorption Probabilities — state.
99. Weak Law of Large Numbers — theorem.
100. Bernoulli's Law of Large Numbers — theorem.
101. Borel's Strong Law of Large Numbers — theorem.
102. Borel-Cantelli Lemma — theorem.
103. De Moivre-Laplace Local Limit Theorem — theorem.
104. De Moivre-Laplace Integral (Central) Limit Theorem — theorem.
105. Normal (Gaussian) Distribution and Density — state.
106. Exponential Distribution (continuous limit) — state.
107. Uniform Distribution (continuous, as limit) — state.
108. Generating Function Method for Recurrences/Renewals — technique.

## VOLUME 2 (continuous / measure-theoretic)

1. Exponential density — axiom.
2. Lack-of-memory property — theorem.
3. Uniform density — axiom.
4. Density (pdf) — axiom.
5. Densities in higher dimensions (joint density) — axiom.
6. Marginal density — state.
7. Change of variables for densities (Jacobian) — technique.
8. Convolution of densities — technique.
9. Normal (Gaussian) density — axiom.
10. Multivariate normal density — axiom.
11. Gamma density — axiom.
12. Beta density — axiom.
13. Cauchy density — axiom.
14. Stable density (definition) — axiom.
15. Chi-square distribution — state.
16. Randomization / mixtures — technique.
17. Compound (subordinated) distribution — state.
18. Bessel function densities — state.
19. Probability measure — axiom.
20. σ-algebra — axiom.
21. Probability space — axiom.
22. Distribution function — axiom.
23. Lebesgue–Stieltjes integral/measure — axiom.
24. Lebesgue decomposition of distributions — theorem.
25. Types of distributions — axiom.
26. Expectation as Stieltjes integral — axiom.
27. Convolution of distributions — technique.
28. Convolution semigroup — state.
29. Weak law of large numbers — theorem.
30. Strong law of large numbers — theorem.
31. Convergence in probability — axiom.
32. Almost sure convergence — axiom.
33. Truncation method — technique.
34. Weak convergence of distributions — axiom.
35. Helly's selection theorem — theorem.
36. Tightness / proper convergence — axiom.
37. Method of moments — technique.
38. Moment problem / determinacy — state.
39. Characteristic function — axiom.
40. Inversion formula (characteristic functions) — theorem.
41. Uniqueness theorem (characteristic functions) — theorem.
42. Continuity theorem (Lévy) — theorem.
43. Central limit theorem (i.i.d.) — theorem.
44. Lindeberg–Feller CLT — theorem.
45. Lindeberg condition — axiom.
46. Lyapunov condition — axiom.
47. Berry–Esseen theorem — theorem.
48. Triangular array — axiom.
49. Null (uniformly asymptotically negligible) array — axiom.
50. Infinitely divisible distribution — axiom.
51. Lévy–Khinchine representation — theorem.
52. Canonical (Lévy) measure — axiom.
53. Kolmogorov canonical representation — theorem.
54. Poisson as infinitely divisible — state.
55. Limit theorems for triangular arrays (general) — theorem.
56. Stable distribution (char. function form) — theorem.
57. Index of stability α — axiom.
58. Domain of attraction — axiom.
59. Domain of normal attraction — axiom.
60. Characterization of domains of attraction (regular variation) — theorem.
61. Stable densities (series/asymptotics) — state.
62. Renewal equation (continuous) — axiom.
63. Renewal function/measure — state.
64. Elementary renewal theorem — theorem.
65. Blackwell's renewal theorem — theorem.
66. Key renewal theorem — theorem.
67. Directly Riemann integrable function — axiom.
68. Defective/excessive renewal equations — state.
69. Regenerative process — axiom.
70. Residual/spent waiting time (inspection paradox) — state.
71. Persistent vs transient renewal — state.
72. Random walk (general, on line) — axiom.
73. Ladder variables (ascending/descending) — axiom.
74. Ladder epochs and ladder heights — state.
75. Wiener–Hopf factorization — theorem.
76. Duality lemma (random walks) — technique.
77. Fluctuation theory / Sparre Andersen — theorem.
78. Spitzer's identity — theorem.
79. Arcsine law (continuous) — theorem.
80. Recurrence/transience of random walks (Chung–Fuchs) — theorem.
81. Laplace transform — axiom.
82. Laplace transform inversion/uniqueness — theorem.
83. Completely monotone function — axiom.
84. Bernstein's theorem — theorem.
85. Tauberian theorem (Hardy–Littlewood–Karamata) — theorem.
86. Abelian theorem — theorem.
87. Regularly varying function — axiom.
88. Slowly varying function — axiom.
89. Karamata's theorem/representation — theorem.
90. Fourier transform/integral — axiom.
91. Parseval relation — theorem.
92. Bochner's theorem — theorem.
93. Positive-definite function — axiom.
94. Pólya's criterion — theorem.
95. Smoothing inequality — technique.
96. Markov process — axiom.
97. Transition probability / Markov semigroup — axiom.
98. Chapman–Kolmogorov equation — theorem.
99. Infinitesimal generator — state.
100. Exponential formula / Hille–Yosida — theorem.
101. Resolvent — state.
102. Kolmogorov forward (Fokker–Planck) equation — theorem.
103. Kolmogorov backward equation — theorem.
104. Diffusion process — state.
105. Brownian motion / Wiener process — state.
106. Independent increments process — axiom.
107. Poisson process — state.
108. Stationary increments — axiom.
109. First-passage/hitting time distributions — state.
110. Convergence of types theorem — theorem.
