  In this work, the probability of return for random walks on $\mathbb{Z}$,
whose increment is given by the $k$-bonacci sequence, is determined. Also, the
Hausorff, packing and box-counting dimensions of the set of these walks that
return an infinite number of times to the origin are given. As an application,
we study the return for tribonacci random walks to the first term of the
tribonacci sequence.
