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Miller K.S. — Complex stochastic processes
Miller K.S. — Complex stochastic processes

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Название: Complex stochastic processes

Автор: Miller K.S.

Аннотация:

When the student of engineering or applied science is first exposed to stochastic processes, or noise theory, he is usually content to manipulate random variables formally as if they were ordinary functions. Sometime later the serious student becomes concerned about such problems as the validity of differentiating random variables and the interpretation of stochastic integrals, to say nothing of the usual worries associated with the interchange of the order of integration in multiple integrals. It is to this class of readers that this book is addressed. We attempt to analyze problems of the type just mentioned at an elementary yet rigorous level, and to adumbrate some of the physical applications of the theory.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1974

Количество страниц: 238

Добавлена в каталог: 13.10.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Additive set function      6
Almost certain convergence      14
Almost sure convergence      14
Approximating sum      102
Autoregressive process      134
Best linear unbiased estimate      214
Bhattacharyya coefficient      80
Bochner’s theorem      42
Borel field      6
Borel — Cantelli lemma      10
Casorati      134
Cauchy criterion      21
Cauchy criterion in distribution      18 72 109
Cauchy, criterion      21
Cauchy, distribution      88
Characteristic function      12 38 82 83
Chi-square distribution      224
Complex, density function      77
Complex, Gaussian density function      78
Complex, linear least squares      (see Least squares)
Complex, random variable      14
Complex, signal      140
Complex, stochastic process      31
Continuity in mean square      55
Continuity of sample functions      48 55
Continuity with probability one      48
Contracting sequence      3
Convergence in mean square      19
Convergence in measure      17
Convergence in P-measure      17
Convergence in probability      17
Convergence in quadratic mean      19
Convergence of stochastic sequences      14 17 19
Convergence of stochastic series      23
Convergence with probability one      14
Convergence, almost everywhere      14
Convergence, Cauchy criterion      21
Correlation function      44
Covariance, function      31 69
Covariance, input-output      125
Covariance, matrix      35
Cramer — Rao lower bound      141
Critical region      188
Density function      13
Density function, complex Gaussian      78
Detection probability      188
Difference equation      (see Linear difference equation)
Differentiability in mean square      62
Differentiability of sample functions      59
Differentiability with probability one      59
Differential equation      (see Linear differential equation)
Differentiation with respect to a matrix      211
Distributed, identically      40
Distribution of amplitudes      87 89 92
Distribution of phases      88 89
Distribution, Cauchy      88
Distribution, convergence in      18 72 109
Distribution, function      10
Distribution, Gaussian      36 78
Distribution, joint      13
Distribution, normal      36 78
Distribution, Rayleigh      87
Distribution, spectral      44 136
Efficient statistic      141 153
Elliptic integral      97
Equivalent, random variables      14
Equivalent, stochastic processes      31
Ergodic property      119
Estimation of deterministic signals      154
Estimation of random signals      150
Estimation of spectral density      161
Estimation of spectral moments      159
Expanding sequence      3
Expectation      11
False alarm probability      188
Fisher information matrix      145
Frequency function      13
Function, characteristic      12 38 82 83
Function, complex Gaussian      78
Function, correlation      44
Function, covariance      31 69
Function, density      13
Function, distribution      10 13
Function, frequency      13
Function, generating      82
Function, left continuous      11
Function, likelihood      140 145 189 190
Function, mean      31
Function, measurable      10
Function, measure      6
Function, moment generating      37 84 85
Function, monotone      10
Function, nonnegative definite      12 31
Function, positive definite      31
Function, probability density      13
Function, regression      214 226
Function, right continuous      11
Function, sample      48
Function, set      6
Function, spectral distribution      44
Function, system      132 137
Gauss — Markov theorem      216
Gaussian distribution      36
Gaussian distribution, multivariate      36 78
Gaussian process, complex      38
Gaussian process, real      38
Generalized variance      229
Generating function      82
Green’s function      120 134
Grettenberg’s theorem      41
Hermitian matrix      32
Hermitian matrix, conjugate      32
Hermitian matrix, nonnegative definite      33
Hermitian matrix, positive definite      33
Hermitian matrix, skew      33
Hilbert space      20
Hypothesis testing      188
Identically distributed      40
Impulsive response      132 137
indefinite      197
Index set      31
Indicial polynomial      125 136
Inequality, Kantorovich      220
Inequality, Markov      118
Inequality, Schwarz      21
Inequality, triangle      21
Inequality, Tschebyscheff      16
Input-output covariance      125
Integral, mean square      103
Integral, Riemann — Stieltjes      11 125
Integral, sample function      103
Integral, Stochastic      103
Integral, stochastic, generalized      111
Integral, stochastic, infinite      114
Jacobi polynomial      86
Jacobi — Anger formula      94
