TIMETABLE:

Wednesday 16.00 -19.00 - Aula Careri

Thursday 16.00-19.00- Aula Careri

Lectures:

01/10/2025  (3 h)

Introduction to complex systems. 

Accompanying slides on the Prolusion [PDF]

02/10/2025  (3 h)

(1) Introduction to probability density functions (pdf). Observables and non-normalizable distributions. Distribution of a dependent variable and relation with the generation of random numbers. The sum of two random variables. Characteristic functions and Convolution theorem. Cumulants for the sum and the average of i.i.d random variables. (2) Moments for the sum and the average of i.i.d. random variables. Derivation of the Central Limit Theorem for distributions with well-defined moments. (3) Introduction to power-laws: normalisation and representations. Generation of random variables with a generic probability distribution: the case of power-laws. Criterion of Maximum Likelihood. Application to the estimation of the exponent of a power-law.

08/10/2025  (3 h)

(1) Frequency-rank plot and its relation with the pdf.  Again, on the criterion of Maximum Likelihood. Estimation of the exponent of a power-law and the error on the estimate. Estimate of the maximum value of a sample drawn from a power-law pdf. (2)  Probability distribution functions for extreme events drawn from a power-law distribution. Fréchet distribution. Cauchy-Lorentz distribution. Calculation of the first and second moments. Computation of the characteristic function and proof that a sum of Cauchy-distributed random variables is again Cauchy-distributed. (3) Lévy's distribution: calculation of the moments. Stable distributions and alpha-stable distributions. Conditions on the exponents of a power-law pdf for the existence of the moments. 

09/10/2025  (3 h)

Central Limit Theorem for long-tailed distributions with well-defined variance. Central Limit Theorem for long-tailed distributions without a well-defined variance: the case of the Cauchy distribution as an alpha-stable distribution. Benford's law. Empirical observations, scaling arguments and explanation based on multiplicative processes. More on multiplicative processes and log-normal distributions.  Zipf's law: empirical observations, normalization depending on the value of the exponent. Relationship between the Zipf's exponent and the exponent of the usual probability distribution function. Heaps' law: empirical observations. Taylor's law. Example of its derivation for a Poissonian process.