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International audience In this paper infinite systems of functional equations in finitely or infinitely many random variables arising in combinatorial enumeration problems are studied. We prove sufficient conditions under which the combinatorial random variables encoded in the generating function of the system tend to a finite or infinite dimensional limiting distribution.

The main goal of this paper is to study the behavior of subsequences u c = {u(⌊n c ⌋) : n ∈ N} of automatic sequences u that are indexed by [n c ] for some c > 1. In particular we show that the densities of the letters of u c are precisely the same as those of the original sequence (provided that c < 7/5). In this sense u c and u behave in the same way. However, the pair correlation might be completely different as we will show in the special case of the Thue-Morse sequence. The proofs use exponential sum estimates like the double large sieve and a discrete Fourier analysis related to automatic sequences.

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Let t = (t n ) n 0 be the classical Thue-Morse sequence defined by t n = s 2 (n) (mod 2), where s 2 is the sum of the bits in the binary representation of n. It is well known that for any integer k 1 the frequency of the letter "1" in the subsequence t 0 , t k , t 2k , . . .is asymptotically 1/2. Here we prove that for any k there is an n k + 4 such that t kn = 1. Moreover, we show that n can be chosen to have Hamming weight 3. This is best in a twofold sense. First, there are infinitely many k such that t kn = 1 implies that n has Hamming weight 3. Second, we characterize all k where the minimal n equals k, k +1, k +2, k +3, or k +4. Finally, we present some results and conjectures for the generalized problem, where s 2 is replaced by s b for an arbitrary base b 2.