- Multifractal spectrum of Cascades
- Stochastic integration II
- Stochastic integration I
- Martingales
- Conditional expectations II
- Conditional expectations I
- Fractal dimensions

- Brownian motion. Statistical scaling, iterative construction (midpoint displacement)
- Dimensions: geometrical scaling. Self-similar paths.
- Hoelder continuity and dimension I. Intro + Ex: path of Brownian motion
- Hoelder continuity and dimension II. Rigorous arguments.
- Box dimension, Diophantine examples
- Self-similar processes, simple properties
- The unique Gaussian self-similar process: fBm (definition through correlation structure)
- Conditional expectation
- Martingales, stochastic integrals I
- Stochastic integrals II, Ito's formula
- Second definition of fBm (as Ito integral of Brownian motion)
- Second order self-similarity
- Synthesis of fBm I: Midpoint, FFT, Cholesky
- Synthesis of fBm II: Crouse, FARIMA, Weierstrass-Mandelbrot. Stable laws.
- Wavelet decomposition of fBm: decorrelation and synthesis.
- Wavelet-based estimation of LRD for self-similar.
- Large deviation principles (LDP), ex: coin toss, appl: buffer-overflow of fGn-based traffic model
- Local singularity exponents, via increments and...
- ...via wavelets. Two-microlocalization.
- Binomial cascade: singularities (via increments) and dyadic expansions,
- Binomial cascade: multifractal structure (via increments and wavelets).
- Multifractal spectra
- Multifractal formalism as a LDP
- Modeling data traffic via cascades, statistical advantages over fGn

The best known fractal process is Brownian motion which can be constructed through a simple iteration, a property shared by many fractal objects which can be studied analytically. Brownian motion strikes through its erratic - fractal - appearance which finds a natural description through self-similarity and fractal dimensions.

With the siblings of Brownian motion, the fractional Brownian motions (fBm) highly irregular behavior becomes linked to the concept of Long Range Dependence (LRD). LRD stands simply for the presence of strong auto-correlations even over large time lags. In the class we will show how LRD relates to the concept of self-similarity, i.e. the ``looking alike'' on all scales. With the example of Internet data traffic it will be developed how self-similarity can be used in modelling, numerically as well as analytically.

Multiplicative measures, in short cascades, were introduced by Mandelbrot in 1974 as models for intermittency in turbulence. Cascades can be viewed as generalizing the self-similarity of fBm and providing a class of processes with greater flexibility. Cascades possess a multi-fractal structure which describe their highly complex dependence structure.

In this course we try to balance an introduction to the mathematical background of fractals and multifractals with applications of theoretical and practical importance, e.g. in Internet traffic modelling and in image processing. Thereby, we will keep things as simple as possible, making the course accessible to a wide audience.

Homework will be assigned on an irregular basis. Students who wish to
claim credit are expected to do a project which may consist of presenting
a research article or programming. The course is fairly distinct of last
year's ELEC 697 and does not assume any pre-knowledge on fractals. *Only
a basic course in probability (ELEC 533) is required.*

The structure of the course is as follows.

- Introduction: Self-similarity, iteration, dimensions
- Brownian motion: a fractal
- fBm: Self-similarity and LRD
- Data traffic modeling (via self-similar processes)
- Cascades: paradigm of multifractal behavior
- Large deviations: the multifractal formalism
- Multifractal properties and LRD of cascades
- Data traffic modelling (via multifractal processes)