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Course syllabus - Advanced Probability Theory, 5 credits

Information about the course

  • Course code: FMAM001
  • Third-cycle subject: Mathematics/Applied Mathematics
  • School: The School for education, culture and communication
  • Valid from: Spring term 2026
  • Established by: Dean of SChool
  • Decision date: 2026-12-16
  • Level of education: Third cycle level

Course objective

The purpose of the course is to provide PhD students with a deep and coherent understanding of probability theory built on a measure-theoretic foundation. The course emphasizes key concepts such as probability measures, random variables, modes of convergence, laws of large numbers, the central limit theorem, conditional expectations, and martingales.

The course aims to strengthen the theoretical and analytical skills needed to understand and use modern research literature where probability theory plays a central role in mathematics and applied mathematics.

Course content

  • Probability spaces, σ-algebras, and probability measures
  • Measurable functions and random variables
  • Expectation as a Lebesgue integral and major convergence theorems (Monotone and Dominated Convergence)
  • Independence and product measures
  • Modes of convergence: almost sure, in probability, in $L^p$, and in distribution
  • Laws of large numbers (weak and strong forms)
  • Central limit theorem and characteristic functions
  • Conditional expectation and its properties
  • Martingales, stopping times, and martingale convergence
  • Applications and examples from stochastic analysis and applied mathematics

Intented learning outcomes

After completing the course, the student shall be able to:

  1. explain the central concepts of measure-theoretical probability,
  2. describe key results such as the laws of large numbers, the central limit theorem, and conditional expectation,
  3. demonstrate understanding of martingales and their role in modern probability theory,
  4. apply probabilistic theorems and methods to solve theoretical problems,
  5. analyze and present mathematical reasoning and proofs,
  6. use probability theory as a tool in related research areas,
  7. assess the correctness and relevance of probabilistic arguments.

The intended qualitative targets in relation to the Higher Education Ordinance, appendix 2.

Knowledge and understanding

For the Degree of Doctor, the doctoral student shall demonstrate:

  • A1: broad knowledge within and a systematic understanding of the research field, as well as deep and up-to-date specialist knowledge in a limited area of that field, and
  • A2: familiarity with scientific methodology in general and with the specific methods of the research field in particular.

Competence and skills

For the Degree of Doctor, the doctoral student shall demonstrate

  • B1: to conduct scientific analysis and synthesis and to independently and critically examine and assess new and complex phenomena, issues, and situations,
  • B2: to critically, independently, creatively, and with scientific precision identify and formulate research questions, plan and carry out research and other qualified tasks within given time frames using appropriate methods, and to review and evaluate such work,
  • B4: to, in both national and international contexts, orally and in writing with authority, present and discuss research and research results in dialogue with the scientific community and society at large.

Teaching formats

Lectures and seminars.

Examination

MUN1, oral examination, 2 cr, concerning learning outcomes 1-7, grade Fail (U) or Pass (G).

INL1, written assignment, 3 cr, concerning learning outcomes 1-7, grade Fail (U) or Pass (G).

Grade

Examinations included in the course are assessed according to a two-grade scale, fail or pass.

A person who has not passed the regular examination shall be given the opportunity to retake the test.

Requirements

To participate in the course and its examinations, the applicant must be admitted to third cycle (doctoral) studies at Mälardalen University.

Selection criteria

Doctoral students admitted in other subjects at Mälardalen University may be admitted to the course subject to availability. The same applies to doctoral students admitted at other higher-education institutions. Selection of applicants is made according to the following order of priority:

  1. Doctoral students in Mathematics/Applied Mathematics at Mälardalen University
  2. Others doctoral students at Mälardalen University
  3. Doctoral students at other higher education institutions in Sweden