Modelling some stationary Markov processes and related characterizations
type: Item
instance of: scholarly article
title: Modelling some stationary Markov processes and related characterizations
zbMATH Open document ID: 0893.62083
DOI: 10.1016/S0378-3758(97)84451-5
author: V. A. Kalamkar
author: Maddipatla Sreehari
published in: Journal of Statistical Planning and Inference
publication date: 1998-08-06T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 62E10
zbMATH DE Number: 1086145
zbMATH Keywords: thinning
zbMATH Keywords: thickening
zbMATH Keywords: exponential distribution
zbMATH Keywords: Poisson distribution
zbMATH Keywords: geometric distribution
reviewed by: Jiří Anděl
MaRDI profile type: Publication
cites work: FIRST-ORDER INTEGER-VALUED AUTOREGRESSIVE (INAR(1)) PROCESS
cites work: An extremal markovian sequence
cites work: On the stationary distribution of some extremal Markovian sequences
cites work: First-Order Integer-Valued Autoregressive (INAR (1)) Process: Distributional and Regression Properties
cites work: An integer-valued pth-order autoregressive structure (INAR(p)) process
cites work: A characterization of the Pareto process among stationary stochastic processes of the form \(X_ n=c\,\min (X_{n-1},Y_ n)\)
cites work: A time-reversibility relationship between two Markov chains with exponential stationary distributions
cites work: Q3883242
cites work: A condition for absolute continuity of infinitely divisible distribution functions
cites work: First-order autoregressive gamma sequences and point processes
cites work: Minification processes with discrete marginals
cites work: Minification processes and their transformations
cites work: Discrete minification processes and reversibility
cites work: Autoregressive moving-average processes with negative-binomial and geometric marginal distributions
cites work: Some ARMA models for dependent sequences of poisson counts
cites work: Semi-Pareto processes
cites work: Q5511781
cites work: Q5526189
cites work: An elementary proof for the Rao-Rubin characterization of the Poisson distribution
cites work: Discrete analogues of self-decomposability and stability
cites work: An exponential Markovian stationary process
cites work: Pareto processes
description: scientific article; zbMATH DE number 1086145