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Estimation of second order properties of stationary doubly stochastic Poisson sequences

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Doubly Stochastic Poisson Processes

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References

  • Barndorff-Nielsen, O. and Yeo, G.F. (1969). Negative binomial processes. J. Appl. Prob. 6, 633–647. Correction in J.A.P. 7, 249.

    Article  MathSciNet  MATH  Google Scholar 

  • Bauer, H. (1968). Wahrscheinlichkeitstheorie und Grundzüge der Masstheorie. Walter de Gruyter & Co. Berlin.

    MATH  Google Scholar 

  • Billingsley, P. (1965). Ergodic theory and information. John Wiley and Sons. New York.

    MATH  Google Scholar 

  • Billingsley, P. (1968). Convergence of probability measures. John Wiley and Sons. New York.

    MATH  Google Scholar 

  • Bingham, N.H. (1971). Limit theorems for occupation times of Markov processes. Z. Wahrscheinlichkeitstheorie verw. Geb. 17, 1–22.

    Article  MathSciNet  MATH  Google Scholar 

  • Cox, D.R. (1955). Some statistical methods related with series of events. J. R. statist. Soc. B, 17, 129–164.

    MATH  Google Scholar 

  • Cox, D.R. and Lewis, P.A.W. (1966). The statistical analysis of events. Methuen. London. and Barnes and Noble. New York.

    Book  MATH  Google Scholar 

  • Cramér, H. (1955). Collective risk theory. The jubilee volume of Försäkringsbolaget Skandia. Stockholm.

    Google Scholar 

  • Cramér, H. and Leadbetter, M.R. (1967). Stationary and related stochastic processes. John Wiley and Sons, New York.

    MATH  Google Scholar 

  • Cramér, H. (1969). On streams of random events. Skand. Aktuar. Tidskrift 52, Suppl., 13–23.

    MathSciNet  MATH  Google Scholar 

  • Daley, D.J. and Vere-Jones, D. (1972). A summary of the theory of point processes. Stochastic point processes: Statistical analysis, theory and applications. Ed. by Lewis, P.A.W., 299–383. Wiley-Interscience. New York.

    Google Scholar 

  • Dobrushin, R.L. (1965). A lemma on the limit of a composite random function (in Russian). Uspehi Mat. Nauk 10, no. 2 (64), 157–159.

    MathSciNet  Google Scholar 

  • Doob, J.L. (1953). Stochastic Processes. John Wiley and Sons, New York.

    MATH  Google Scholar 

  • Feller, W. (1971). An introduction to probability and its applications. Vol. 11. 2nd ed. John Wiley and Sons. New York.

    MATH  Google Scholar 

  • Gaver, D.P. (1963). Random hazard in reliability problems. Technometrics 5, 211–226.

    Article  MathSciNet  Google Scholar 

  • Grandell J. (1971). On stochastic processes generated by a stochastic intensity function. Skand. Aktuar. Tidskrift 54, 204–240.

    MathSciNet  MATH  Google Scholar 

  • Grandell, J. (1972:1). On the estimation of intensities in a stochastic process generated by a stochastic intensity sequence. J. Appl. Prob. 9, 542–556.

    Article  MathSciNet  MATH  Google Scholar 

  • Grandell, J. (1972:2). Statistical inference for doubly stochastic Poisson processes. Stochastic point processes: Statistical analysis, theory and applications. Ed. by Lewis, P.A.W., 90–121. Wiley-Interscience. New York.

    Google Scholar 

  • Grandell, J. (1973). A note on characterization and convergence of non-atomic random measures. Int. conf. on prob. theory and math. stat., Abstracts of communications T. 1., 175–176, Vilnius.

    Google Scholar 

  • Grenander, U. (1951). On Toeplitz forms and stationary processes. Arkiv för matematik 1. 555–571.

    Article  MathSciNet  MATH  Google Scholar 

  • Grenander, U. and Rosenblatt, M. (1956). Statistical analysis of stationary time series. Almqvist & Wiksell, Stockholm, and John Wiley and Sons. New York.

    MATH  Google Scholar 

  • Grenander, U. and Szegö, G. (1958). Toeplitz forms and their applications. Univ. of California Press, Berkeley and Los Angeles.

    MATH  Google Scholar 

  • Hannan, E.J. (1960). Time series analysis. Methuen & Co. London.

    MATH  Google Scholar 

  • Hannan, E.J. (1970). Multiple time series. John Wiley and Sons. New York.

    Book  MATH  Google Scholar 

  • Jagers, P. (1973). On Palm probabilities. Z. Wahrscheinlichkeits-theorie verw. Geb. 26, 17–32.

    Article  MathSciNet  MATH  Google Scholar 

  • Jagers, P. (1974). Aspects of random measures and point processes. Advances in probability and related topics. 3. Ed. by Ney, P., 179–239. Marcel Dekker, New York.

    Google Scholar 

  • Jung, J. and Lundberg, O. (1969). Risk processes connected with the compound Poisson process. Skand. Aktuar. Tidskrift, Suppl., 118–131.

