We consider excited random walks on the integers with a bounded number of i.i.d. cookies per site which may induce drifts both to the left and to the right. We extend the criteria for recurrence and transience by M. Zerner and for positivity of speed by A.-L. Basdevant and A. Singh to this case and also prove an annealed central limit theorem. The proofs are based on results from the literature concerning branching processes with migration and make use of a certain renewal structure.
"Positively and negatively excited random walks on integers, with branching processes." Electron. J. Probab. 13 1952 - 1979, 2008. https://doi.org/10.1214/EJP.v13-572