A reversibility relationship (Q1374628)

Define \(\text{Ber}(\rho)\) the distribution of a Bernoulli r.v. such that \(\text{P}(B=1)=\rho\), \(\text{P}(B=0)=1-\rho\). Let \(B_0, B_1,\dots\) be i.i.d. \(\text{Ber}(\rho)\) r.v's. The operation \(\rho*X=\sum _{i=1}^X B_i\) is known as thinning. The author defines the discrete autoregressive process \(X_n =\rho*I_{n-1}X_{n-1}+\xi _n\) where \(I_{n-1}\sim\text{Ber}(1-\rho)\) are i.i.d. r.v.'s and \(\{\xi _n\}\) is also a sequence of i.i.d. r.v.'s. The operation \(\rho*I\) is called zero enhanced thinning (ZET). The zero adapted discrete minification process \(\{Y_t\}\) is given by \(Y_n=\rho\setminus _I \min (Y_{n-1},\eta _n)\) where \(\rho\setminus _I\) is the left inverse of the ZET and \(\{\eta _n\}\) is a sequence of i.i.d. r.v.'s. The author gives a condition under which \(\{X_n\}\) and \(\{Y_n\}\) are mutually time reversed, i.e., the relation \(\{X_{n-1},X_n\}\buildrel d\over = \{Y_n, Y_{n-1}\}\) holds.