"For instance, the fact that most of the mass of a unit ball in high dimensions lurks near the boundary of the ball can be interpreted as a manifestation of the law of large numbers"
As usual Terry Tao's comment is wonderfully illuminating (at least for a non-mathematician like me). It's common knowledge that the ratio of the volume of a n-sphere to its circumscribed n-cube goes to zero as n->0 (http://en.wikipedia.org/wiki/N-sphere#Volume_and_surface_are...), but I never thought of this as another result of the law of large numbers.
That's actually what I considered to be the cause for the law of large numbers, when I tried to trace the normal distribution to axioms of three-dimensional geometry. I failed to get much beyond that, for lack of ability to reason about exponential functions, and abandoned the pursuit.
As usual Terry Tao's comment is wonderfully illuminating (at least for a non-mathematician like me). It's common knowledge that the ratio of the volume of a n-sphere to its circumscribed n-cube goes to zero as n->0 (http://en.wikipedia.org/wiki/N-sphere#Volume_and_surface_are...), but I never thought of this as another result of the law of large numbers.