Probability and Stochastic Processes


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


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Defended Theses

  • Structure, algorithmics and dynamics of endomorphisms for certain classes of groups
      André da Cruz Carvalho (April 2023)
      Pedro V. Silva
  • A point-free study of z-embeddings, more general classes of localic maps, and uniform continuity
      Ana Belén Avilez García (April 2023)
      Jorge Picado
  • Analysis of equations of motion of inextensible strings and networks
      Ayk Telciyan (April 2023)
      Dmitry Vorotnikov
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