Everybody has a personal bank account. Every month the salary is added to the balance. Now suppose that you are a big spender and you spend most of the money every month. It would be unpleasant if the balance would go down to zero during the month but this has never happened. You could be afraid of this event to happen and you would like to know the probability that in fact the balance woul be depleted during a month. This is a difficult statistical problem since you want to estimate the probability of an event that has never happened and this seems impossible. Other similar examples are: Banks and insurance companies want to (have to) assess the probability that they go bankrupt in some given period of time. The regulator forces them to do so. A communication tower is much affected by wind storms, but the tower has never collapsed. Since much depends on this tower, one needs to know the probability of collapse. Problems of this kind can be attacked using a special branch of the area of mathematical statistics called extreme value theory. The theory has been developed over the last 70 years by scientist mainly from Europe and notably by Professor Tiago de Oliveira from this university. We are now able to provive a reliable answer to the mentioned questions.
Introduction to Random TIME and Quantum Randomness
For the first author of these nots, the concept of Random Time is the single tool without wich the theory of Markovian Stochastic processes would lose much of its strength and depth.
In Physics there is only one theory where probabilities are regarded as really irreducible, this is Quantum Mechanics. But, curiously, these probabilities are lying more in the laboratory than in the theory itself. Why?
In what sense Random Times constitute the heart of the Probabilistic Reasoning? And why is that the role of probability theory, in Quantum Physics, is still so controversial 70 years after the constrution of this theory? Can we hope to reinterpret at least part of Quantum Theory in a way to introduce physically meaningful Random Times?
These are some of the questions investigated in these lectures notes, in a semi-expository style accessible to graduate students of applied Mathematics and Physics.