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Statistics for stochastic processes


Statistics for stochastic processes


Academic year 2021/2022

Course ID
Teaching staff
Prof. Elvira Di Nardo (Lecturer)
Luis Alberiko Gil-Alana (Lecturer)
1st year
Teaching period
Second semester
D.M. 270 TAF B - Distinctive
Course disciplinary sector (SSD)
MAT/06 - probabilita' e statistica matematica
Type of examination
Good knowledge of probability theory and the basics of stochastic processes. In more details
- laws of large numbers and central limit theorems
- measure theory
- conditional expectations
- L^p spaces with respect to a probability measure
- Hilbert spaces (some introductory material on this topic is present in the text books)

Sommario del corso


Course objectives

The goal of lectures is to introduce statistical inference for time series taking into account both the theoretical/mathematical aspects and their practical application to data analysis.

Time series are considered, aiming to characterize properties, asymptotic behavior, estimations and forecasting,  spectral analysis as well as decomposition in trend and seasonal components. Such concepts are applied to the analysis of simulated data or existing databases in order to infer and validate a model supporting the data.


Results of learning outcomes

Knowledge and understanding

By the end of the course, the student is able to transform a real problem into a statistical one and interpret results in an effective way for phenomena evolving during the time.  Moreover it is expected that the student is able to employ mathematical/statistical models for a better identification of the dependence and for forecasting the behaviour of the stochastic dynamic system under observation.  Computational skills are acquired by means of the open source software R. 

Applying knowledge and understanding

The student is requested to be able to set out statistical models in order to make evidence of relations among variable both for individual data and time series and devise appropriate computational algorithms for the models. In particular, by the end of the course, the student will know 

  • how to use R for the analysis of a time series including descriptive analysis and inferential tools to recognize patterns in the datasets;
  • how to select a theoretical model, including parameter estimation;
  • how to validate the theoretical model by using statistical test;
  • how to forecast and predict the patterns with an estimation of the errors. 

Making judgements

By comparing the results obtained in performing the statistical analysis, the student has to be able to select which model better describes the dependence among the observed phenomena, when they are correlated by a temporal evolution.

Communication skills

The student must be able to communicate the information got from the qualitative and quantitative analysis by using the most appropriate terminology and the most useful graphical tools, aiming to avoid possible distortions, to optimize their employment and to  validate the analysis.

Learning skills

The skills acquired will give students the opportunity of improving and deepening their knowledge of the different aspects of stochastic modeling of observed time series also by using the computational skills acquired in the Lab.


Course delivery

The course is structured in 48 hours of frontal teaching, divided into lessons of 2 hours according to academic calendar.  Classes are delivered in presence and with live streaming through the link published on the course's Moodle page (see button below) Please enroll on Moodle page for updating and getting further teaching material. 

Start date: February 22, 2022. Class schedule: Tuesday/Thursday 11:15am (new)


Learning assessment methods

The final assessment is foreseen to take place in presence. An online procedure with Webex video surveillance will be reserved for those whose absence is justified.

Who wants to be examined on the syllabus of the course given

  • before the a.y.<2015-16
    • send an e-mail to Elvira Di Nardo, one week before the practical session, to organize the methods
  • during the a.y. 2015-16 
    1. a practical session on the analysis of a dataset in the computer lab, including the descriptive analysis and a critical discussion (1hr);
    2. a short essay on one of the arguments introduced by Prof.Sirovich (immediately after the dataset analysis in the Lab) to verify the correct use of terminology and the hability to present a clear and concise exposition of the topics (30 mms);
    3. the final evaluation with an oral examination and a discussion on the practical session a couple of days later (20mns).
  • after the a.y. 2015-16 (including the current a.y.)
    1. a practical session including exercises, the employment of R and the analysis of a dataset in the computer lab (1hr)
    2. oral exam a couple of days later. 

For part 1. there is no mark, just an evaluation which can be: excellent, very good, good, quite good, sufficient and not sufficient sent by e-mail through This evaluation will be added to the oral examination mark to obtain the final mark.  



Support activities

Computer sessions.



  1. Introduction to Time series.
  • Weak and strong stationarity.
  • Autocovariance and autocorrelation functions: characterizations. 
  • IID sequences and WN sequences.

     2. Estimation.

  • Sample mean, variance and autocovariance function.
  • Ergodicity, application of Kolmogorov 0-1 law, generalizations of law of large numbers, Birkhoff theorem,
  • L^2 ergodicity and relation with properties of  autocovariance  functions.
  • Moving average filters: estimation of trend and seasonal component.

     3. Transformation of time series.

  • Linear processes.  Backward shift and difference operators.
  • Time invariant linear filters: convergence a.s. and in mean square of partial summations to Laurent series of WN, autocovariance function.
  • Not uniqueness of modeling. Invertibility. 
  • Q-dependence and q-correlation: Moving Average models of order q and of order infinite.
  • Causality: AutoRegressive models of order p and their multivariate representation as AR(1) model. 

     4. ARMA models. 

  • How to construct ARMA models whose solution is causal, invertible and not redundant. 
  • Autocovariance function of ARMA models and homogeneous linear difference equations. 
  • Yule-Walker equations for AR(p) model and Yule-Walker estimators.
  • Partial autocorrelation function.

     5. Forecasting.

  • Conditional mean and best linear predictor.
  • The n-step ahead predictor when the covariance matrix is non-singular and singular.
  • Perfect predictable time series and the Wold decomposition. 
  • Choosing p and q from data:  Akaike's criterion.
  • The ARIMA procedure in R. 

     6. Spectral representation of simple processes.

  • Spectral density.
  • Bochner Theorem
  • Computing the spectral density for various models.
  • Relation with Wold's decomposition theorem.
  • Short memory and long memory processes. 
  • Periodogram.

      6. Computer sessions

  • Simulation and statistical analysis of time series with R. 
  • Estimation of the parameters and model selection.
  • Diagnostic tools.
  • ARIMA and SARIMA models.

Suggested readings and bibliography


Lectures in the classroom refers to

  • Brockwell and Davis, Introduction to Time Series and Forecasting, Second Edition. Springer texts in statistics. 2002

Lectures in the LAB refers to

  • Shumway and Stoffer, Time series Analysis and Its Applications, Springer, 2011
  • R-procedures in

For details on some proofs refer to

  • Brockwell and Davis, Time Series, theory and methods, Springer (collana SSS), New York, 1991

References for each topic will be made available during the lectures: for students not attending the course, a detailed summary.txt file is availabe under the web-page Teaching materials of the course. 

Additional materials are made available by the teacher to supplement the textbooks under the web-page Teaching materials of the course. 


Class schedule

Last update: 15/02/2022 17:59
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