## The Annals of Probability

### The maximum of the periodogram for a heavy-tailed sequence

#### Abstract

We consider the maximum of the periodogram based on an infinite variance heavy-tailed sequence. For $\alpha < 1$ we show that the maxima constitute a weakly convergent sequence and find its limiting distribution. For $1 \leq \alpha < 2$ we show that the sequence of the maxima is not tight and find a normalization that makes it tight.

#### Article information

Source
Ann. Probab., Volume 28, Number 2 (2000), 885-908.

Dates
First available in Project Euclid: 18 April 2002

https://projecteuclid.org/euclid.aop/1019160264

Digital Object Identifier
doi:10.1214/aop/1019160264

Mathematical Reviews number (MathSciNet)
MR1782277

Zentralblatt MATH identifier
1044.62097

Subjects
Primary: 62M15: Spectral analysis
Secondary: 60F05: Central limit and other weak theorems 60G10: Stationary processes 60G55.

#### Citation

Mikosch, Thomas; Resnick, Sidney; Samorodnitsky, Gennady. The maximum of the periodogram for a heavy-tailed sequence. Ann. Probab. 28 (2000), no. 2, 885--908. doi:10.1214/aop/1019160264. https://projecteuclid.org/euclid.aop/1019160264

#### References

• Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
• Bingham, N., Goldie, C. and Teugels, J. (1987). Regular Variation. Cambridge Univ. Press.
• Brockwell, P. and Davis, R. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
• Chen,-G. and Hannan, E. (1980). The distribution of the periodogram ordinates. J. Time Ser. Anal. 1 73-82.
• Davis, R. and Mikosch, T. (1999). The maximum of the periodogram of a non-Gaussian sequence. Ann. Probab. 27 522-536.
• Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York.
• Fisher, R. A. (1929). Tests of significance in harmonic analysis. Proc. Roy. Statist. Soc. Ser. A 125 54-59.
• Freedman, D. and Lane, D. (1980). The empirical distribution of Fourier coefficients. Ann. Statist. 8 1244-1251.
• Freedman, D. and Lane, D. (1981). The empirical distribution of the Fourier coefficients of a sequence of independent, identically distributed long-tailed random variables.Wahrsch. Verw. Gebiete 58 21-39.
• Kl ¨uppelberg, C. and Mikosch, T. (1993). Spectral estimates and stable processes. Stochastic Process. Appl. 47 323-344.
• Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces: Isoperimetry and Processes. Springer, New York.
• Priestley, M. (1981). Spectral Analysis and Time Series I, II. Academic Press, New York.
• Resnick, S. (1986). Point processes. Regular variation and weak convergence. Adv. in Appl. Probab. 18 66-138.
• Resnick, S. (1987). Extreme Values. Regular Variation and Point Processes. Springer, New York.
• Samorodnitsky, G. and Taqqu, M. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall. New York.
• Turkman, K. F. and Walker, A. (1984). On the asymptotic distributions of maxima of trigonometric polynomials with random coefficients. Adv. in Appl. Probab. 16 819-842.
• Weyl, H. (1916). ¨Uber die Gleichverteilung von Zahlen mod Eins. Math. Ann. 77 313-352.