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Title:
Intermediate Probability A Computational Approach
Marc S. Paolella
Swiss Banking Institute, University of Zurich, Switzerland
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Chapter Listing
Preface
Part I Sums of Random Variables
1 Generating functions
2 Sums and other functions of several random variables
3 The multivariate normal distribution
Part II Asymptotics and Other Approximations
4 Convergence concepts
5 Saddlepoint approximations
6 Order statistics
Part III More Flexible and Advanced Random Variables
7 Generalizing and mixing
8 The stable Paretian distribution
9 Generalized inverse Gaussian and generalized hyperbolic distributions
10 Noncentral distributions
Appendix
A Notation and distribution tables
References
Index
Contents
Preface xi
Part I Sums of Random Variables 1
1 Generating functions 3
1.1 The moment generating function 3
1.1.1 Moments and the m.g.f. 4
1.1.2 The cumulant generating function 7
1.1.3 Uniqueness of the m.g.f. 11
1.1.4 Vector m.g.f. 14
1.2 Characteristic functions 17
1.2.1 Complex numbers 17
1.2.2 Laplace transforms 22
1.2.3 Basic properties of characteristic functions 23
1.2.4 Relation between the m.g.f. and c.f. 25
1.2.5 Inversion formulae for mass and density functions 27
1.2.6 Inversion formulae for the c.d.f. 36
1.3 Use of the fast Fourier transform 40
1.3.1 Fourier series 40
1.3.2 Discrete and fast Fourier transforms 48
1.3.3 Applying the FFT to c.f. inversion 50
1.4 Multivariate case 53
1.5 Problems 55
2 Sums and other functions of several random variables 65
2.1 Weighted sums of independent random variables 65
2.2 Exact integral expressions for functions of two continuous random
variables 72
viii Contents
2.3 Approximating the mean and variance 85
2.4 Problems 90
3 The multivariate normal distribution 97
3.1 Vector expectation and variance 97
3.2 Basic properties of the multivariate normal 100
3.3 Density and moment generating function 106
3.4 Simulation and c.d.f. calculation 108
3.5 Marginal and conditional normal distributions 111
3.6 Partial correlation 116
3.7 Joint distribution of X and S2 for i.i.d. normal samples 119
3.8 Matrix algebra 122
3.9 Problems 124
Part II Asymptotics and Other Approximations 127
4 Convergence concepts 129
4.1 Inequalities for random variables 130
4.2 Convergence of sequences of sets 136
4.3 Convergence of sequences of random variables 142
4.3.1 Convergence in probability 142
4.3.2 Almost sure convergence 145
4.3.3 Convergence in r-mean 150
4.3.4 Convergence in distribution 153
4.4 The central limit theorem 157
4.5 Problems 163
5 Saddlepoint approximations 169
5.1 Univariate 170
5.1.1 Density saddlepoint approximation 170
5.1.2 Saddlepoint approximation to the c.d.f. 175
5.1.3 Detailed illustration: the normal–Laplace sum 179
5.2 Multivariate 184
5.2.1 Conditional distributions 184
5.2.2 Bivariate c.d.f. approximation 186
5.2.3 Marginal distributions 189
5.3 The hypergeometric functions 1F1 and 2F1 193
5.4 Problems 198
Contents ix
6 Order statistics 203
6.1 Distribution theory for i.i.d. samples 204
6.1.1 Univariate 204
6.1.2 Multivariate 210
6.1.3 Sample range and midrange 215
6.2 Further examples 219
6.3 Distribution theory for dependent samples 230
6.4 Problems 231
Part III More Flexible and Advanced Random Variables 237
7 Generalizing and mixing 239
7.1 Basic methods of extension 239
7.1.1 Nesting and generalizing constants 240
7.1.2 Asymmetric extensions 244
7.1.3 Extension to the real line 247
7.1.4 Transformations 249
7.1.5 Invention of flexible forms 252
7.2 Weighted sums of independent random variables 254
7.3 Mixtures 254
7.3.1 Countable mixtures 255
7.3.2 Continuous mixtures 258
7.4 Problems 269
8 The stable Paretian distribution 277
8.1 Symmetric stable 277
8.2 Asymmetric stable 281
8.3 Moments 287
8.3.1 Mean 287
8.3.2 Fractional absolute moment proof I 288
8.3.3 Fractional absolute moment proof II 293
8.4 Simulation 296
8.5 Generalized central limit theorem 297
9 Generalized inverse Gaussian and generalized hyperbolic distributions 299
9.1 Introduction 299
9.2 The modified Bessel function of the third kind 300
9.3 Mixtures of normal distributions 303
x Contents
9.3.1 Mixture mechanics 303
9.3.2 Moments and generating functions 304
9.4 The generalized inverse Gaussian distribution 306
9.4.1 Definition and general formulae 306
9.4.2 The subfamilies of the GIG distribution family 308
9.5 The generalized hyperbolic distribution 315
9.5.1 Definition, parameters and general formulae 315
9.5.2 The subfamilies of the GHyp distribution family 317
9.5.3 Limiting cases of GHyp 327
9.6 Properties of the GHyp distribution family 328
9.6.1 Location–scale behaviour of GHyp 328
9.6.2 The parameters of GHyp 329
9.6.3 Alternative parameterizations of GHyp 330
9.6.4 The shape triangle 332
9.6.5 Convolution and infinite divisibility 336
9.7 Problems 338
10 Noncentral distributions 341
10.1 Noncentral chi-square 341
10.1.1 Derivation 341
10.1.2 Moments 344
10.1.3 Computation 346
10.1.4 Weighted sums of independent central χ2 random variables 347
10.1.5 Weighted sums of independent χ2 (ni, θi ) random variables 351
10.2 Singly and doubly noncentral F 357
10.2.1 Derivation 357
10.2.2 Moments 358
10.2.3 Exact computation 360
10.2.4 Approximate computation methods 363
10.3 Noncentral beta 369
10.4 Singly and doubly noncentral t 370
10.4.1 Derivation 371
10.4.2 Saddlepoint approximation 378
10.4.3 Moments 381
10.5 Saddlepoint uniqueness for the doubly noncentral F 382
10.6 Problems 384
A Notation and distribution tables 389
References 401
Index 413
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