TUM - IAS Primary Sources Presenting

Sometimes we are rudely awakened to this fact when a really catastrophic event happens such as an earthquake with unexpected strength at an unexpected location. That is the gist of non-traditional stochastic modeling: accounting for " unexpected events." Although such events cannot be predicted, they can be modeled. measurement errors, leading to the binomial distribution and in the limit, the Gaussian distribution. Many random physical phenomena are indeed governed by such distributions, a lot of theory has been developed for them and they have invaded engineering ranging from the design of electronic amplifiers and telecommunication links to helicopter control via Kalman filtering. Somehow, and thanks to the famous (Lindeberg-L ´ evy) central limit theorem, there is a belief that when you have the sum of a large number of presumably small effects, you automatically get a near-Gaussian distribution and that therefore all stochastic disturbances in engineering can be meaningfully modeled by Gaussian models.. . The truth, however, is much more complex. Many natural and human-made probabilistic phenomena do not adhere to a Gaussian distribution. This is already pretty evident from the properties of the latter. It has a very pronounced average value; the distribution is symmetric around that value and tapers off exponentially. Three or four " sigmas " away (sigma is a number equal to the standard deviation of the random variable) and the probability that the process has such a value is virtually nil. Already early on in the development of probability theory, the famous French mathematician L ´ evy defined a continuous distribution function for a new class of positive random variables, characteristic of some physical effects in spectroscopy and deviating from the Gaussian case. That new distribution has then been found useful for the modeling of economic effects such as the return on shares in the stock market. Pareto drew the attention of the economic community to these so-called " power law probability distributions, " a class of distributions that are described by a power law and hence have a " fat tail, " leading to statements such as " 80% of world's wealth is in 20% of the hands. " It seems that many meaningful distributions obey such laws: e.g., size of islands, Internet traffic, 1/f noise, size of sand particles, you name it – reason enough to give these distributions a great deal of attention! Probability theorists generalized the distributions discovered by L ´ …