Simulation has now become an integral part of research and development across many fields of study. Despite the large amounts of literature in the field of simulation and modeling, one recurring problem is the issue of accuracy and confidence level of constructed models.
By outlining the new approaches and modern methods of simulation of stochastic processes, this book provides methods and tools in measuring accuracy and reliability in functional spaces.
The authors explore analysis of the theory of Sub-Gaussian (including Gaussian one) and Square Gaussian random variables and processes and Cox processes. Methods of simulation of stochastic processes and fields with given accuracy and reliability in some Banach spaces are also considered.
1. Introduction - p. ix 2. 1: The Distribution of the Estimates for the Norm of Sub-Gaussian Stochastic Processes - p. 1 3. Abstract - p. 1 4. 1.1 The space of sub-Gaussian random variables and sub-Gaussian stochastic processes - p. 2 5. 1.2 The space of strictly sub-Gaussian random variables and strictly sub-Gaussian stochastic processes - p. 14 6. 1.3 The estimates of convergence rates of strictly sub-Gaussian random series in L2(T) - p. 24 7. 1.4 The distribution estimates of the norm of sub-Gaussian stochastic processes in Lp(T) - p. 27 8. 1.5 The distribution estimates of the norm of sub-Gaussian stochastic processes in some Orlicz spaces - p. 30 9. 1.6 Convergence rate estimates of strictly sub-Gaussian random series in Orlicz spaces - p. 34 10. 1.7 Strictly sub-Gaussian random series with uncorrelated or orthogonal items - p. 42 11. 1.8 Uniform convergence estimates of sub-Gaussian random series - p. 48 12. 1.9 Convergence estimate of strictly sub-Gaussian random series in C(T) - p. 58 13. 1.10 The estimate of the norm distribution of Lp-processes - p. 69 14. 2: Simulation of Stochastic Processes Presented in the Form of Series - p. 71 15. Abstract - p. 71 16. 2.1 General approaches for model construction of stochastic processes - p. 71 17. 2.2 Karhunen–Loéve expansion technique for simulation of stochastic processes - p. 73 18. 2.3 Fourier expansion technique for simulation of stochastic processes - p. 84 19. 2.4 Simulation of stationary stochastic process with discrete spectrum - p. 93 20. 2.5 Application of Fourier expansion to simulation of stationary stochastic processes - p. 102 21. 3: Simulation of Gaussian Stochastic Processes with Respect to Output Processes of the System - p. 105 22. Abstract - p. 105 23. 3.1 The inequalities for the exponential moments of the quadratic forms of Gaussian random variables - p. 107 24. 3.2 The space of square-Gaussian random variables and square-Gaussian stochastic processes - p. 116 25. 3.3 The distribution of supremums of square-Gaussian stochastic processes - p. 117 26. 3.4 The estimations of distribution for supremum of square-Gaussian stochastic processes in the space [0, T]d - p. 126 27. 3.5 Accuracy and reliability of simulation of Gaussian stochastic processes with respect to the output process of some system - p. 133 28. 3.6 Model construction of stationary Gaussian stochastic process with discrete spectrum with respect to output process - p. 143 29. 3.7 Simulation of Gaussian stochastic fields - p. 157 30. 4: The Construction of the Model of Gaussian Stationary Processes - p. 169 31. 5: The Modeling of Gaussian Stationary Random Processes with a Certain Accuracy and Reliability - p. 181 32. Abstract - p. 181 33. 5.1 Reliability and accuracy in Lp(T), p ≥ 1 of the models for Gaussian stationary random processes - p. 181 34. 5.2 The accuracy and reliability of the model stationary random processes in the uniform metric - p. 202 35. 5.3 Application of Subφ(Ω) space theory to find the accuracy of modeling for stationary Gaussian processes - p. 222 36. 5.4 Generalized model of Gaussian stationary processes - p. 241 37. 6: Simulation of Cox Random Processes - p. 251 38. Abstract - p. 251 39. 6.1 Random Cox processes - p. 251 40. 6.2 Simulation of log Gaussian Cox processes as a demand arrival process in actuarial mathematics - p. 253 41. 6.3 Simplified method of simulating log Gaussian Cox processes - p. 268 42. 6.4 Simulation of the Cox process when density is generated by a homogeneous log Gaussian field - p. 280 43. 6.5 Simulation of log Gaussian Cox process when the density is generated by the inhomogeneous field - p. 286 44. 6.6 Simulation of the Cox process when the density is generated by the square Gaussian random process - p. 292 45. 6.7 Simulation of the square Gaussian Cox process when density is generated by a homogeneous field - p. 298 46. 6.8 Simulation of the square Gaussian Cox process when the density is generated by an inhomogeneous field - p. 301 47. 7: On the Modeling of Gaussian Stationary Processes with Absolutely Continuous Spectrum - p. 305 48. 8: Simulation of Gaussian Isotropic Random Fields on a Sphere - p. 315 49. Bibliography - p. 325 50. Index - p. 333
Simulation has now become an integral part of research and development across many fields of study. Despite the large amounts of literature in the field of simulation and modeling, one recurring problem is the issue of accuracy and confidence level of constructed models.
