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| Lukacs E. — Characterisic functions |
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| Предметный указатель |
 -value 109
![$N_a[f(t)], N_a(f)$](/math_tex/b0ae142b50cdef0bb07e535feba0b13f82.gif) 109
Absolute moment 17
Absolutely continuous 12
Absolutely continuous d. 12 67
Absolutely monotonic 196
Achieser, N.I. 209 210
Adjoint polynomial 161
Algebraic moment 17
Almost periodic 26 43
Analytic characteristic functions 108 130
Analytic characteristic functions, factorization of 170 ff.
Analytic characteristic functions, i.d.a.c.f. 187 ff.
Approximation theorem of Weierstrass 35 77 208
Arithmetic of d.f. 78 108 180 182
Bergstroem, H. 105 210
Beta d. 15 26
Binomial d. 14 21 26
Bochner, S. 62 145 210
Bochner’s theorem 62 145
Bohr, H. 26 43 210
Borel, E. 156
Bounded d. (to left, right) 106 139 140
Cancellation law 92
Canonical product 156
Canonical representation 85 ff.
Canonical representation of stable c.f. 101
Canonical representation, Kolmogorov’s 90
Canonical representation, Levy — Khinchine’s 89
Canonical representation, Levy’s 90 119 122
Canonical representation, table of 93
Cantor’s (ternary) set 15 16
Cauchy distribution 15 19 26 72 80 93 103 106 206
Characteristic function 18 22
Characteristic function, decomposable c.f. 78
Characteristic function, i.d.c.f. 79 80 92 187
Characteristic function, indecomposable c.f. 78 81 122 123 180
Characteristic function, periodic c.f. 158
Characteristic function, table of c.f. 26
Characteristic, second 33 182
Characterization 184
Characterization of normal d. 184
Characterization of normal, Poisson, conjugate Poisson d. 185
Chebyshev polynomials 150
Chi-square d. 15
Chung, K.L. 205 210
Closure theorems 82 84
Complex conjugate 22
Components of a d.f. 45
composition 45
Concave 205
Conjugate complex 22
Conjugate d. 36 110
Continuity interval 10
Continuity point 10 36
Continuity theorem 53 54 74 77
Continuity theorem for bounded non-decreasing functions 57
Continuous (to the right) 10
Continuous part of d.f. 11
Convergence in the mean 67 68
Convergence theorem 172 199 201
Convergence theorem, dominated 199 201
Convergence theorem, monotone 172
Convergence, weak 48 49 54
Convergent sequence of d.f. 47 48
Convex 70 205
Convexity properties of a.c.f. 135 136
Convolution 45 47
Convolution theorem 43 ff.
Convolution, v-fold 200
Copson, E.T. 134 156 210
Cramer, H. 9 18 20 21 29 44 67 112 173 210
Cramer’s criterion 65 145
Cramer’s theorem 173 174 176 182 184 196
Criteria for c.f. 59 145
Cumulant generating function 33
Cumulants 33
Darboux sums 44 84 170 171
Decomposable c.f. 78 173 182
Decomposition of d. 11
Decomposition theorem 112
Decomposition theorem, general 112
DeFinetti’s theorem 83
Degenerate d. 13 25 26 80 93 108
Degree of freedom 15
Density function 12 40
Derivative, symmetric 33
Differential equations 160 161
Differential equations, c.f. as solutions of 161
Differential equations, positive definite 161
Dirichlet’s conditions 74
Discontinuity point 10
Discontinuous part of a d.f. 11
Discrete part 11 12
Distribution function of a.c.f. 136 ff.
Distribution function, absolutely continuous 14 67
Distribution function, discrete 13 14 43 46
Distribution function, indecomposable 123 125
Distribution function, singular 15
Distribution, Beta d. 15 26
Distribution, Binomial d. 14 21 26
Distribution, bounded 106 139 140
Distribution, Cauchy d. 15 19 26 72 80 93 103 106 206
Distribution, Chi-square d. 15
Distribution, Compound Poisson d. 200
Distribution, conjugate 36 110
Distribution, Degenerate d. 13 25 26 80 93 108
Distribution, Exponential d. 15 21
Distribution, finite 106 124 141 161 188
Distribution, Gamma d. 15 21 26 80 81 91 93 122 130
Distribution, Gaussian (Gauss — Laplace) d. 14
Distribution, Generalized Poisson d. 200
Distribution, Geometric d. 14 84 85
Distribution, Hypergeometric d. 14
Distribution, i.d. 79
Distribution, Laplace d. 15 21 26 72 74 81 94
Distribution, lattice 14 24 158 176 197
Distribution, Negative Binomial d. 14 21 26 80 93
Distribution, Normal d. 15 21 26 80 93 102 106 130 147 169 173 182 184
Distribution, one-sided 106 139
Distribution, Pascal d. 14
Distribution, Poisson d. 14 21 26 80 93 127 130 160 174
Distribution, Rectangular d. 15 21 26 81 129
Distribution, Simpson’s d. 15
Distribution, stable 97 ff.
