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The importance of infinite variance


Work at IBM: randomness - background
Benoît Mandelbrot Mathematician
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But the first thing I wanted to do when I came to IBM was to pursue more seriously this idea of power-law distributions and their meaning in the world. And how did I know of them? I knew of them because of distribution of income, which I mentioned before, which follows a power-law, the probability exceeding u is u to the power -alpha where alpha is some number, and that was discovered by Pareto, and there have been many attempts to find something more normal than that. Enormous scorn was piled on this law by everybody; in particular the scorn was based in the fact that if alpha is less than 2 - which was the case, at least once upon a time - the variant second moment is infinite, and everybody said an infinite second moment is something totally unacceptable. Now, I never understood that. I was never a very obedient follower of fashion, and it seemed to me that an infinite second moment can be unacceptable when it is for example, energy in physics, when physics is finite; but in the case of income it is just a number which has no reality, no interpretation, nothing, just a way of describing scatter. Therefore infinite expectation? Well it's somewhat - you can think of it as a somewhat dangerous idea because if it has no second moment it's something which is unattainable or, I mean, it's too rough? too difficult to keep in mind. Yes. Precisely. That's exactly the word. I didn't use the word 'rough' in that case, I used the world 'wild', which is the same idea. And since you press me, and thank you for pressing me, I would like, before I describe the beginning, to describe the end. After many, many years, in my recent book on fractals and finance I was led to put down in a very schematic fashion the distinction I saw between two kinds of randomness, in fact more than two. I even called them not kinds, but states. And so I argue as follows, formally in 1997 and more or less informally in late '50s. I argued that the laws of probability theory are unique, in the same way as the laws of mechanics are unique, but everybody knows, not just the physicists, that depending upon conditions of temperature and pressure, the same body can be a solid, a liquid or a gas, and that one cannot say, once can't work with those without specialising as to what they are. It is simply ridiculous to try to reduce oneself to laws of mechanics every time so there are special laws for gases, for liquids and for crystals, for solids. Now I became convinced from early on and increasingly so - and I stated that only recently - that randomness is the same situation; as when people spoke of randomness before and referred themselves to axiomatics and so on, they were in a certain sense obfuscating. They did not need all these axiomatics because actually the only randomness studied was very moderate. For example, when Laplace, who in addition to being a great mind was a great writer, descried randomness, it's clear that what he had in mind were fluctuations around an equilibrium. Something that was defined but had irregularities, which make it go up and down, around it, but clearly there is an equilibrium. The average is something very, very significant, and the variance is in a certain sense a description of a kind of serpent, a kind of strip in which variations stay all the time. That kind of randomness was understood almost thoroughly. It had tools beyond belief, books beyond counting, and also pretensions beyond reason because it did not, how to should I put it, use its expertise - which did not give it an idea of its own limitations, and which invite the self, imposing itself upon phenomena to which it didn't fit. And so, when I had to face this question of infinite variance for prices, I had to think very hard and say we must accept being freed from both the constraints and the help that probability, as applying to mild fluctuations, imposes and just go on our own way and start from the beginning. Now needless to say, for me, this was not a scary thought, it was an exhilarating thought. I'd been hoping that fate would bring me something novel to study, and hopefully not only one phenomenon like word frequencies, which was after all self-terminating more or less, but something more important, more in the centre of some discipline, in which a step beyond that tool box would be indispensable and in which I could hope to be the person who'd continued this step myself.

Benoît Mandelbrot (1924-2010) discovered his ability to think about mathematics in images while working with the French Resistance during the Second World War, and is famous for his work on fractal geometry - the maths of the shapes found in nature.

Listeners: Daniel Zajdenweber Bernard Sapoval

Daniel Zajdenweber is a Professor at the College of Economics, University of Paris.

Bernard Sapoval is Research Director at C.N.R.S. Since 1983 his work has focused on the physics of fractals and irregular systems and structures and properties in general. The main themes are the fractal structure of diffusion fronts, the concept of percolation in a gradient, random walks in a probability gradient as a method to calculate the threshold of percolation in two dimensions, the concept of intercalation and invasion noise, observed, for example, in the absorbance of a liquid in a porous substance, prediction of the fractal dimension of certain corrosion figures, the possibility of increasing sharpness in fuzzy images by a numerical analysis using the concept of percolation in a gradient, calculation of the way a fractal model will respond to external stimulus and the correspondence between the electrochemical response of an irregular electrode and the absorbance of a membrane of the same geometry.

Duration: 5 minutes, 44 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008