## Limit theory for bilinear processes with heavy-tailed noise.(English)Zbl 0879.60053

The authors consider a simple first-order bilinear time series $$\{X_t\}$$ defined as a stationary solution to the equations $$X_t=cX_{t-1}Z_{t-1}+ Z_t$$, where $$\{Z_t\}$$ is an i.i.d. sequence of random variables with regularly varying tail probabilities. Using the weak convergence techniques based on the point process method, they give a complete analysis of weak limit behaviour of $$\{X_t\}$$. Some corollaries of the obtained convergence results, with emphasis on the limiting behaviour of the extremes, partial sums and sample correlations, are presented. Note that, unlike the linear process case, the sample correlations of the bilinear process converge in distribution to nondegenerate limit random variables.

### MSC:

 60G70 Extreme value theory; extremal stochastic processes 60E07 Infinitely divisible distributions; stable distributions 60F17 Functional limit theorems; invariance principles 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
Full Text:

### References:

 [1] BILLINGSLEY, P. 1968. Convergence of Probability Measures. Wiley, New York. Z. · Zbl 0172.21201 [2] BINGHAM, N., GOLDIE, C. and TEUGELS, J. 1987. Regular variation. Ency clopedia of Mathematics and Its Applications. Cambridge Univ. Press. Z. · Zbl 0617.26001 [3] BREIMAN, L. 1965. On some limit theorems similar to the arc-sin law. Theory Probab. Appl. 10 323 331. Z. · Zbl 0147.37004 [4] BROCKWELL, P. and DAVIS, R. 1991. Time Series: Theory and Methods, 2nd ed. Springer, New York. Z. · Zbl 0709.62080 [5] CLINE, D. 1983. Estimation and linear prediction for regression, autoregression and ARMA with infinite variance data. Thesis, Dept. Statistics, Colorado State Univ. Z. [6] 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. · Zbl 0837.60017 [7] 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. · Zbl 0562.60026 [8] 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. · Zbl 0572.62075 [9] 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. · Zbl 0605.62092 [10] 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. [11] 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. [12] FEIGIN, P. and RESNICK, S. 1994. Limit distributions for linear programming time series estimators. Stochastic Process. Appl. 51 135 166. Z. · Zbl 0819.62070 [13] FEIGIN, P. and RESNICK, S. 1996. Pitfalls of fitting autoregressive models for heavy-tailed time series. Unpublished manuscript. Available at http: www.orie.cornell.edu trlist trlist.html as TR1163.ps.Z. Z. URL: · Zbl 0933.62087 [14] FEIGIN, P., KRATZ, M. and RESNICK, S. 1996. Parameter estimation for moving averages with positive innovations. Ann. Appl. Probab. 6 1157 1190. · Zbl 0881.62093 [15] 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. · Zbl 0837.60035 [16] GELUK, J. and DE HAAN, L. 1987. Regular Variation, Extensions and Tauberian Theorems. CWI Tract 40. Center for Mathematics and Computer Science, Amsterdam. Z. · Zbl 0624.26003 [17] KALLENBERG, O. 1983. Random Measures, 3rd ed. Akademie-Verlag, Berlin. Z. · Zbl 0544.60053 [18] 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. · Zbl 0691.62079 [19] RESNICK, S. 1986. Point processes, regular variation and weak convergence. Adv. in Appl. Probab. 18 66 138. Z. JSTOR: · Zbl 0597.60048 [20] RESNICK, S. 1987. Extreme Values, Regular Variation, and Point Processes. Springer, New York. Z. Z · Zbl 0633.60001 [21] RESNICK, S. 1995. Heavy tail modelling and teletraffic data. Unpublished manuscript. Availa. ble at http: www.orie.cornell.edu trlist trlist.html as TR1134.ps.Z. Z. URL: [22] WILLINGER, W., TAQQU, M., SHERMAN, R. and WILSON, D. 1995. Self-similarity through highvariability: statistical analysis of ethernet LAN traffic at the source level. [23] ITHACA, NEW YORK 14853 E-MAIL: sid@orie.cornell.edu
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.