Anders Karlsson
KTH, Stockholm

A general noncommutative law of large numbers


Abstract

In the 1950s and early 1960s probabilists started asking for
extensions of the law of large numbers to cases where the random
variables take values in a general group instead of the reals.
Grenander argued in his book "Probabilities on Algebraic Structures"
that a noncommutative theory extending classical probability would be
of much practical use. Of particular importance is the case of
products of random matrices, where much work has been done since then,
notably the important multiplicative ergodic theorem of Oseledec from
1968.

In a joint work with F. Ledrappier we prove a rather general
noncommutative law of large numbers. It specializes to Oseledec's
theorem in the case of invertible matrices (actually it might give
finer information in more specialized situations). It also applies to
the asymptotic behaviour of random walks on infinite groups (or
transient graphs, or Brownian motion on universal covers of compact
manifolds).  Intuitively our theorem asserts that whenever a random
walk escapes at a linear rate from the origin it converges in
direction.