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Point processes, regular variation and weak convergence

Published online by Cambridge University Press:  01 July 2016

Sidney I. Resnick
Sidney I. Resnick*
Affiliation:
Colorado State University
*
∗Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523, USA.
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Abstract

A method is reviewed for proving weak convergence in a function-space setting when regular variation is a sufficient condition. Point processes and weak convergence techniques involving continuity arguments play a central role. The method is dimensionless and holds computations to a minimum. Many applications of the methods to processes derived from sums and maxima are given.

Keywords

EXTREME VALUESPARTIAL SUM PROCESSESSTABLE PROCESSESPOISSON PROCESSEXTREMAL PROCESSESΠ-VARIATIONSUBORDINATIONINVARIANCE PRINCIPLESCONTINUITY OF FUNCTIONALS
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 1 , March 1986 , pp. 66 - 138
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

Partially supported by NSF Grant MCS 78-00915 and MCS-820235. Portions of the initial version were completed while supported by a Lady Davis Fellowship at the Technion. Grateful acknowledgement is made to the Faculty of Industrial and Management Engineering, Technion, Haifa, Israel, for their hospitality.

References

Adler, R., Monrad, D., Scissors, R. and Wilson, R. (1982) Representations, decompositions and sample function continuity of random fields with independent increments. Stoch. Proc. Appl. 15, 3–30.CrossRefGoogle Scholar
Arov, D. Z. and Bobrov, A. A. (1960) The extreme terms of a sample and their role in the sum of independent variables. Theory Prob. Appl. 5, 377–396.CrossRefGoogle Scholar
Berman, S. (1963) Limiting distribution of the studentized largest observation. Skand. Aktuarietidskr. , 154–161.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Bingham, N., Goldie, C. and Teugels, J. (1986) Regular Variation. To appear.CrossRefGoogle Scholar
Breiman, L. (1965) On some limit theorems similar to the arc-sin law. Theory Prob. Appl. 10, 323–331.CrossRefGoogle Scholar
Callaert, H. and Cohen, J. W. (1972) A lemma on regular variation of a transient renewal function. Z. Wahrscheinlichkeitsth. 24, 275–278.CrossRefGoogle Scholar
Chow, T. L. and Teugels, J. (1978) The sum and maximum of i.i.d. random variables. Proc. 2nd Prague Symp. Asymptotic Statistics. Google Scholar
Cinlar, E. (1976) Random measures and dynamic point processes II: Poisson random measures. Discussion Paper Number 11, Center for Statistics and Probability, Northwestern University, Evanston, Illinois.Google Scholar
Cline, D. (1983) Estimation and Linear Prediction for Regression, Autoregression and ARMA with Infinite Variance Data. Ph.D. Thesis, Dept. of Statistics, Colorado State University, Fort Collins, CO 80523, USA.Google Scholar
Cohen, J. W. (1973) Some results on regular variation for distributions in queueing and fluctuation theory. J. Appl. Prob. 10, 343–353.CrossRefGoogle Scholar
Darling, D. A. (1952) The influence of the maximum term in the addition of independent random variables. Trans. Amer. Math. Soc. 73, 95–107.Google Scholar
Darling, D. A. (1975) A note on a limit theorem. Ann. Prob. 3, 876–888.Google Scholar
Davis, R. (1983) Stable limits for partial sums of dependent random variables. Ann. Prob. 11, 262–269.Google Scholar
David, R. and Resnick, S. (1984a) Limit theory for the sample covariance and correlation functions of moving averages. Ann. Statist. To appear.CrossRefGoogle Scholar
Davis, R. and Resnick, S. (1984b) More limit theory for the sample correlation function of moving averages. Stoch. Proc. Appl. To appear.CrossRefGoogle Scholar
Deken, J. (1976) On Records: Scheduled Maxima Sequences and Longest Common Subsequences. Ph.D. Thesis, Department of Statistics, Stanford University.Google Scholar
Durrett, R. and Resnick, S. I. (1978) Functional limit theorems for dependent variables. Ann. Prob. 6, 829–846.Google Scholar
Dwass, ?. (1966) Extremal processes II. Illinois J. Math. 10, 381–391.CrossRefGoogle Scholar
Embrechts, P. and Goldie, C. (1980) On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. A 29, 243–256.CrossRefGoogle Scholar
Embrechts, P. and Goldie, C. (1982) On convolution tails. Stoch. Proc. Appl. 13, 263–278.Google Scholar
Embrechts, P. and Omey, E. (1983) On subordinated distributions and random record processes. Proc. Camb. Phil. Soc. 93, 339–353.Google Scholar
Embrechts, P., Goldie, C. and Veraverbeke, N. (1979) Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications. Vol. II, 2nd edn. Wiley, New York.Google Scholar
Freedman, D. (1971) Markov Chains. Holden-Day, San Francisco.Google Scholar
Gnedenko, B. and Kolmogorov, A. (1968) Limit Distributions for Sums of Independent Random Variables , translated by Chung, K. L. Addison-Wesley, Menlo Park, Ca.Google Scholar
Greenwood, P. (1973) Asymptotics of randomly stopped sequences with independent increments. Ann. Prob. 1, 317–321.CrossRefGoogle Scholar
