Maxima of random algebraic curves

Authors:
M. Das and S. S. Bhatt

Journal:
Trans. Amer. Math. Soc. **243** (1978), 195-212

MSC:
Primary 60G99

DOI:
https://doi.org/10.1090/S0002-9947-1978-0494482-X

MathSciNet review:
0494482

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a sequence of independent and identically distributed random variables with common characteristic function where , and . Then we show that the numbers of maxima of the curves have expectation , as , where and

**[1]**Minaketan Das,*The average number of maxima of a random algebraic curve*, Proc. Cambridge Philos. Soc.**65**(1969), 741–753. MR**0239669****[2]**Minaketan Das,*Real zeros of a class of random algebraic polynomials*, J. Indian Math. Soc. (N.S.)**36**(1972), 53–63. MR**0322960****[3]**B. V. Gnedenko and A. N. Kolmogorov,*Limit distributions for sums of independent random variables*, Addison-Wesley Publishing Company, Inc., Cambridge, Mass., 1954. Translated and annotated by K. L. Chung. With an Appendix by J. L. Doob. MR**0062975****[4]**G. H. Hardy, J. E. Littlewood, and G. Pólya,*Inequalities*, Cambridge, at the University Press, 1952. 2d ed. MR**0046395****[5]**M. Kac,*On the average number of real roots of a random algebraic equation*, Bull. Amer. Math. Soc.**49**(1943), 314–320. MR**0007812**, https://doi.org/10.1090/S0002-9904-1943-07912-8**[6]**K. Knopp,*Theory and application of infinite series*, Blackie, London, 1957.**[7]**B. F. Logan and L. A. Shepp,*Real zeros of random polynomials*, Proc. London Math. Soc. (3)**18**(1968), 29–35. MR**0234512**, https://doi.org/10.1112/plms/s3-18.1.29**[8]**B. F. Logan and L. A. Shepp,*Real zeros of random polynomials. II*, Proc. London Math. Soc. (3)**18**(1968), 308–314. MR**0234513**, https://doi.org/10.1112/plms/s3-18.2.308

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0494482-X

Keywords:
Random variables,
distribution,
normally distributed random variables,
sequence,
mathematical expectation,
variance,
characteristic function,
random algebraic curves

Article copyright:
© Copyright 1978
American Mathematical Society