# Objective

This course lays out the fundamental concepts in the probability calculus. It shows how the tools of the theory of measure, introduced in the course "Fondements mathématiques des probabilités" (mathematical foundations of probability), can be adapted to the probabilistic model. Many concrete cases illustrate the problem of probabilistic modelling. Next, the common techniques are explained, and conditional and convergence laws are studied in detail. The fundamental limit theorems are demonstrated.

# Planning

1. Reminders of theory of measure and theory of integration– Model of probability space; measurable applications; image measure and transfer theorem; measures that admit a density in relation to another.
2. Random variables: characteristics, moments and changes of variable– Introduction: concept of a random variable; Determining and characterising the laws of random variables; Study of the moments of a random variable; The variable change problem; Various applications.
3. Study of normal laws– Normal laws over \$mathbb{R}\$; Normal laws over \$mathbb{R}^n\$; Usual laws derived from normal laws; Algebraic and geometric properties of normal laws.
4. Characteristic functions– Definition and initial properties of characteristic functions of real random variables; Analytical properties of characteristic functions; Inversion and injectivity theorems of characteristic functions and reciprocity formulae; Characteristic functions of random variables with values in \$mathbb{R}\$.
5. Pointwise and functional convergence– Almost sure convergence; Uniform almost sure convergence (or in \$L_infty\$); Convergence in probability (or stochastic processes); Convergence in \$L^p\$ spaces.
6. Convergence in distribution– Definition of convergence in distribution; Usual criteria for convergence in distribution; The Paul Levy theorem; Properties of convergence in distribution; use of Taylor expansions.
7. Asymptotic theory– Laws of large numbers; Exceptions to the law of large numbers; Central limit theorem: usual version and various extensions.
8. Conditioning and conditional expected value– Conditioning by an event in the elementary case; "Geometric" theory of conditional expected value; Extension: general theory of conditional expected value; General theory of the conditional laws of probability; Reinterpretation of conditional expected value based on the conditional law of probability.

# Références

BASS, Eléments de calcul des probabilités, Masson, [16 BAS 00 A]BILINGSLEY, Probability and Measure, second edition, Wiley, [16 BIL 00 B],
COTTREL et Cie, Exercices de probabilités, Cassini, [16 COT 00 B]DACUNHA-CASTELLE, REVUZ et SCHREIBER, Recueil de problèmes de calcul des probabilités, Masson, [16 DAC 00 C]DACUNHA-CASTELLE ET DUFLO, Probabilités et Statistiques, Tome 1: problèmes à temps fixe, Masson, [16 DAC 00 A]OUVRARD, J-Y.  Probabilités 2. Master et agrégation, Cassini. [16 OUV 00A2]

Poly de cours de Benjamin Jourdain