Jacobi’s theorem      82
Kantorovich’s inequality      220
Karhunen — Loeve expansion      113
Lagrange multiplier      214
Laguerre polynomial      85
Least squares, dependent variables      212
Least squares, efficiency      219
Least squares, equivalence of estimates      217
Least squares, estimate      212
Least squares, Gauss — Markov theorem      216
Least squares, independent variables      212
Least squares, Markov estimate      215
Least squares, normal equation      213
Least squares, regression function      214 226
Least squares, residual      214
Likelihood, equation      140
Likelihood, function      140 145 189 190
Likelihood, ratio statistic      191
Limit in the mean      20
Limit, inferior      2
Limit, superior      2
Linear difference equation      134
Linear difference equation, Casorati      134
Linear difference equation, Green’s function      134
Linear difference equation, impulsive response      137
Linear difference equation, indicial polynomial      136 137
Linear difference equation, stochastic      134
Linear difference equation, system function      137
Linear differential equation      120
Linear differential equation, Green’s function      120
Linear differential equation, impulsive response      132
Linear differential equation, indicial polynomial      125 132
Linear differential equation, stochastic      122
Linear differential equation, system function      132
Linear differential equation, Wronskian      121
Linear index set      31
Loeve criterion      21
Markov estimate      215
Markov inequality      118
Mathematical expectation      11
Matrix, covariance      35
Matrix, differentiability with respect to      211
Matrix, Fisher information      145
Matrix, hermitian      32
Matrix, indefinite      197
Matrix, negative definite      193
Matrix, negative semidefinite      200
Matrix, nonnegative definite      33 200
Matrix, nonpositive definite      200
Matrix, nonsingular      32
Matrix, positive definite      33
Matrix, positive semidefinite      200
Matrix, semidefinite      200
Matrix, skew-symmclric      2
Matrix, symmetric      32 34
Matrix, unitary      33
Maximum likelihood estimator      140 222
Mean, frequency      153
Mean, function      31
Mean, square continuity      55
Mean, square differentiability      62
Mean, square integral      103
Mean, square regression plane      227
Mean, vector      35
Measurable function      10
Measure function      6
Mercer’s theorem      112
Missed signal probability      188
Moment generating function      37 84 85
Moments of phases      94
Moments, nonintegral      92
Moments, spectral      153 159
Neumann addition formula      92
Neumann factor      92
Neyman — Pearson lemma      189
Nonintegral moments      92
Nonnegative definite      12 31 33 200
Nonpositive definite      200
Norm      102
Normal distribution      36
Normal distribution, multivariate      36 78
Normal equation      212 213
One-sided Green’s function      120 134
Parameter set      31
Parseval’s theorem      180
Partitfon      102
Positive definite      31 33
Power of the test      188
Power spectrum      44
Price’s theorem      81
Probability density function      13
Probability measure      6
Probability space      6
Process, stochastic      (see Stochastic process)
Pulse-pair      188
Quadratic form      33
Quadratic form, nonnegative definite      33
Quadratic form, positive definite      33
Random process      (see Stochastic process)
Random variable      10
Random variable, complex      14
Random variable, continuity      48
Random variable, differentiability      58
Random variable, equivalent      14
Rayleigh distribution      87
Regression, coefficients      226
Regression, curves      226
Regression, function      214 226
Regression, plane      227
Residual      214
Riemann — Stieltjes integral      11 125
Sample function      48
Sample function, continuity      48
Sample function, differentiability      59
Sample function, integral      103
Schwarz inequality      21
Semidefinite      200
Sequences of random variables      14
Sequences of random variables, Cauchy criterion      21
Sequences of random variables, Loeve criterion      21
Sequences of sets      2
Sequences of sets, contracting      3
Sequences of sets, expanding      3
Sequences of sets, limit      4
Sequences of sets, limit inferior      2
Sequences of sets, limit superior      2
Set, class      2
Set, empty      2
Set, function      6
Set, index      31
Set, sequence      2
Sigma-algebra      6
Simple hypothesis      188
Skew — Hermitian matrix      33
Space, abstract      6
Space, probability      6
Spectral density      44
Spectral distribution function      44 136
Spectral moment estimation, addition of noise      167
Spectral moment estimation, coherent      161 169
Spectral moment estimation, covariance approach      164
Spectral moment estimation, mathematical justification      179
Spectral moment estimation, noncoherent      169
Spectral moment estimation, spectral approach      164
Spectral moment estimation, statistics      174
Spectral moments      153 159
SPREAD      153
Stationary in the strict sense      32
Stationary in the wide sense      32
Stationary stochastic process      32
Stationary to the second order      32
Statistic, efficient      141 153
Statistic, spectral moment      174
Statistic, sufficient      141 223
Stochastic, autoregressive      134
Stochastic, complex      31
Stochastic, convergence      17
Stochastic, difference equation      134
Stochastic, differential equation      122
Stochastic, examples      25
Stochastic, Gaussian      38
Stochastic, generalized      111
Stochastic, infinite      114
Stochastic, integral      103
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