Speed of Excited Random Walks with Long Backward Steps

We study a model of multi-excited random walk with non-nearest neighbour steps on Z, in which the walk can jump from a vertex x to either x+ 1 or x- i with i∈ { 1 , 2 , ⋯ , L} , L≥ 1. We first point out the multi-type branching structure of this random walk and then prove a limit theorem for a related multi-type Galton–Watson process with emigration, which is of independent interest. Combining this result and the method introduced by Basdevant and Singh (Probab Theory Relat Fields 141:3–4, 2008), we extend their result (w.r.t. the case L= 1 ) to our model. More specifically, we show that in the regime of transience to the right, the walk has positive speed if and only if the expected total drift δ> 2. This confirms a special case of a conjecture proposed by Davis and Peterson.