Algebra and Number Theory Seminar: On consecutive values of random completely multiplicative functions

We study the behaviour of consecutive values of random completely multiplicative functions (X_n)_{n >= 1} whose values are i.i.d. at primes. We prove that for X_2 uniform on the unit circle, or uniform on the set of roots of unity of a given order, and for fixed k >= 1, X_{n+1},..., X_{n+k} are independent if n is large enough. Moreover, with the same assumption, we prove the almost sure convergence of the empirical measure (1/N) sum of Dirac masses at (X_{n+1},...,X_{n+k})  when N goes to infinity, with an estimate of the rate of convergence.