Modelling some stationary Markov processes and related characterizations (Q1372420)

Some models of integer-valued time series are based on operations called thinning and thickening. If \(X\) is a non-negative integer-valued random variable and \(\rho\in[0,1]\) then the usual definition of the thinning operator \(\rho\ast\) is \(\rho\ast X=B_1+\cdots+B_X\), where \(B_i\)'s are iid Bernoulli\((\rho)\) variables independent of \(X\). The authors define a new thickening operator \(\theta\circ\) for \(\theta\in (0,1)\) by \(\theta\circ X=X+\theta\ast X_1\), where \(X_1\) is independent of \(X\) and \({\mathcal L}(X_1)={\mathcal L}(X)\). Some properties of \(\rho\ast\) and \(\theta\circ\) are derived and characterizations of Poisson, exponential, and geometric distributions are given. The results are related to AR(1) processes with these marginal distributions.