    Google Scholar 

  • Kallenberg, O. (1971). Lecture at the Gothenburg conference on point processes.

    Google Scholar 

  • Kallenberg O. (1973:1). Characterization and convergence of random measures and point processes. Z. Wahrscheinlichkeitstheorie verw. Geb. 27. 9–21.

    Article  MathSciNet  MATH  Google Scholar 

  • Kallenberg, O. (1973:2). Characterization of continuous random processes and signed measures. Studia Sci. Math. Hungarica 8, 473–477.

    MathSciNet  MATH  Google Scholar 

  • Kallenberg, O. (1975:1). Limits of compound and thinned point processes. J. Appl. Prob. 12, 269–278.

    Article  MathSciNet  MATH  Google Scholar 

  • Kallenberg, O. (1975:2). Random measures. Schriftenreihe des Zentral-instituts für Mathematik und Mechanik der ADW der DDR, Akademie-Verlag, Berlin.

    MATH  Google Scholar 

  • Kallenberg, O. (1976). On the structure of stationary flat processes. Tech. Rep., Dept. of math., Gothenburg.

    MATH  Google Scholar 

  • Kerstan, J., Matthes, K. and Mecke, J. (1974). Unbegrenzt teilbare Punktprozesse. Akademie-Verlag, Berlin.

    MATH  Google Scholar 

  • Khintchine, A.Y. (1960). Mathematical methods in the theory of queuing. Charles Griffin, London.

    MATH  Google Scholar 

  • Kingman, J.F.C. (1964). On doubly stochastic Poisson processes. Proc. Camb. Phil. Soc. 60, 923–930.

    Article  MathSciNet  MATH  Google Scholar 

  • Kingman, J.F.C. (1972). Regenerative phenomena. John Wiley and Sons, New York.

    MATH  Google Scholar 

  • Kolmogorov, A. N. (1939). Sur l'interpolation et extrapolation des suites stationnaires. C. R. Acad. Sc. Paris 208, 2043–2045.

    MATH  Google Scholar 

  • Krickeberg, K. (1972). The Cox process. Symposia Mathematica IX. 151–167.

    MathSciNet  MATH  Google Scholar 

  • Kummer, G. and Matthes, K. (1970). Verallgemeinerung eines Satzes von Sliwnjak III. Rev. Roum. math. pure et appl. 15: 10, 1631–1642.

    MathSciNet  MATH  Google Scholar 

  • Lamperti, J. (1962). Semi-stable stochastic processes. Trans. Amer. Math. Soc. 104, 62–78.

    Article  MathSciNet  MATH  Google Scholar 

  • Lawrance, A.J. (1972) Some models for stationary series of events. Stochastic point processes: Statistical analysis, theory and applications. Ed. by Lewis, P.A.W., 199–256, Wiley-Interscience. New York.

    Google Scholar 

  • Lindvall, T. (1973:1). Weak convergence of probability measures and random functions in the function space D[0,∞). J. Appl. Prob. 10, 109–121.

    Article  MathSciNet  MATH  Google Scholar 

  • Lindvall, T. (1973:2). Weak convergence in the function space D[0,∞) and diffusion approximations of certain Galton-Watson branching processes. Tech. Rep., Dept. of math., Gothenburg.

    Google Scholar 

  • Lundberg, O. (1940). On random processes and their application to sickness and accident statistics 2nd ed. 1964, Almqvist & Wiksell. Uppsala.

    MATH  Google Scholar 

  • Macchi, O. (1971). Distribution statistique des instants d'émission des photoélectrons d'une lumière thermique. C. R. Acad. Sc. Paris 272, ser A, 437–440.

    MathSciNet  MATH  Google Scholar 

  • Macchi, O. and Picinbono, B. (1972). Estimation and detection of weak optical signals. IEEE Trans. Inform. Theory 18, 562–573.

    Article  MATH  Google Scholar 

  • Marcus, M. and Minc, H. (1965). Permanents. Amer. Math. Monthly 72, 577–591.

    Article  MathSciNet  MATH  Google Scholar 

  • Mecke, J. (1967). Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen. Z. Wahrscheinlichkeitstheorie verw. Geb. 9, 36–58.

    Article  MathSciNet  MATH  Google Scholar 

  • Mecke, J. (1968). Eine charakteristische Eigenschaft der doppelt stochastischen Poissonschen Prozesse. Z. Wahrscheinlichkeitstheorie verw. Geb. 11, 74–81.

    Article  MathSciNet  MATH  Google Scholar 

  • Mecke, J. (1972). Zufällige Masse auf lokalkompakten Hausdorffschen Räumen. Beiträge zur Analysis 3. 7–30.

    MATH  Google Scholar 

  • Mönch, G. (1971). Verallgemeinerung eines satzes von A. Rényi. Studia Sci. Math. Hungar. 6, 81–90.

    MathSciNet  MATH  Google Scholar 

  • Neuts, M.F. (1971). A queue subject to extraneous phase changes. Adv. Appl. Prob. 3, 78–119.

