Random walks with similar transition probabilities

We consider random walks on the nonnegative integers with a possible absorbing state at −1. A random walk X̃ is called α-similar to a random walk X if there exist constants Cij such that for the corresponding n-step transition probabilities P̃ij(n)=α−nCijPij(n), i,j⩾0, hold. We give necessary and sufficient conditions for the α-similarity of two random walks both in terms of the parameters and in terms of the corresponding spectral measures which appear in the spectral representation of the n-step transition probabilities developed by Karlin and McGregor.