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On the lower bound of the number of real roots of a random algebraic equation with infinite variance


Authors: G. Samal and M. N. Mishra
Journal: Proc. Amer. Math. Soc. 33 (1972), 523-528
MSC: Primary 60E05
DOI: https://doi.org/10.1090/S0002-9939-1972-0295411-4
MathSciNet review: 0295411
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {N_n}$ be the number of real roots of a random algebraic equation $ \Sigma _0^n{\xi _v}{x^v} = 0$, where the coefficients $ {\xi _v}$'s are independent random variables with common characteristic function $ \exp ( - C\vert t{\vert^\alpha })$, C being a positive constant and $ \alpha \geqq 1$. It is proved that

$\displaystyle {N_n} \geqq (\mu \log n)/(\log \log n).$

The measure of the exceptional set tends to zero as n tends to infinity.

References [Enhancements On Off] (What's this?)

  • [1] 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
  • [2] J. E. Littlewood and A. C. Offord, On the number of real roots of a random algebraic equation. II, Proc. Cambridge Philos. Soc. 35 (1939), 133-148.
  • [3] G. Samal, On the number of real roots of a random algebraic equation, Proc. Cambridge Philos. Soc. 58 (1962), 433–442. MR 0139221

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0295411-4
Keywords: Random variables, stable distribution, infinite variance, random algebraic equations
Article copyright: © Copyright 1972 American Mathematical Society