    Article  MathSciNet  MATH  Google Scholar 

  • Parzen, E. (1959). Statistical inference on time series by Hilbert space methods. I. Published in Parzen, E. (1967). Time series analysis papers. Holden Day, San Francisco.

    Google Scholar 

  • Rodhe, H. and Grandell, J. (1972). On the removal time of aerosol particles from the atmosphere by precipitation scavenging. Tellus 24, 443–454.

    Google Scholar 

  • Rootzén, H. (1975). A note on the central limit theorem for doubly stochastic Poisson processes. Tech. report, The university of North Carolina.

    Google Scholar 

  • Rosenblatt, M. (1959). Statistical analysis of stochastic processes with stationary residuals. Probability and statistics—The Harald Cramér volume. Ed. by Grenander, U., 246–257. Almqvist & Wiksell, Stockholm, and John Wiley and Sons, New York.

    Google Scholar 

  • Rozanov, Yu. A. (1960). On stationary sequences forming a basis. Soviet Math.-Doklady 1, 155–158.

    MathSciNet  MATH  Google Scholar 

  • Rozanov, Yu. A. (1967). Stationary random processes. Holden-Day. San Francisco.

    MATH  Google Scholar 

  • Rubin, I. (1972). Regular point processes and their detection. IEEE Trans. Inform. Theory 18, 547–557.

    Article  MathSciNet  MATH  Google Scholar 

  • Rudemo, M. (1972). Doubly stochastic Poisson processes and process control. Adv. Appl. Prob. 4, 318–338.

    Article  MathSciNet  MATH  Google Scholar 

  • Rudemo, M. (1973:1) State estimation for partially observed Markov chains. J. Math. Anal. Appl. 44, 581–611.

    Article  MathSciNet  MATH  Google Scholar 

  • Rudemo, M. (1973:2). Point processes generated by transitions of Markov chains. Adv. Appl. Prob. 5, 262–286.

    Article  MathSciNet  MATH  Google Scholar 

  • Rudemo, M. (1975). Prediction and smothing for partially observed Markov chains. J. Math. Anal. Appl. 49, 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  • Ryll-Nardzewski, C. (1961). Remarks on processes of calls. Proc. 4th Berkeley Symp. 2, 465–471.

    MathSciNet  MATH  Google Scholar 

  • Serfozo, R. (1972:1). Conditional Poisson processes. J. Appl. Prob. 9, 288–302.

    Article  MathSciNet  MATH  Google Scholar 

  • Serfozo, R. (1972:2). Processes with conditional independent increments. J. Appl. Prob. 9, 303–315.

    Article  MathSciNet  MATH  Google Scholar 

  • Siegert, A.J.F. (1957). A systematic approach to a class of problems in the theory of noise and other random phenomena: Part II. IRE Trans. Inform. Theory 3, 37–43.

    Article  Google Scholar 

  • Skorohod, A.V. (1957). Limit theorems for stochastic processes with independent increments. Theory Prob. Applications 11, 138–171.

    Article  MathSciNet  Google Scholar 

  • Snyder, D.L. (1972:1). Filtering and detection for doubly stochastic Poisson processes. IEEE Trans. Inform. Theory 18, 91–102.

    Article  MathSciNet  MATH  Google Scholar 

  • Snyder, D. L. (1972:2). Smoothing for doubly stochastic Poisson processes. IEEE Trans. Inform. Theory 18, 558–562.

    Article  MathSciNet  MATH  Google Scholar 

  • Snyders D.L. (1975). Random point processes. John Wiley and Sons. New York.

    Google Scholar 

  • Snyders, J. (1972). Error formulae for optimal linear filtering, prediction and interpolation of stationary time series. Ann. Math. Statist. 43, 1935–1943.

    Article  MathSciNet  MATH  Google Scholar 

  • Stone, C. (1963). Weak convergence of stochastic processes defined on a semifinite time interval. Proc. Amer. Math. Soc. 14, 694–696.

    Article  MathSciNet  MATH  Google Scholar 

  • van Trees, H.L. (1968). Detection, estimation, and modulation theory. Part 1. John Wiley and Sons, New York.

    MATH  Google Scholar 

  • Waldenfels, M.V. (1968). Charakteristische Funktionale zufälliger Masse. Z. Wahrscheinlichkeitstheorie verw. Geb. 10, 279–283.

    Article  MATH  Google Scholar 

  • Westcott, M. (1972). The probability generating functional. J. Aust. Math. Soc. 14, 448–466.

    Article  MathSciNet  MATH  Google Scholar 

  • Whitt, W. (1970). Weak convergence of probability measures on the function space D[0,∞). Tech. report, Yale university.

    Google Scholar 

  • Whitt, W. (1972). Continuity of several functions on the function space D. A. revised version is sometimes referred to as ‘to appear in Ann. Prob.'.

    Google Scholar 

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Grandell, J. (1976). Estimation of second order properties of stationary doubly stochastic Poisson sequences. In: Doubly Stochastic Poisson Processes. Lecture Notes in Mathematics, vol 529. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0077765

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