Output sum of transducers: limiting distribution and periodic fluctuation (Q2344821)

Summary: As a generalization of the sum of digits function and other digital sequences,~sequences defined as the sum of the output of a~transducer~are asymptotically analyzed. The input of the transducer is a random integer in \([0, N)\). Analogues in higher~dimensions are also considered. Sequences defined by a certain class of~recursions can be written in this framework.{ }{ }Depending on properties of the transducer, the~main term, the periodic fluctuation and an error term of the~expected value and the variance of this sequence are established. The periodic fluctuation of the expected~value is Hölder continuous and, in many cases, nowhere differentiable. A general formula for~the Fourier coefficients of this periodic function is derived. Furthermore, it~turns out that the sequence is asymptotically normally~distributed for many transducers. As an example, the abelian~complexity function of the paperfolding sequence is analyzed. This sequence has recently been~studied by Madill and Rampersad.