Autumn semester

Credits
5 ECTS

Teaching method
Classroom instruction and problem sets.
2x2 lectures per week for 7 weeks. 

Language
English

Examination
Individual oral examination, no electronic aids allowed:

The course is evaluated through an individual oral examination. The exam is based on core concepts from the course, representative problem-set material, and the student’s ability to explain how the main ideas of the course fit together.

At the exam, students may be asked to define key concepts, prove or outline representative results, solve standard exercises, and interpret assumptions.

The duration of the oral exam is 30 minutes; in addition, the exam includes preparation time (also 30 minutes). Electronic aids or use of GAI tools it not allowed.

Prerequisites for examination participation
Prior to the exam, a number of homework assignments must be handed in. The solutions handed in must demonstrate a reasonable attempt at solving the assignments.

Assessment
Pass/No pass, internal co-examination

Lecturer
Mikkel Bennedsen

Course description
This course provides a rigorous but accessible foundation in the mathematical and probabilistic tools used throughout PhD-level quantitative coursework. It is designed for PhD students in economics, econometrics, finance, business intelligence, operations research, and related quantitative fields, and it prepares students for later technical courses without duplicating the advanced theory covered there.

The course deliberately prioritizes depth, pacing, and reusable reasoning patterns over breadth. The central themes are mathematical language and proof habits; deterministic limits and convergence; approximation of functions; projection and least-squares geometry; compactness, existence, and basic optimization; probability and conditional expectation; and stochastic convergence. 

The course also introduces deterministic ideas that prepare students for more advanced econometric theory. In particular, students study pointwise and uniform convergence of functions, the supremum norm, equicontinuity, compactness, and the Weierstrass theorem. These topics provide useful intuition for later courses where uniform laws of large numbers, stochastic equicontinuity, and extremum-estimator arguments are developed formally.

Academic prerequisites
No prior measure-theoretic probability is assumed. Students should be comfortable with basic calculus and linear algebra at the level of an undergraduate economics, business economics, mathematics-for-economists, or quantitative methods sequence. The course starts from first principles of mathematical language and proof-writing and builds gradually toward probability, stochastic convergence, and large-sample approximation.

For PhD students at AU (ECON), this course has been pre-approved as an internal BSS PhD course equivalent to 5 ECTS.  

Contents
The course covers the following core topics:

• Mathematical language, quantifiers, proof methods, sequences, deterministic limits, inequalities, bounding arguments, and deterministic big-O/little-o notation.
• Vectors and matrices, norms and inner products, orthogonality, projections, least-squares geometry, projection matrices, residual orthogonality, quadratic forms, positive semidefiniteness, and eigenvalues for symmetric matrices.
• Limits, continuity, differentiability, gradients, Hessians, Jacobians, local linear approximation, mean-value-type bounds, Taylor expansions, and remainder control.
• Sequences of functions, pointwise and uniform convergence, the supremum norm, uniform approximation, equicontinuity, compactness in finite-dimensional spaces, and the Weierstrass theorem.
• Optional application: contraction mappings and fixed-point theory as an application of convergence and iteration.
• Probability spaces, random variables, distributions, joint distributions, independence, expectation, variance, covariance, conditional expectation, and conditional expectation as best prediction/projection.
• Markov, Chebyshev, and Jensen inequalities; the Weak Law of Large Numbers; and the main modes of stochastic convergence: almost sure convergence, convergence in probability, and convergence in distribution.
• The Central Limit Theorem, Continuous Mapping Theorem, Slutsky’s theorem, and the Delta method.
• Final synthesis: ordinary least squares as projection, optimization, and large-sample approximation.

If time permits, additional examples may be included from econometrics, finance, business analytics, operations research, logistics, and quantitative decision-making.

Description of qualifications
After completing the course, the student will gain knowledge and understading of, skills to, and competencies in, the following.

Knowledge and understanding:

• The mathematical language, notation, and proof structures used in PhD-level quantitative courses.
• Deterministic convergence of sequences and functions, including pointwise convergence, uniform convergence, supremum-norm arguments, and equicontinuity.
• Linear algebra concepts central to quantitative methods, including norms, inner products, orthogonality, projections, least-squares geometry, quadratic forms, eigenvalues, and positive semidefiniteness.
• Continuity, differentiability, local approximation, Taylor expansions, and the role of gradients, Hessians, and Jacobians in optimization and statistical approximation.
• Compactness and the Weierstrass theorem as tools for establishing existence of solutions in finite-dimensional optimization problems.
• Probability, expectation, conditional expectation, inequalities, laws of large numbers, central limit theory, and the main modes of stochastic convergence.
• The relationship between deterministic approximation arguments and stochastic large-sample arguments used in later econometrics and quantitative methods courses.

Skills:

• Read, interpret, and write standard mathematical arguments using quantifiers, proof templates, inequalities, and bounding strategies.
• Work with deterministic limits, stochastic limits, and order notation in simple but rigorous arguments.
• Use linear algebra and projection geometry to derive and interpret least-squares estimators.
• Use gradients, Hessians, Taylor expansions, and first-order conditions to analyze simple optimization problems.
• Distinguish pointwise and uniform convergence of functions and verify simple uniform convergence and equicontinuity results.
• Use compactness and continuity to establish existence of optimizers in finite-dimensional problems.
• Define and compare almost sure convergence, convergence in probability, and convergence in distribution, and explain the relationships between these modes of convergence.
• Apply Markov, Chebyshev, and Jensen inequalities to obtain useful bounds, including a proof of a basic Weak Law of Large Numbers.
• Use the Law of Large Numbers, Central Limit Theorem, Continuous Mapping Theorem, Slutsky’s theorem, and the Delta method in basic large-sample approximations.
• Explain the ordinary least-squares estimator as a unifying example involving projection, optimization, and stochastic convergence.

Competences:

• Following and participating in rigorous PhD-level courses in econometrics, finance, business analytics, operations research, and related quantitative fields.
• Recognizing recurring mathematical proof patterns across deterministic analysis, probability, and asymptotic arguments.
• Connecting geometric, optimization-based, and probabilistic interpretations of standard quantitative methods.
• Assessing the role of assumptions such as continuity, compactness, finite moments, independence, and positive definiteness in mathematical and statistical arguments.
• Preparing for later coursework involving asymptotic theory, extremum estimators, stochastic equicontinuity, uniform convergence, and advanced statistical inference.

Schedule autumn 2026
Weeks 43-49: Tuesdays 10:00-12:00, room 1816-128
Weeks 43 & 45: Thursdays 10:00-12:00, room 1832-115
Weeks 44, 46-47 & 49: Thursdays 10:00-12:00, room 1816-128 
Week 48: Thursday 10:00-12:00, room 1816-114

Registration
Registration deadline: September 10^th at noon. The deadline has expired. Any further inquiries should be directed to ahoff@econ.au.dk.