By outlining the new approaches and modern methods of simulation of stochastic processes, this book provides methods and tools in measuring accuracy and reliability in functional spaces.
The authors explore analysis of the theory of Sub-Gaussian (including Gaussian one) and Square Gaussian random variables and processes and Cox processes. Methods of simulation of stochastic processes and fields with given accuracy and reliability in some Banach spaces are also considered.
1. Introduction - p. ix 2. 1: The Distribution of the Estimates for the Norm of Sub-Gaussian Stochastic Processes - p. 1 3. Abstract - p. 1 4. 1.1 The space of sub-Gaussian random variables and sub-Gaussian stochastic processes - p. 2 5. 1.2 The space of strictly sub-Gaussian random variables and strictly sub-Gaussian stochastic processes - p. 14 6. 1.3 The estimates of convergence rates of strictly sub-Gaussian random series in L2(T) - p. 24 7. 1.4 The distribution estimates of the norm of sub-Gaussian stochastic processes in Lp(T) - p. 27 8. 1.5 The distribution estimates of the norm of sub-Gaussian stochastic processes in some Orlicz spaces - p. 30 9. 1.6 Convergence rate estimates of strictly sub-Gaussian random series in Orlicz spaces - p. 34 10. 1.7 Strictly sub-Gaussian random series with uncorrelated or orthogonal items - p. 42 11. 1.8 Uniform convergence estimates of sub-Gaussian random series - p. 48 12. 1.9 Convergence estimate of strictly sub-Gaussian random series in C(T) - p. 58 13. 1.10 The estimate of the norm distribution of Lp-processes - p. 69 14. 2: Simulation of Stochastic Processes Presented in the Form of Series - p. 71 15. Abstract - p. 71 16. 2.1 General approaches for model construction of stochastic processes - p. 71 17. 2.2 Karhunen–Loéve expansion technique for simulation of stochastic processes - p. 73 18. 2.3 Fourier expansion technique for simulation of stochastic processes - p. 84 19. 2.4 Simulation of stationary stochastic process with discrete spectrum - p. 93 20. 2.5 Application of Fourier expansion to simulation of stationary stochastic processes - p. 102 21. 3: Simulation of Gaussian Stochastic Processes with Respect to Output Processes of the System - p. 105 22. Abstract - p. 105 23. 3.1 The inequalities for the exponential moments of the quadratic forms of Gaussian random variables - p. 107 24. 3.2 The space of square-Gaussian random variables and square-Gaussian stochastic processes - p. 116 25. 3.3 The distribution of supremums of square-Gaussian stochastic processes - p. 117 26. 3.4 The estimations of distribution for supremum of square-Gaussian stochastic processes in the space [0, T]d - p. 126 27. 3.5 Accuracy and reliability of simulation of Gaussian stochastic processes with respect to the output process of some system - p. 133 28. 3.6 Model construction of stationary Gaussian stochastic process with discrete spectrum with respect to output process - p. 143 29. 3.7 Simulation of Gaussian stochastic fields - p. 157 30. 4: The Construction of the Model of Gaussian Stationary Processes - p. 169 31. 5: The Modeling of Gaussian Stationary Random Processes with a Certain Accuracy and Reliability - p. 181 32. Abstract - p. 181 33. 5.1 Reliability and accuracy in Lp(T), p ≥ 1 of the models for Gaussian stationary random processes - p. 181 34. 5.2 The accuracy and reliability of the model stationary random processes in the uniform metric - p. 202 35. 5.3 Application of Subφ(Ω) space theory to find the accuracy of modeling for stationary Gaussian processes - p. 222 36. 5.4 Generalized model of Gaussian stationary processes - p. 241 37. 6: Simulation of Cox Random Processes - p. 251 38. Abstract - p. 251 39. 6.1 Random Cox processes - p. 251 40. 6.2 Simulation of log Gaussian Cox processes as a demand arrival process in actuarial mathematics - p. 253 41. 6.3 Simplified method of simulating log Gaussian Cox processes - p. 268 42. 6.4 Simulation of the Cox process when density is generated by a homogeneous log Gaussian field - p. 280 43. 6.5 Simulation of log Gaussian Cox process when the density is generated by the inhomogeneous field - p. 286 44. 6.6 Simulation of the Cox process when the density is generated by the square Gaussian random process - p. 292 45. 6.7 Simulation of the square Gaussian Cox process when density is generated by a homogeneous field - p. 298 46. 6.8 Simulation of the square Gaussian Cox process when the density is generated by an inhomogeneous field - p. 301 47. 7: On the Modeling of Gaussian Stationary Processes with Absolutely Continuous Spectrum - p. 305 48. 8: Simulation of Gaussian Isotropic Random Fields on a Sphere - p. 315 49. Bibliography - p. 325 50. Index - p. 333
