How misleading can sample ACFs of stable MAs be? (Very!). (English) Zbl 0959.62076

Summary: For the stable moving average process \[ X_t=\int^\infty_{-\infty}f(t+x)M(dx),\;t=1,2,\dots, \] we find the weak limit of its sample autocorrelation function as the sample size \(n\) increases to \(\infty\). It turns out that, as a rule, this limit is random! This shows how dangerous it is to rely on sample correlation as a model fitting tool in the heavy tailed case. We discuss for what functions \(f\) this limit is nonrandom for all (or only some – this can be the case, too!) lags.


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G32 Statistics of extreme values; tail inference
60E07 Infinitely divisible distributions; stable distributions
60G70 Extreme value theory; extremal stochastic processes
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[1] Ahlfors, L. V. (1979). Complex Analy sis: An Introduction to the Theory of Analy tic Functions of One Complex Variable, 3rd ed. McGraw-Hill, New York.
[2] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York. · Zbl 0822.60002
[3] Brockwell, P. and Davis, R. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York. · Zbl 0709.62080
[4] Cohen, J., Resnick, S. and Samorodnitsky, G. (1998). Sample correlations of infinite variance time series models: an empirical and theoretical study. J. Appl. Math. Stochastic Anal. 11 255-282. · Zbl 0919.62104 · doi:10.1155/S1048953398000227
[5] Davis, R. and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13 179-195. · Zbl 0562.60026 · doi:10.1214/aop/1176993074
[6] Davis, R. and Resnick, S. (1996). Limit theory for bilinear processes with heavy tailed noise. Ann. Appl. Probab. 6 1191-1210. · Zbl 0879.60053 · doi:10.1214/aoap/1035463328
[7] Duffy, D., McIntosh, A., Rosenstein, M. and Willinger, W. (1993). Analy zing telecommunications data from working common channel signaling subnetworks. In Proceedings of the 25th Interface 156-165. Interface Foundation of North America, San Diego.
[8] Duffy, D., McIntosh, A., Rosenstein, M. and Willinger, W. (1994). Statistical analysis of CCSN/SS7 traffic data from working CCS subnetworks. IEEE J. Selected Areas Comm. 12 544-551.
[9] Durrett, R. (1996). Probability: Theory and Examples, 2nd ed. Duxbury Press, Belmont, CA. · Zbl 0709.60002
[10] Meier-Hellstern, K., Wirth, P., Yan, Y. and Hoeflin, D. (1991). Traffic models for ISDN data users: office automation application, teletraffic and datatraffic in period of change. In Proceedings of the 13th ITC (A. Jensen and V. B. Iversen, eds.) 167-192. North-Holland, Amsterdam.
[11] Resnick, S. (1997). Heavy tail modeling and teletraffic data. Ann. Statist. 25 1805-1869. · Zbl 0942.62097 · doi:10.1214/aos/1069362376
[12] Resnick, S. (1998). Why non-linearities can ruin the heavy tailed modeler’s day. In A Practical Guide to Heavy Tails: Statistical Techniques for Analy zing Heavy Tailed Distributions (R. Adler, R. Feldman and M. Taqqu, eds.) 219-240. Birkhäuser, Boston. · Zbl 0954.62107
[13] Resnick S. and Van Den Berg, E. (1998). Sample correlation behavior for the heavy tailed general bilinear process. TR 1210, http://www.orie.cornell.edu.trlist/. URL: · Zbl 0955.60028
[14] Rosi ński, J. (1998). Structure of stationary stable processes. In A Practical Guide to Heavy Tails: Statistical Techniques for Analy sing Heavy Tailed Distributions (R. Adler, R. Feldman and M. Taqqu, eds.) 461-472. Birkhäuser, Boston. · Zbl 0927.60051
[15] Samorodnitsky, G. and Szulga, J. (1989). An asy mptotic evaluation of the tail of a multiple sy mmetric -stable integral. Ann. Probab. 17 1503-1520. · Zbl 0689.60051 · doi:10.1214/aop/1176991170
[16] Samorodnitsky, G. and Taqqu, M. (1994). Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance. Chapman and Hall, New York. · Zbl 0925.60027
[17] Willinger, W., Taqqu, M., Sherman, R. and Wilson, D. (1997). Self-similarity through highvariability: statistical analysis of Ethernet LAN traffic at the source level. IEEE/ACM Trans. Networking 5 71-96.
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