Processus stochastiques (statistique) [ LMAT2470 ]
5.0 crédits ECTS
30.0 h
1q
| Teacher(s) |
Johannes Jan ;
|
| Language |
French
|
Place
of the course |
Louvain-la-Neuve
|
| Main themes |
Discrete time Martingales (sub-martingales and super-martingales), stationary processes, exchangeable processes, conditionally i.i.d. processes and Markov processes.
|
| Aims |
Presentation of the main discrete time stochastic processes with an introduction to their statistical analysis.
|
| Content |
Content
1. Discrete time stochastic processes. Generalities, asymptotic, invariant and exchangeable sigma-algebra's. Stopping times and hitting times.
2. Martingales, sub-martingales and super-martingales. Optional stopping theorems, almost sure and in mean convergences. Inverse martingales, strong law of large numbers, 0-1 laws of Kolmogorov and Hewitt-Savage.
3. Stationary processes. Ergodic theorem, almost surely invariant sigma-algebra, almost sure and in mean convergences. Recurrence theorems of Poincaré.
4. Exchangeable processes. Almost surely exchangeable sigma-algebra, De Finetti's theorem, representation of an exchangeable process as a conditionally i.i.d. process.
5. Conditionally i.i.d. processes. Identification and almost sure equality between the invariant and the conditioning sigma-algebra. Optional stopping theorems and hitting times.
6. Conditional Markov processes. Strong Markov property. Homogeneous and stationary Markov processes. Characterization of the invariant sigma-algebra and ergodicity conditions.
7. Regular Markov processes. Stationary initial distribution, ergodicity and Dublin condition. Ergodic theorems for a non-stationary Markov process.
Methods
The course is given in two hours lectures during 14 weeks.
|
| Other information |
Prerequisite :
The courses MAT1322 Théorie de la mesure and MAT1371 Probabilités are an essential prerequisite.
References :
NEVEU, J., Martingales à temps discret, Masson, 1972.
BREIMAN, L., Probability, Addison-Wesley, 1968.
CHOW, Y.S. and M. TEICHER, Probability Theory: Independence, Interchangeability, Martingales, Springer-Verlag, 1987.
CHUNG K.L., A Course in Probability Theory. Harcourt, Brace & World Inc., 1968.
KARLIN S. and H.M. TAYLOR, A First Course in Stochastic Processes, Academic Press, 1975.
Evaluation :
Each student receives 5 exercices to solve. He writes up the solutions and orally presents them to the professor. who may ask theoretical questions related to the subject of the proposed exercices.
|
Cycle et année
d'étude |
> Master [120] in Mathematics
> Master [120] in Statistics: General
> Master [120] in Physics
|
Faculty or entity
in charge |
> MATH
|
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