Stochastic Processes

I Introduction.- 1. Examples.- 2. Definitions.- 3. Classification according to the nature of E.- 4. Classification according to the nature of the set T of instants where the system is observed.- II Poisson processes.- A. Generalities - Point processes.- 1. Definition.- 2. Point processes.- 3. Definition of a point process.- B. The uniform Poisson process.- 4. Definition of the Poisson point process.- 5. Distribution of the random interval between two consecutive events.- 6. Conclusion.- 7. The Gamma probability distributions.- 8. Distribution of " survival " of an interval that obeys ?1.- 9. Distribution of the interval embracing (k+1) consecutive events.- 10. Probability distribution of the number of events occurring in a given interval of time.- 11. Second stochastical definition of the Poisson process.- 12. Conditional distributions.- 13. Number of occurrences in an interval of random length.- C. Probability distributions associated to Poisson processes.- 14. Poisson distribution.- 15. Gamma distribution.- 16. Beta distributions.- D. Extensions: Poisson-type-processes.- 17. Continuous, non-uniform Poisson processes.- 18. Poisson-type processes.- 19. Parameter-distribution.- 20. Decomposition ot the parameter distribution.- 21. Interval distribution and conditional distributions.- 22. Poisson cluster process.- Problems for solution.- Random functions associated to a Poisson process.- III Numerical processes with independent random increments.- 1. Definition.- 2. Mean values - zero mean process.- 3. Variance distribution.- 4. Cumulants and second characteristic functions.- 5. Indicator function of a Poisson process.- 6. The Wiener-Levy process.- 7. Laplace processes with independent increments.- 8. General form of the random functions X (t) with independent increments.- 9. Infinitely divisible distributions.- 10. Stable distributions.- Problems for solution.- IV Markov processes.- A. Generalities and definitions.- 1. Generalities.- 2. Notation.- 3. Representation by temporal development.- 4. Markov sequences.- 5. Probability law of states.- 6. Homogeneous Markov sequences.- B. Study by means of convexity.- 7. Limit for k = ? of Mk when all the terms of M are positive.- 8. Regular positive case.- C. Study by means of spectral analysis.- 9. Definitions.- 10. Eigenvalues and eigenvectors.- 11. Properties of the eigenvalues of a Markov matrix.- 12. Canonical representation.- D. Direct algebraic study.- 13. Classification of states.- 14. Order on classes: transient classes and final classes.- 15. Examples.- 16. Summary.- E. Reaching delays and sojourn duration problems.- 17. Introduction.- 18. Example.- 19. General method.- F. Miscellaneous.- 20. Ergodicity:.- 21. Inversion of a homogeneous Markov sequence.- Problems for solution.- V Laplace processes and second order processes.- 1. Introduction.- A. Second order properties.- 2. Second order processes.- 3. Stationary second order processes.- B. Laplace processes.- 4. Definition.- 5. Reminder of a few definitions and properties concerning Laplace random sets (or Laplace random vectors) of k dimensions.- 6. Covariance Function of a Laplace process.- 7. Discrete process - Levy canonical decomposition.- Problems for solution.- VI Some Markov processes on continuous-time.- 1. Homogeneous Laplace-Markov sequences.- 2. Stationary Laplace-Markov sequence.- 3. Estimation problems.- 4. Interpolation - permanent process.- 5. Non homogeneous standardized Laplace-Markov processes.- 6. General form of Laplace-Markov processes.- 7. Wiener-Levy processes.- 8. Poisson-Markov processes.- Problems for solution.- Answers to problems.