Distribution, Student’s d. 15 19 21
Distribution, symmetric 36 37 190
Distribution, Triangular d. 15
Distribution, Uniform d. 15
Distribution, unimodal 205
Dominated Convergence Theorem 199 201
Dugue, D. 74 128 196 198 210
Elliptic function 160
Entire characteristic functions 134 137 142 173
Entire characteristic functions of infinite order 144
Entire characteristic functions, determination by properties of factors of 182
Entire characteristic functions, factorization of 173
Entire characteristic functions, order of 134 135 142 173 195
Entire function 134
Equivalence (of c.f. or second c.f.) 182
Evans, G.C. 21 210
Exponent of convergence 156
Exponent of stable d. 102 104 106
Exponential d. 15 21
Exponential type, entire function of 141
Extensions of factorization theorems for a.c.f. 190
Extremity (left, right) of a d.f. 106
Faa di Bruno’s formula 34
Factor of a.c.f. 170
Factor of c.f. 78
| Factor, indecomposable 78 112 119
Factor-closed family 175
Factorization problems 78 92 108 130 170
Factorization theorem, Hadamard’s 141 156 174 175
Factorization theorem, Weierstrass’ 74
Faltung 45
Fatou’s Lemma 162 191 192
Feller, W. 200 210
Finetti’s theorem 83
Finite d. 106 124 141 161 188
Fourier coefficients 43 74
Fourier integral 132 133
Fourier inversion theorem 71
Fourier series 74
Fourier transform 18
Freedom, degree of 15
Frequency function 12 15
Frequency, Poisson 181
Gamma d. 15 21 26 80 81 91 93 122 130
Gamma d., canonical representation of 93
Gauss — Laplace (law of) 14
Gaussian d. 14
Generating function, cumulant 33
Generating function, moment 18 135 136 178
Generating function, probability 18 124
Genus 146
Geometric d. 14 84 85
Girault, M. 25 27 74 160 205 210
Gnedenko, B.V. 97 102 106 107 205 210
Hadamard, J. 125
Hadamard’s factorization theorem 141 156 174 175
Hahn, H. 76 210
Hardy, G. H„ 70 210
Helly’s first theorem 49 54 63 115
Helly’s second theorem 51 63 88
Helly’s second theorem, extension of 52 54 88
Hermitian 62 68
Hincin see "Khinchine"
Hobson, E.W. 10 210
Hypergeometric d. 14
Ibragimov, I.A. 205 210
Imaginary part (Im) 131
Increase, point of 10
Indecomposable (absolutely continuous d.f.) 125
Indecomposable c.f. 78 81 122 123 180
Indecomposable d. 123 125
Indecomposable factor 78 112 119
Infinitely divisible a.c.f. 187 ff.
Infinitely divisible c.f., definition 79
Infinitely divisible c.f., examples 80
Infinitely divisible c.f., factorization of 92
Infinitely divisible d.f. 79
Integrable, quadratically 67
Integral functions see "Entire functions"
Integral representation 130
Integral transforms 17
Inversion formula for absolutely integrable c.f. 40
Inversion formulae 37
Inversion theorem 38
Iterated exponentials 146
Jessen, B. 25 27 210
Jordan, Ch. 34 210
Jump 10
Kernel 17 75
Kershner, R. 25 210
Khinchine, A.Ya. 74 112 205
Khinchine’s criterion 69 145
Kolmogorov (canonical) representation 90 93 205
Kolmogorov (canonical) representation of a.c.f. 189
Kolmogorov, A.N. 97 102 106 107 205 210
Krasner, M. 125 210
L-class 107
Laha, R.G. 190
Laplace 18
Laplace d. 15 21 26 72 74 81 94
Laplace integral 132
Laplace, first law of 15
Laplace, second law of 14
Laplace, transform 132
Lattice d. 14 24 158 176 197
Lattice points 14 158
Lebesgue decomposition of c.f. 26
Lebesgue decomposition theorem 12
Left extremity 106
Levy (canonical) representation 90 93 101 119 122
Levy — Khinchine representation 89 93
Levy, P. 102 106 107 128 145 173 176 178 179 180 187 188 211
Lienard, A. 125
Lim (weak limit) 49
Limit in the mean (l.i.m.) 67
Limit theorem 79 95 97
Limits of d.f. 47 ff.
Linnik, Yu.V. 105 160 161 176 180 196 211 212
Littlewood, J.E. 70 210
Livingstone, A.E. 71 211
Loeffel, H. 205 211
Loeve, M. 9 12 19 102 107 172 199 211
Lukacs, E. 146 158 211
Maclaurin 30 130
Marcinkiewicz, J. 147 155 158 211
Marcinkiewicz, theorem of 147 155 158
Maximum modulus 134 196
Medgyessy, P. 106 211
Median 109
Mixtures of d. 199
Modulus, maximum 134 196
Moivre — Laplace, law of 14
moment 17 19 27 32 47 136
Moment generating function 18 135 136 178
Moment, absolute 17
Moment, algebraic 17
Moment, factorial 17
Moment, symmetric 32
Moments, table of 21
Monotone Convergence Theorem 172
Natanson, I.P. 209 211
Negative binomial d.f. 14 21 26 80 93
Noack, A. 21 211
Non-degenerate d. 141
Non-negative definite function 61
Normal d. 15 21 26 80 93 102 106 130 147 169 173 182 184
O, o 207
One-sided d. 106 139
Order (of entire function) 134 135 142 173 195
Paley, R.E.A.C. 67 211
Parseval’s theorem 68
Parts of a d.f. 12
Pascal d. 14
Periodic c.f. 158
Pitman, E.J.G. 33 211
Plancherel’s theorem 67 68
Point of increase 10
Poisson d. 14 21 26 80 93 127 130 160 174.
Poisson d., compound 200
Poisson d., generalized 200
Poisson frequencies 181
Poisson spectrum 180 181
Poisson spectrum, (bounded, denumerable, finite, negative, positive) 180 181
Poisson type, d. 83 84
Pollard, H. 105 211
Polya type c.f. 72
Polya’s condition 70 72
Polynomial, trigonometric 35 208
Positive definite differential equation 161
Prime factor 112 123
Pringsheim, A. 71
Probability density function 12
Probability generating function 18 124
Product representation, trivial 78
Pure d.f. 12 24
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