Course Syllabus for

# Stationary Stochastic Processes Stationära stokastiska processer

## FMS010F, 10 credits

Valid from: Spring 2013
Decided by: FN1/Anders Gustafsson
Date of establishment: 2013-05-21

## General Information

Division: Mathematical Statistics
Course type: Third-cycle course
Teaching language: English

## Aim

The aim of the course is to increase the understanding of the basic principles and results in the theory of stationary stochastic processes and to add to the arsenal of useful tools for their application. The course givea increased knowledge and skills in stationary stochastic processes on top of basic courses in the subject in the engineering programs: to present and train how stationary process models are constructed and how their mathematical, probabilistic, and statistical properties can be analysed. It is intended as a "second course" in stationary processes; some previous knowledge from mathematical statistics or signal processing helps.

## Goals

Knowledge and Understanding

For a passing grade the doctoral student must Know and understand: how to define a stochastic process from finite-dimensional distributions, conditions for different analytical properties, relation between covariance function and spectrum, sprectral representation of a stationary process and its role in linear filters, sampling of stationary process, envelope, statistical formulations of ergodicity, models for random fields, Rice formula

Competences and Skills

For a passing grade the doctoral student must Relate covariance function to spectrum and find continuity and differentiability properties from covariance and spectrum; use the spectral representation of the process to derive filter properties and other relations; use Rice's formula to evaluate extremal- and crossings properties

Judgement and Approach

For a passing grade the doctoral student must Find suitable process models for specific scientific and engineering applications; understand limitations of stationary models; interpret high frequency and low frequency parts of a spectrum in terms of smoothness and long range dependence

## Course Contents

1. How to define a stochastic process; sample space, ensemble, distribution 2. Analytic properties of sample functions 3. Covariance function and its spectral representation 4. Spectral representation of a stationary process 5. Linear filters and their spectral properties, white noise 6. Hilbert transform, envelope, Karhunen-Loève expansion 7. Classical ergodic theory, mixing conditions 8. Multivariate processes and cross-correlation properties 9. Spectral properties of random fields 10. Level crossings, extremes and excursions

## Course Literature

Lindgren, G.: Stationary Stochastic Processes: Theory and Applications. Chapman & Hall/CRC, 2012. ISBN 9781466557796.

## Instruction Details

Types of instruction: Lectures, exercises

## Examination Details

Examination formats: Written exam, oral exam, written assignments
Examiner:

Assumed prior knowledge: Introduction to Stationary stochastic processes corresponding to FMSF10 or similar.

## Further Information

The course is a modernized version of a PhD course with the same title that has been given regularly since 2000.

## Contact and Other Information

Course coordinators: