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The growth of the expected number of real zeros of a random polynomial with dependent coefficients

Published online by Cambridge University Press:  24 October 2008

Richard Glendinning
Richard Glendinning
Affiliation:
School of Mathematical Sciences, University of Bath, Bath BA2 7AY
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Abstract

We determine the rate of growth of the expected number of real zeros of a class of random polynomials whose coefficients form a stationary uniformly mixing sequence.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 104 , Issue 3 , November 1988 , pp. 547 - 559
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Adler, R. J.. The Geometry of Random Fields (John Wiley & Sons, 1981).Google Scholar
[2]Bharucha-Reid, A. T. and Sambandham, M.. Random Polynomials (Academic Press, 1986).Google Scholar
[3]Christensen, M. J. and Sambandham, M.. An improved lower bound for the expected number of real zeros of a random polynomial. Stochastic Anal. Appl. 2 (1984), 431–436.CrossRefGoogle Scholar
[4]Dunnage, J. E. A.. The number of real zeros of a random trigonometric polynomial. Proc. London Math. Soc. 16 (1966), 53–84.CrossRefGoogle Scholar
[5]Erdös, P. and Offord, A. C.. On the number of real roots of a random algebraic equation. Proc. London Math. Soc. 6 (1956), 139–160.CrossRefGoogle Scholar
[6]Farahmand, K.. On the average number of real roots of a random algebraic equation. Ann. Probab. 14 (1986), 702–709.CrossRefGoogle Scholar
[7]Glendinning, R. H.. Random Polynomials. Ph.D thesis, London University (1986).Google Scholar
[8]Glendinning, R. H.. The growth of the expected number of real zeros of a random polynomial. J. Austral. Math. Soc. Ser. A (to appear).Google Scholar
[9]Glendinning, R. H.. The rate of convergence in the central limit theorem for non-stationary dependent random vectors. J. Multivariate Anal. (to appear).Google Scholar
[10]Holgate, P.. Studies in the history of probability and statistics XLI. Waring and Sylvester on random algebraic equations. Biometrika 73 (1986), 228–231.CrossRefGoogle Scholar
[11]Ibragimov, I. A. and Linnik, Yu. V.. Independent and Stationary Sequences of Random Variables (Wolters-Noordhoff, 1971).Google Scholar
[12]Ibragimov, I. A. and Maslova, N. B.. On the expected number of real zeros of random polynomials (I). Coefficients with zero means. Theory Probab. Appl. 16 (1971), 228–248.CrossRefGoogle Scholar
[13]Ibragimov, I. A. and Maslova, N. B.. On the expected number of real zeros of random polynomials (II). Coefficients with non-zero means. Theory Probab. Appl. 17 (1971), 485–493.CrossRefGoogle Scholar
[14]Ibragimov, I. A. and Maslova, N. B.. Average number of real roots of random polynomials. Soviet Math. Dokl. 12 (1971), 1004–1008.Google Scholar
[15]Iosifescu, M.. La loi du logarithme itéré pour une classe de variables aléatoires dépendent. Theor. Verojanost. i. Premenen 13 (1968), 315–325.Google Scholar
[16]Jamron, B. R.. The average number of real zeros of random polynomials. Soviet Math. Dokl. 13 (1972), 1381–1383.Google Scholar
[17]Kac, M.. On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Soc. 49 (1943), 314–320.CrossRefGoogle Scholar
[18]Kac, M.. On the average number of real roots of a random algebraic equation (II). Proc. London Math. Soc. 50 (1948), 390–408.CrossRefGoogle Scholar
[19]Littlewood, J. E. and Offord, A. C.. On the number of real roots of a random algebraic equation. J. London Math. Soc. 13 (1938), 288–295.CrossRefGoogle Scholar
[20]Logan, B. F. and Shepp, L. A.. Real zeros of random polynomials. Proc. London Math. Soc. 18 (1968), 29–39.CrossRefGoogle Scholar
[21]Logan, B. F. and Shepp, L. A.. Real zeros of random polynomials (II). Proc. London Math. Soc. 19 (1968), 308–314.CrossRefGoogle Scholar
[22]Sambandham, M.. On the average number of real zeros of a class of random algebraic curves. Pacific J. Math. 81 (1979), 207–215.CrossRefGoogle Scholar
[23]Shenker, M.. The mean number of real roots for one class of random polynomials. Ann. Probab. 8 (1981), 510–512.Google Scholar
[24]Stevens, D. C.. The average number of real zeros of a random polynomial. Comm. Pure Appl. Math. 22 (1969), 457–477.CrossRefGoogle Scholar
[25]Yoshihara, Ken-Ichi. Moment inequalities for mixing sequences. Kodai Math. J. 1 (1978), 316–318.CrossRefGoogle Scholar