The Annals of Applied Probability

Limit theory for bilinear processes with heavy-tailed noise

Richard A. Davis and Sidney I. Resnick

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We consider a simple stationary bilinear model $X_t = cX_{t-1} Z_{t-1} + Z_t, t = 0, \pm 1, \pm 2, \dots,$ generated by heavy-tailed noise variables ${Z_t}$. A complete analysis of weak limit behavior is given by means of a point process analysis. A striking feature of this analysis is that the sample correlation converges in distribution to a nondegenerate limit. A warning is sounded about trying to detect nonlinearities in heavy-tailed models by means of the sample correlation function.

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Ann. Appl. Probab., Volume 6, Number 4 (1996), 1191-1210.

First available in Project Euclid: 24 October 2002

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Zentralblatt MATH identifier

Primary: 60E07: Infinitely divisible distributions; stable distributions 60F17: Functional limit theorems; invariance principles 60G55: Point processes 60G70: Extreme value theory; extremal processes 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Extreme value theory Poisson processes bilinear and nonlinear processes stable laws point processes stationary processes sample correlation


Davis, Richard A.; Resnick, Sidney I. Limit theory for bilinear processes with heavy-tailed noise. Ann. Appl. Probab. 6 (1996), no. 4, 1191--1210. doi:10.1214/aoap/1035463328.

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  • BILLINGSLEY, P. 1968. Convergence of Probability Measures. Wiley, New York. Z.
  • BINGHAM, N., GOLDIE, C. and TEUGELS, J. 1987. Regular variation. Ency clopedia of Mathematics and Its Applications. Cambridge Univ. Press. Z.
  • BREIMAN, L. 1965. On some limit theorems similar to the arc-sin law. Theory Probab. Appl. 10 323 331. Z.
  • BROCKWELL, P. and DAVIS, R. 1991. Time Series: Theory and Methods, 2nd ed. Springer, New York. Z.
  • CLINE, D. 1983. Estimation and linear prediction for regression, autoregression and ARMA with infinite variance data. Thesis, Dept. Statistics, Colorado State Univ. Z.
  • DAVIS, R. A. and HSING, T. 1995. Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab. 23 879 917. Z.
  • DAVIS, R. A. and RESNICK, S. 1985a. Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13 179 195. Z.
  • DAVIS, R. A. and RESNICK, S. 1985b. More limit theory for the sample correlation function of moving averages. Stochastic Process. Appl. 20 257 279. Z.
  • DAVIS, R. A. and RESNICK, S. 1986. Limit theory for the sample covariance and correlation functions of moving averages. Ann. Statist. 14 533 558. Z.
  • DUFFY, D., MCINTOSH, A., ROSENSTEIN, M. and WILLINGER, W. 1993. Analy zing telecommunications traffic data from working common channel signaling subnetworks. In Proceedings of the 25th Interface Conference, San Diego, CA 156 165. Interface Foundation of North America. Z.
  • DUFFY, D., MCINTOSH, A., ROSENSTEIN, M. and WILLINGER, W. 1994. Statistical analysis of CCSN SS7 traffic data from working CCS subnetworks. IEEE Journal on Selected Areas in Communications 12 544 551. Z.
  • FEIGIN, P. and RESNICK, S. 1994. Limit distributions for linear programming time series estimators. Stochastic Process. Appl. 51 135 166. Z.
  • FEIGIN, P. and RESNICK, S. 1996. Pitfalls of fitting autoregressive models for heavy-tailed time series. Unpublished manuscript. Available at http: trlist trlist.html as Z.
  • FEIGIN, P., KRATZ, M. and RESNICK, S. 1996. Parameter estimation for moving averages with positive innovations. Ann. Appl. Probab. 6 1157 1190.
  • FEIGIN, P., RESNICK, S. and STARICA, C. 1995. Testing for independence in heavy tailed and positive innovation time series. Stochastic Models 11 587 612. Z.
  • GELUK, J. and DE HAAN, L. 1987. Regular Variation, Extensions and Tauberian Theorems. CWI Tract 40. Center for Mathematics and Computer Science, Amsterdam. Z.
  • KALLENBERG, O. 1983. Random Measures, 3rd ed. Akademie-Verlag, Berlin. Z.
  • LIU, J. 1989. On the existence of a general multiple bilinear time series. J. Time Series Anal. 10 341 355. Z. MEIER-HELLSTERN, K., WIRTH, P., YAN, Y. and HOEFLIN, D. 1991. Traffic models for ISDN data users: office automation application. In Teletraffic and Datatraffic in a Period of Z. Change. Proceedings of the 13th ITC A. Jensen and V. B. Iversen, eds. 167 192. NorthHolland, Amsterdam. Z.
  • RESNICK, S. 1986. Point processes, regular variation and weak convergence. Adv. in Appl. Probab. 18 66 138. Z.
  • RESNICK, S. 1987. Extreme Values, Regular Variation, and Point Processes. Springer, New York. Z. Z
  • RESNICK, S. 1995. Heavy tail modelling and teletraffic data. Unpublished manuscript. Availa. ble at http: trlist trlist.html as Z.
  • WILLINGER, W., TAQQU, M., SHERMAN, R. and WILSON, D. 1995. Self-similarity through highvariability: statistical analysis of ethernet LAN traffic at the source level. Preprint.