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On association and other forms of positive dependence for Feller processes

Part of: Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  30 July 2019

Eddie Tu
Eddie Tu*
Affiliation:
Dickinson College
*
*Postal address: Department of Mathematics and Computer Science, Dickinson College, PO Box 1773, Carlisle, PA 17013, USA.
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Abstract

We characterize various forms of positive dependence, such as association, positive supermodular association and dependence, and positive orthant dependence, for jump-Feller processes. Such jump processes can be studied through their state-space dependent Lévy measures. It is through these Lévy measures that we will provide our characterization. Finally, we present applications of these results to stochastically monotone Feller processes, including Lévy processes, the Ornstein–Uhlenbeck process, pseudo-Poisson processes, and subordinated Feller processes.

Keywords

Associationsupermodular dependencesupermodular associationorthant dependenceFeller processLévy processintegro-differential operatorsymbols

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions 60E15: Inequalities; stochastic orderings 60J25: Continuous-time Markov processes on general state spaces 60J35: Transition functions, generators and resolvents 60J75: Jump processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 2 , June 2019 , pp. 624 - 646
Copyright
© Applied Probability Trust 2019 

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References

Applebaum, D. (2007). On the infinitesimal generators of Ornstein–Uhlenbeck processes with jumps in Hilbert space. Potential Anal. 26, 79–100.CrossRefGoogle Scholar
Bäuerle, N., Blatter, A. and Müller, A. (2008). Dependence properties and comparison results for Lévy processes. Math. Meth. Oper. Res. 67, 161–186.CrossRefGoogle Scholar
Böttcher, B. (2010). Feller processes: The next generation in modeling: Brownian motion, Lévy processes and beyond. PLoS ONE 5 (12), 1–8.CrossRefGoogle ScholarPubMed
Böttcher, B. (2014). Feller evolution systems: Generators and approximation. Stoch. Dyn. 14 (3), 1–15.CrossRefGoogle Scholar
Böttcher, B., Schilling, R. and Wang, J. (2013). Lévy Matters III. Springer.CrossRefGoogle Scholar
Burton, R., Dabrowski, A. and Dehling, H. (1986). An invariance principle for weakly associated random vectors. Stoch. Proc. Appl. 23, 301–306.CrossRefGoogle Scholar
Chen, M. and Wang, F. (1993). On order-preservation and positive correlations for multidimensional diffusion processes. Prob. Theory Rel. Fields 95, 421–428.CrossRefGoogle Scholar
Christofides, T. and Vaggelatou, E. (2004). A connection between supermodular ordering and positive/negative association. J. Multivar. Anal. 88, 138–151.CrossRefGoogle Scholar
Courrège, P. (1965). Sur la forme intégro-différentielle des opérateurs de C∞ k dans C satisfaisant au principe du maximum. Sém. Théorie du Potentiel 2, 1–38.Google Scholar
Dynkin, E. (1965). Markov Processes. Springer.CrossRefGoogle Scholar
Esary, J., Proschan, F. and Walkup, D. (1967). Association of random variables, with applications. Ann. Math. Statist. 38 (5), 1466–1474.CrossRefGoogle Scholar
Harris, T. (1977). A correlation inequality for Markov processes in partially ordered state spaces. Ann. Prob. 5 (3), 451–454.CrossRefGoogle Scholar
Herbst, I. and Pitt, L. (1991). Diffusion equation techniques in stochastic monotonicity and positive correlations. Prob. Theory Rel. Fields 87, 275–312.CrossRefGoogle Scholar
Houdré, C., Pérez-abreu, V. and Surgailis, D. (1998). Interpolation, correlation identities, and inequalities for infinitely divisible variables. Fourier Anal. Appl. 4, 651–668.CrossRefGoogle Scholar
Hu, T. (2000). Negatively superadditive dependence of random variables with applications. Chinese J. Appl. Probab. Statist. 16 (2), 133–144.Google Scholar
Kühn, F. and Schilling, R. (2019). On the domain of fractional Laplacians and related generators of Feller processes. J. Funct. Anal. 276 (8), 2397–2439.CrossRefGoogle Scholar
Lehmann, E. (1966). Some concepts of dependence. Ann. Math. Statist. 37 (5), 1137–1153.CrossRefGoogle Scholar
Liggett, T. (1985). Interacting Particle Systems. Springer.CrossRefGoogle Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. Wiley.Google Scholar
Phillips, R. (1952). On the generation of semigroups of linear operators. Pacific J. Math. 2, 343–369.CrossRefGoogle Scholar
Pitt, L. (1982). Positive correlated normal random variables are associated. Ann. Prob. 10, 496–499.CrossRefGoogle Scholar
Resnick, S. (1988). Association and extreme value distributions. Austral. J. Statist. 30A, 261–271.CrossRefGoogle Scholar
Rüschendorf, L. (2008). On a comparison result for Markov processes. J. Appl. Prob. 45, 279–286.CrossRefGoogle Scholar
Rüschendorf, L., Schnurr, A. and Wolf, V. (2016). Comparison of time-inhomogeneous Markov processes. Adv. Appl. Prob. 48 (4), 1015–1044.CrossRefGoogle Scholar
Samorodnitsky, G. (1995). Association of infinitely divisible random vectors. Stoch. Proc. Appl. 55, 45–55.CrossRefGoogle Scholar
Schilling, R. (1998). Conservativeness and extensions of Feller semigroups. Positivity 2, 239–256.CrossRefGoogle Scholar
Schilling, R. (1998). Growth and Hölder conditions for the sample paths of Feller processes. Prob. Theory Rel. Fields 112, 565–611.CrossRefGoogle Scholar
Schnurr, A. (2017). The fourth characteristic of a semimartingale. Available at arXiv:1709.06756v3.Google Scholar
Szekli, R. (1995). Stochastic Ordering and Dependence in Applied Probability. Springer.CrossRefGoogle Scholar
Tu, E. (2017). Dependence structures in Lévy-type Markov processes. Doctoral thesis, University of Tennessee, Knoxville.Google Scholar
Tu, E. (2018). Association and other forms of positive dependence for Feller evolution systems. Available at arXiv:1805.03080.Google Scholar
Wang, J. M. (2009). Stochastic comparison and preservation of positive correlations for Lévy-type processes. Acta Math. Sinica 25 (5), 741–758.CrossRefGoogle Scholar