On neighbour-dependent shifts preserving renewal process

Let T = {0 = T0 < T1 < T2 <...} denote a renewal process on the real line, so that the increments are i.i.d. strictly positive ran-dom variables τk = Tk − Tk−1 with distribution F. Choose random points Xk between each pair of neighbouring points Tk−1, Tk as follows: Xk = Tk−1 + bk(Tk − Tk−1), where bk, k = 1, 2,... are i.i.d. with a given distribution G supported by [0, 1]. Now shift these points so that they start also from the origin, this defines an operator ΦG such that T ′ = ΦG(T) = {0 = T ′0 < T ′1 < T ′2 <...} with T ′k = Xk+1 −X1, k ∈ N. Generally speaking, the resulting process T ′ is no longer a renewal process. However, apart from degenerate cases when τk’s and bk’s are con-stants, ΦG(T) is a distributional copy of T if and only if τk’s are Gamma-distributed and bk’s are Beta-distributed. In particular, a homogeneous Poisson process is invariant with respect to (and only to) B(a, 1−a)-based shifts for some 0 ≤ a < 1