[Persi Diaconis][1] gave a talk about probabilities of carries at MIT about half a year ago. Unfortunately I was so busy trying to spot symmetric functions in the talk that most of it drifted past my mind, but here are a couple references:

[Persi Diaconis and Jason Fulman, *Carries, Shuffling and An Amazing Matrix*, arXiv:0806.3583][2]

[Noga Alon, *Minimizing the number of carries in addition*, arXiv:1209.1131][3]

This all seems to be more about the idea of elementary school addition rather than actual digits of numbers, but the OP seems to not be excluding this.


  [1]: http://www-stat.stanford.edu/~cgates/PERSI/papers.html
  [2]: http://arxiv.org/abs/0806.3583
  [3]: http://arxiv.org/abs/1209.1131v1