Rice University

*This section of the course provides an introduction to the concepts
of*

*Long Range Dependence and statistical self-similarity and develops
both,*

*models and estimators in the context of real world data*

Multiscale Statistics, R. Riedi

Classical limit theorems typically assume strong properties such as finite second moments and independence, or at least weak dependence, to arrive at strong results. In real world situations one may be confronted with data for which strong-holds such as the CLT do not seem to be verifiable empirically. A famous example is found with the classical Nil river annual water level. Violations of the CLT can be explained via long range dependence (LRD) or via infinite variance. In this mini course, we will formalize the concept of LRD and give simple time series models with LRD, such as generalized AR processes and fractional Brownian motion. We derive inference tools for LRD. This will lead us to wavelets, a powerful tool in multiscale statistics, to which a short and very intuitive introduction will be given. We will end with a tail estimator based on wavelets which pertains to the issue of infinite variance. If time permits or interest is indicated, cascade processes and Large Deviation Principles will be discussed.

[Outline] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests][Knowledge Milestones]

**Instructor**

Dr. Rudolf Riedi

Duncan Hall 2082, 713 / 348 3020,

MWF 09:00 AM - 9:50 AM Duncan Hall 1044

see also the official Rice for times and the Rice map for how to find the location

Monday 1-2 pm pm (DH 2082), or by appointment

**Outline and suggested topics**

- Long Range Dependence
- Statistical Self-similarity
- Auto-regressive time series models
- Hierarchical time series models
- Time domail and spectral domain estimators
- Wavelets, wavelet based estimation of LRD and heavy tails
- EXAM (open-notes)

**Textbook**

Long Range Dependence (Doukhan et al, editors)alternative texts will be indicated on request

**Grading**

60% EXAM

40% Homework

[Outline] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests][Knowledge Milestones]

**Classes and Reading assignments**

This doubles as a calendar for the course.

2006 |
Covered material | Reading: |

October 2 |
Overview | Slide Talk [pdf] |

October 4 |
Overview (Cont) | |

October 6 | Statistical Self-similarity | Ch. 1, LRD (Doukhan et al eds) |

October 9 | F.d.d. and Consistency | Ch. 1 |

October 11 | fBm (Gaussian self-sim process) | Ch. 1 |

October 13 | fGn (increments of fBm) | Ch. 1 |

October 16 |
Recess | |

October 18 | LRD and 2nd order self-similarity | Ch. 1 |

October 20 | FARIMA (times series model of LRD) | Ch. 1 |

October 23 | Spectral Estimation, Wiener-Kinchin | Slides [pdf] |

October 25 | Wavelets and spectral estimation | Slides [pdf] |

October 30 | Proj 1: Fractional Stochastic Calculus and Stock Markets | Slides [ppt] |

TBA | Proj 2+3: Estimation | |

TBA | Proj 4+5: LRD in Markets, Simulation |

**Homework**

(tex-source and solutions restricted to Rice University)

Homework sheet | Due date (in class) | Solutions |

Problem Set 1 [ps] [pdf] [tex] | Oct 30, 2006 | posted Nov 1, 2006 |

Homework is due at the beginning of class on the due date. After the due date, but before solutions are handed out, homework can be turned in for 50% credit. In this case, please slip your homework under the instructors's office door, or bring it to class. After solutions are handed out, 0% credit will be issued. You are encouraged to work in groups for homeworks but you will hand in your own solution which you are expected to understand.

Any student with a documented disability needing academic adjustments or accommodations is requested to speak with me during the first two weeks of class. All discussions will remain confidential. Students with disabilities should also contact Disabled Student Services in the Ley Student Center.

August 29, 2006