•

the Blog

## Infinite expectations

By Guillaume Filion, filed under
black swans,
diffusion,
brownian motion,
probability.

• 06 January 2014 •

In the first days of my PhD, I sincerely believed that there was a chance I would find a cure against cancer. As this possibility became more and more remote, and as it became obvious that my work would not mark a paradigm shift, I became envious of those few people who *did* change the face of science during their PhD.
One of them is Andrey Kolmogorov, whose PhD work was nothing less than the modern theory of probability. His most famous result was the strong law of large numbers, which essentially says that random fluctuations become infinitesimal on average. Simply put, if you flip a fair coin a large number of times, the frequency that ‘tails’ turn up will be very close to the expected value 1/2.

### The chaos of large numbers

Most fascinating about the strong law of large numbers is that it is a theorem, which means that it comes with *hypotheses* that do *not* always hold. There *are* cases that repeating a random experiment a very large number of times does not guarantee that you will get closer to the expected value — I wrote the gory detail on Cross Validated, for those interested...