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Foreword ix
Preface xiii
1 Basic notions of probability theory 1
1.1 Random variables, their distributions and moments ........... 1
1.2 Generating and characteristic functions ........1..... .... 11
1.3 Random vectors. Stochastic independence ................ 21
1.4 Weak convergence of random variables and distribution functions . 24
1.5 Poisson theorem ............................. . 30
1.6 Law of large numbers. Central limit theorem. Stable laws ........ 35
1.7 The Berry-Esseen inequality ..... ............ 45
1.8 Asymptotic expansions in the central limit theorem ........... 47
1.9 Elementary properties of random sums ................... 56
1.10 Stochastic processes............................. 62
2 Poisson process 69
2.1 The definition and elementary properties of a Poisson process . ... 69
2.2 Poisson process as a model of chaotic displacement of points in time . . 72
2.3 The asymptotic normality of a Poisson process .... .. ........ 74
2.4 Elementary rarefaction of renewal processes ................ 76
3 Convergence of superpositions of independent stochastic processes 83
3.1 Characteristic features of the problem . ....... ........... 83
3.2 Approximation of distributions of randomly indexed random sequences
by special mixtures.............................. 85
3.3 The transfer theorem. Relations between the limit laws for random
sequences with random and non-random indices ............. 89
3.4 Necessary and sufficient conditions for the convergence of distributions
of random sequences with independent random indices ......... 91
3.5 Convergence of distributions of randomly indexed sequences to identi-
fiable location or scale mixtures. The asymptotic behavior of extremal
random sums ................................. 98
3.6 Convergence of distributions of random sums. The central limit theorem
and the law of large numbers for random sums ............. 105
3.7 A general theorem on the asymptotic behavior of superpositions of in-
dependent stochastic processes . ...................... 115
3.8 The transfer theorem for random sums of independent identically dis-
tributed random variables in the double array limit scheme ....... 117
4 Compound Poisson distributions 123
4.1 Mixed and compound Poisson distributions ................ 123
4.2 Discrete compound Poisson distributions .................. 129
4.3 The asymptotic normality of compound Poisson distributions. The
Berry-Esseen inequality for Poisson random sums. Non-central Lya-
punov fractions.............................. .. 133
4.4 Asymptotic expansions for compound Poisson distributions ........ 139
4.5 The asymptotic expansions for the quantiles of compound Poisson dis-
tributions.............................. ...... 151
4.6 Exponential inequalities for the probabilities of large deviations of Pois-
son random sums. An analog of Bernshtein-Kolmogorov inequality . . 155
4.7 The application of Esscher transforms to the approximation of the tails
of compound Poisson distributions ..... ................ 157
4.8 Estimates of convergence rate in local limit theorems for Poisson random
sums ....................................... 166
5 Classical risk processes 181
5.1 The definition of the classical risk process. Its asymptotic normality . .181
5.2 The Pollaczek-Khinchin-Beekman formula for the ruin probability in
the classical risk process ..... . ..................... 185
5.3 Approximations for the ruin probability with small safety loading . . . 189
5.4 Asymptotic expansions for the ruin probability with small safetyloading 191
5.5 Approximations for the ruin probability .. .............. 203
5.6 Asymptotic approximations for the distribution of the surplus in general
risk processes .......................................... 213
5.7 A problem of inventory control ........................ 221
5.8 A non-classical problem of optimization of the initial capital ....... 227
6 Doubly stochastic Poisson processes (Cox processes) 233
6.1 The asymptotic behavior of random sums of random indicators ..... 233
6.2 Mixed Poisson processes ........................... 238
6.3 The modified Pollaczek-Khinchin-Beekman formula ........... 249
6.4 The definition and elementary properties of doubly stochastic Poisson
processes .............. ...................... 252
6.5 The asymptotic behavior of Cox processes ................. 256
7 Compound Cox processes with zero mean 265