Greenwood, P. and Monroe, I. (1977) Random stopping preserves regular variation of process distributions. Ann. Prob. 5, 42–51.Google Scholar
Haan, L. De (1970) On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Mathematical Centre Tract 32, Mathematics Centre, Amsterdam.Google Scholar
Haan, L. De (1976) An Abel-Tauber theorem for Laplace transforms. J. London Math. Soc. 13, 537–542.Google Scholar
Haan, L. De and Resnick, S. I. (1977) Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitsth. 40, 317–337.Google Scholar
Haan, L. De and Resnick, S. I. (1979a) Derivatives of regularly varying functions in Rd and domains of attraction of stable distributions. Stoch. Proc. Appl. 8, 349–355.Google Scholar
Haan, L. De and Resnick, S. I. (1979b) Coniugate ?-variation and process inversion. Ann. Prob. 7, 1028–1035.Google Scholar
Haan, L. De and Resnick, S. I. (1981) On the observation closest to the origin. Stoch. Proc. Appl. 11, 301–308.Google Scholar
Haan, L. De and Resnick, S. I. (1984) Stochastic compactness and point processes. J. Austral. Math. Soc. Google Scholar
Haan, L. De, Omey, E. and Resnick, S. I. (1984) Domains of attraction and regular variation in Rd. J. Mult. Anal. 14, 17–33.Google Scholar
Hudson, W. and Tucker, H. G. (1979). Asymptotic independence in the multivariate central limit theorem. Ann. Prob. 7, 662–671.Google Scholar
Hüsler, J. (1979) The limiting behavior of the last exit time for sequences of independent identically distributed random variables. Z. Wahrscheinlichkeitsth. 50, 159–164.Google Scholar
Ito, K. (1969) Stochastic Processes. Lecture Notes Series No. 16, Mathematics Institute, Aarhus Universitet.Google Scholar
Jain, N. and Orey, S. (1979) Vague convergence of sums of independent random variables. Israel J. Math. 33, 317–348.Google Scholar
Kallenberg, O. (1976) Random Measures. Akademie-Verlag, Berlin.Google Scholar
Lamperti, J. (1964) On extreme order statistics. Ann. Math. Statist. 35, 1726–1737.Google Scholar
Lepage, R., Woodroofe, M. and Zinn, J. (1981) Convergence to a stable distribution via order statistics. Ann. Prob. 9, 624–632.CrossRefGoogle Scholar
Lindvall, T. (1973) Weak convergence of probability measures and random functions in the function space D[0, 8). J. Appl. Prob. 10, 109–121.Google Scholar
Logan, B., Mallows, C., Rice, S. O. and Shepp, L. (1973) Limit distributions of self-normalized sums. Ann. Prob. 1, 788–809.Google Scholar
Mori, T. and Oodaira, H. (1976) A functional law of the iterated logarithm for sample sequences. Yokohama Math. J. 24, 35–49.Google Scholar
Neveu, J. (1976) Processes ponctuels. École d’Été de Probabilités de Saint-Flour. Lecture Notes in Mathematics 598, Springer-Verlag, Berlin.Google Scholar
Resnick, S. I. (1975) Weak convergence to extremal processes. Ann. Prob. 3, 951–960.Google Scholar
Resnick, S. I. (1982) Extremal processes. In Encyclopedia of Statistical Science ed. Johnson, N. L. and Kotz, S. Wiley, New York.Google Scholar
Resnick, S. I. and Greenwood, P. (1979) A bivariate stable characterization and domains of attraction. J. Multivariate Anal. 9, 206–221.CrossRefGoogle Scholar
Resnick, S. I. and Rubinovitch, M. (1973) The structure of extremal processes. Adv. Appl. Prob. 5, 287–307.Google Scholar
Rogozin, B. A. (1971) The distribution of the first ladder moment and height and fluctuation of a random walk. Theory Prob. Appl. 16, 575–595.CrossRefGoogle Scholar
Seneta, E. (1976) Regularly Varying Functions. Lecture Notes in Mathematics 508, Springer-Verlag, Berlin.Google Scholar
Serfozo, R. (1982) Functional limit theorems for extreme values of arrays of independent random variables. Ann. Prob. 10, 172–177.Google Scholar
Simons, G. and Stout, W. (1978) A weak invariance principle with applications to domains of attraction. Ann. Prob. 6, 294–315.Google Scholar
Stam, A. J. (1973) Regular variation of the tail of a subordinated probability distribution. Adv. Appl. Prob. 5, 287–307.Google Scholar
Stam, A. J. (1977) Regular variation in Rd+ and the Abel-Tauber theorem. Preprint, Mathematisch Institut, Rijksuniversiteit Groningen.Google Scholar
Straf, ?. (1972) Weak convergence of stochastic processes with several parameters. Proc. 6th Berkeley Symp. Math. Statist. Prob. 2, 187–222.Google Scholar
Tucker, H. G. (1968) Convolutions of distributions attracted to stable laws. Ann. Math. Statist. 39, 1381–1390.Google Scholar
Vervaat, W. (1979) On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.CrossRefGoogle Scholar
Veraverbeke, N. (1977) Asymptotic behavior of Wiener-Hopf factors of a random walk. Stoch. Proc. Appl. 5, 27–37.CrossRefGoogle Scholar
Weissman, I. (1975) On weak convergence of extremal processes. Ann. Prob. 4, 470–473.Google Scholar
Whitt, W. (1980) Some useful functions for functional limit theorems. Math. Operat. Res. 5, 67–85.Google Scholar
Wichura, M. (1969) Inequalities with applications to the weak convergence of random processes with multidimensional time parameters. Ann. Math. Statist. 40, 681–687.Google Scholar