Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A q-analogue of de Finetti’s theorem. | A q-analogue of de Finetti s theorem Alexander Gnedin Department of Mathematics Utrecht University the Netherlands Grigori Olshanski Institute for Information Transmission Problems Moscow Russia and Independent University of Moscow Russia olsh2007@ Submitted May 13 2009 Accepted Jun 15 2009 Published Jul 2 2009 Mathematics Subject Classification 60G09 60J50 60C05 Abstract A q-analogue of de Finetti s theorem is obtained in terms of a boundary problem for the q-Pascal graph. For q a power of prime this leads to a characterisation of random spaces over the Galois field Fq that are invariant under the natural action of the infinite group of invertible matrices with coefficients from Fq. 1 Introduction The infinite symmetric group Sx consists of bijections 1 2 . 1 2 . which move only finitely many integers. The group Sx acts on the product space 0 1 by permutations of the coordinates. A random element of this space that is a random infinite binary sequence is called exchangeable if its probability law is invariant under the action of 6 . De Finetti s theorem asserts that every exchangeable sequence can be generated in a unique way by the following two-step procedure first choose at random the value of parameter p from some probability distribution on the unit interval 0 1 then run an infinite Bernoulli process with probability p for 1 s. One approach to this classical result as presented in Feller 3 Ch. VII 4 is based on the following exciting connection with the Hausdorff moment problem. By exchangeability the law of a random infinite binary sequence is determined by the array vn k Supported by a grant from the Utrecht University by the RFBR grant 08-01-00110 and by the project SFB 701 Bielefeld University . THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R78 1 where vn k equals the probability of every initial sequence of length n with k 1 s. The rule of addition of probabilities yields the backward recursion Vn k Vn 1 k Vn 1 k 1 0 k n n 0 1
