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.