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Lukacs E. — Characterisic functions
Lukacs E. — Characterisic functions

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Название: Characterisic functions

Автор: Lukacs E.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1960

Количество страниц: 215

Добавлена в каталог: 09.03.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$N_a$-value      109
$N_a[f(t)], N_a(f)$      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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