Absolute continuity of one-sided random translations

Let X = Xkk ⩾ 1 be an i.i.d. random sequence and Y =Ykk ⩾ 1 be an independent random sequence which is also independent of X. We suppose X and Y take values in the sequence space SN, where S is either N0, the space of non-negative integers, or R+, the space of non-negative numbers. Then X and X + Y = Xk + Ykk ⩾ 1 induce probability measures μX and μX + Y on SN, respectively. We shall give necessary or sufficient conditions for μX ∼ μX + Y (equivalence = mutual absolute continuity) under assumptions on the distribution of X1. In particular, we consider the case where X1 obeys a Poisson or an exponential law.