*Tid:***23 januari 2006 kl 1515-1700 **

*Plats :***Seminarierummet 3733**, Institutionen för
matematik, KTH, Lindstedts väg 25, plan 7. Karta!

*Föredragshållare:***
Anders Karlsson, Matematik, KTH
**

**Titel: **
A law of large numbers for random
walks.

* Sammanfattning: *
In the 1950s and early 1960s probabilists started asking for
extensions of the law of large numbers when the random
variables take values in a general group instead of the real numbers.
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 transitive infinite graphs (or fintely generated groups, 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.