Modelling some stationary Markov processes and related characterizations
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.
scientific article; zbMATH DE number 1086145
scientific article