7.1 Definition. Examples .................. ........... 265
7.2 Conditions of convergence of the distributions of compound Cox process-
es with zero mean. Limit laws ............ . .......... 266
7.3 Convergence rate estimates ......................... 269
7.4 Asymptotic expansions for the distributions of compound Cox processes
with zero mean ................................. 273
7.5 Asymptotic expansions for the quantiles of compound Cox processes
with zero mean ............................ 281
7.6 Exponential inequalities for the probabilities of large deviations of com-
pound Cox processes with zero mean ................... 283
7.7 Limit theorems for extrema of compound Cox processes with zero mean 285
7.8 Estimates of the rate of convergence of extrema of compound Cox pro-
cesses with zero mean ........................ 287
8 Modeling evolution of stock prices by compound Cox processes 291
8.1 Introduction .............. ................ 291
8.2 Normal and stable models ................. 292
8.3 Heterogeneity of operational time and normal mixtures .... 294
8.4 Inhomogeneous discrete chaos and Cox processes ..... ....... 297
8.5 Restriction of the class of mixing distributions . ............ 303
8.6 Heavy-tailedness of scale mixtures of normals ............... 307
8.7 The case of elementary increments with non-zero means . .... . 308
8.8 Models within the double array limit scheme ............. 310
8.9 Quantiles of the distributions of stock prices ........... . 313
9 Compound Cox processes with nonzero mean 317
9.1 Definition. Examples...... .... ... ....... 317
9.2 Conditions of convergence of compound Cox processes with nonzero
mean. Limit laws ............................... 318
9.3 Convergence rate estimates for compound Cox processes with nonzero
m ean .... .. ... ..................... ... 322
9.4 Asymptotic expansions for the distributions of compound Cox processes
with nonzero mean ................... .... ... 326
9.5 Asymptotic expansions for the quantiles of compound Cox processes
with nonzero mean .............................. 338
9.6 Exponential inequalities for the negative values of the surplus in collec-
tive risk models with stochastic intensity of insurance payments . .. 339
9.7 Limit theorems for extrema of compound Cox processes with nonzero
m ean .............................. . .. . 342
9.8 Convergence rate estimates for extrema of compound Cox processes with
nonzero mean .................... ...... 347
9.9 Minimum admissible reserve of an insurance company with stochastic
intensity of insurance payments ...... .............. 350
9.10 Optimization of the initial capital of an insurance company in a static
insurance model with random portfolio size ............. . 351
10 Functional limit theorems for compound Cox processes 357
10.1 Functional limit theorems for non-centered compound Cox processes .. 357
10.2 Functional limit theorems for nonrandomly centered compound Cox pro-
cesses. .. .. .. .. .. .. ...... .. .. .. .. . . . . . . . . . 363
11 Generalized risk processes 373
11.1 The definition of generalized risk processes . ............. 373
11.2 Conditions of convergence of the distributions of generalized risk pro-
cesses. ..................... ....... 375
11.3 Convergence rate estimates for generalized risk processes ........ 378
11.4 Asymptotic expansions forthe distributions ofgeneralized risk processes 381
11.5 Asymptotic expansions for the quantiles of generalized risk processes . 384
11.6 Exponential inequalities for the probabilities of negative values of gen-
eralized risk processes ................386
12 Statistical inference concerning the parameters of risk processes 391
12.1 Statistical estimation of the ruin probability in classical risk processes 391
12.2 Specific features of statistical estimation of ruin probability for general-
ized risk processes ............................... 395
12.3 A nonparametric estimator of the ruin probability for a generalized risk
process ..................................... 398
12.4 Interval estimator of the ruin probability for a generalized risk process 404
12.5 Computational aspects of the construction of confidence intervals for the
ruin probability in generalized risk processes .............. 412
Bibliography 415
Index 431
Library of Congress Subject Headings for this publication: Insurance Mathematical models, Finance Mathematical models, Poisson processes, Poisson distribution