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Probability and Stochastic Processes

Program

1 Preliminaries
1.1 Probability spaces
1.2 Integration
1.3 Absolute continuity
1.4 Notions of convergence and Slutsky's theorem
2 Random variables and Stochastic processes
2.1 Distributions and Skhorokhod's representation
2.2 Kolmogorov's existence theorem
2.3 Independence
2.4 Borel-Cantelli Lemmas
2.5 Kolmogorv's 0-1 Law
2.4 Conditional expectation
3 Martingales
3.1 Definitions and properties
3.2 Stopping times and inequalities
3.3 (Sub)martingale convergence theorem
3.4 Central limit theorem
3.5* Application to mixing stationary processes (the Gordin approximation)
4 Brownian motion
4.1 Continuity of paths and their irregularity
4.2 Strong Markov property and reflection principle
4.3 Skorohod's Embedding
5 Weak convergence
5.1 Portmanteau theorem
5.2 Tightness and Prokhorov's theorem
5.3 Weak convergence in C[0,1]
5.4 Donsker's theorem and Invariance principle

Research and Events

Events

  • Research Seminar Program (RSP)
    2024/2025
    University of Coimbra, FCTUC, Department of Mathematics, room 2.5
    December 16, 2024
  • PhD Defense
    Rafael Henriques - Numerical methods for the robust reconstruction of elasticity
    Sala José Anastácio da Cunha, Mathematical Building, Largo D. Dinis, Coimbra (14h30)
    January 6, 2025
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Defended Theses

  • Contributions to regularity theory in the calculus of variations
      Vincenzo Bianca (July 2024)
      José Miguel Urbano
  • Higgs Bundles and Geometric Structures
      Pedro Miguel Silva (April 2024)
      Peter Gothen
  • Some aspects of descent theory and applications
      Rui Rodrigues de Abreu Fernandes Prezado (January 2024)
      Maria Manuel Clementino
      Fernando Lucatelli Nunes
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