Abstract
The aim of this paper is to convey the basic ideas regarding some local hitting and conditioning properties of random measures. Some very general results are obtainable in the special case of simple point processes. Though much of this theory has no counterpart for general random measures, similar results do exist in two cases of special interest—for local time random measures and for Dawson–Watanabe superprocesses. This is an informal account of some of the basic hitting and conditioning properties common to all three cases. Precise statements and proofs will be provided elsewhere.
Similar content being viewed by others
References
Bruckner, A.M., Bruckner, J.B., Thomson, B.S.: Real Analysis. Prentice-Hall, Upper Saddle River (1997)
Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Springer, New York (1988)
Dawson, D.A.: The critical measure diffusion. Z. Wahrsch. verw. Geb. 40, 125–145 (1997)
Dawson, D.A.: Measure-valued Markov processes. In: École d’Été de Probabilités de Saint-Flour XXI–1991. Lect. Notes in Math., vol. 1541, pp. 1–260. Springer, Berlin (1993)
Dawson, D.A., Perkins, E.A.: Historical processes. Mem. Am. Math. Soc. 93, #454 (1991)
Dawson, D.A., Iscoe, I., Perkins, E.A.: Super-Brownian motion: path properties and hitting probabilities. Probab. Theory Relat. Fields 83, 135–205 (1989)
Dynkin, E.B.: Branching particle systems and superprocesses. Ann. Probab. 19, 1157–1194 (1991)
Etheridge, A.M.: An Introduction to Superprocesses. University Lecture Series, vol. 20. AMS, Providence (2000)
Georgii, H.O.: Gibbs Measures and Phase Transitions. De Gruyter, Berlin (1988)
Gorostiza, L.G., Wakolbinger, A.: Persistence criteria for a class of critical branching particle systems in continuous time. Ann. Probab. 19, 266–288 (1991)
Iscoe, I.: On the supports of measure-valued critical branching Brownian motion. Ann. Probab. 16, 200–221 (1988)
Ivanoff, G., Merzbach, E.: Set-Indexed Martingales. Chapman & Hall, Boca Raton (2000)
Jagers, P.: On Palm probabilities. Z. Wahrsch. verw. Geb. 26, 16–32 (1973)
Kallenberg, O.: Stability of critical cluster fields. Math. Nachr. 77, 7–43 (1977)
Kallenberg, O.: Splitting at backward times in regenerative sets. Ann. Probab. 9, 781–799 (1981)
Kallenberg, O.: An informal guide to the theory of conditioning in point processes. Int. Stat. Rev. 52, 151–164 (1984)
Kallenberg, O.: Random Measures, 4th edn. Akademie-Verlag and Academic Press, Berlin and London (1986) (1st edn. 1975, 3rd extended edn. 1983)
Kallenberg, O.: Palm measure duality and conditioning in regenerative sets. Ann. Probab. 27, 945–969 (1999)
Kallenberg, O.: Local hitting and conditioning in symmetric interval partitions. Stoch. Process. Appl. 94, 241–270 (2001)
Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, New York (2002)
Kallenberg, O.: Palm distributions and local approximation of regenerative processes. Probab. Theory Relat. Fields 125, 1–41 (2003)
Kallenberg, O.: Probabilistic Symmetries and Invariance Principles. Springer, New York (2005)
Kaplan, E.L.: Transformations of stationary random sequences. Math. Scand. 3, 127–149 (1955)
Khinchin, A.Y.: Mathematical Methods in the Theory of Queuing (1955) (Russian). Engl. transl.: Griffin, London (1960)
Kingman, J.F.C.: An intrinsic description of local time. J. Lond. Math. Soc. 6(2), 725–731 (1973)
König, D., Matthes, K.: Verallgemeinerung der Erlangschen Formeln, I. Math. Nachr. 26, 45–56 (1963).
Leadbetter, M.R.: On basic results of point process theory. Proc. 6th Berkeley Symp. Math. Stat. Probab. 3, 449–462 (1972)
Le Gall, J.F.: A lemma on super-Brownian motion with some applications. In: The Dynkin Festschrift, pp. 237–251. Birkhäuser, Boston (1994)
Matthes, K.: Stationäre zufällige Punkfolgen, I. Jahresber. Dtsch. Math.-Ver. 66, 66–79 (1963)
Matthes, K., Kerstan, J., Mecke, J.: Infinitely Divisible Point Processes. Wiley, Chichester (1978) (German edn. 1974, Russian edn. 1982)
Matthes, K., Warmuth, W., Mecke, J.: Bemerkungen zu einer Arbeit von Nguyen Xuan Xanh und Hans Zessin. Math. Nachr. 88, 117–127 (1979)
Palm, C.: Intensity variations in telephone traffic. Ericsson Technics 44, 1–189 (1943) (German). Engl. transl.: North-Holland Studies in Telecommunication, vol. 10. Elsevier (1988)
Perkins, E.: Dawson–Watanabe superprocesses and measure-valued diffusions. In: École d’Été de Probabilités de Saint-Flour XXIX–1999. Lect. Notes in Math., vol. 1781, pp. 125–329. Springer, Berlin (2002)
Ryll-Nardzewski, C.: Remarks on processes of calls. Proc. 4th Berkeley Symp. Math. Stat. Probab. 2, 455–465 (1961)
Slivnyak, I.M.: Some properties of stationary flows of homogeneous random events. Theor. Probab. Appl. 7, 336–341 (1962)
Thorisson, H.: Coupling, Stationarity, and Regeneration. Springer, New York (2000)
von Neumann, J.: Algebraische Repräsentanten der Funktionen “bis auf eine Menge vom Masse Null”. J. Crelle 165, 109–115 (1931)
Zähle, U.: Self-similar random measures, I. Notion, carrying Hausdorff dimension and hyperbolic distribution. Probab. Theory Relat. Fields 80, 79–100 (1988)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kallenberg, O. Some Problems of Local Hitting, Scaling, and Conditioning. Acta Appl Math 96, 271–282 (2007). https://doi.org/10.1007/s10440-007-9103-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-007-9103-4