On the lower bound of the number of real roots of a random algebraic equation with infinite variance. II
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- by G. Samal and M. N. Mishra PDF
- Proc. Amer. Math. Soc. 36 (1972), 557-563 Request permission
Abstract:
Let ${N_n}$ be the number of real roots of a random algebraic equation $\sum \nolimits _{v = 0}^n {{\xi _v}{x^v} = 0}$ where the ${\xi _v}$’s are independent random variables with a common characteristic function \[ \exp ( - C|t{|^\alpha }),\quad \alpha > 1,\] and C, a positive constant. Then for $n > {n_0}$, \[ {N_n} > (\mu \log n)/(\log \log n)\] outside a set of measure at most \[ \mu ’/{\{ \log ((\log {n_0})/(\log \log {n_0}))\} ^{\alpha - 1}}.\]References
- E. A. Evans, On the number of real roots of a random algebraic equation, Proc. London Math. Soc. (3) 15 (1965), 731–749. MR 180997, DOI 10.1112/plms/s3-15.1.731
- B. V. Gnedenko and A. N. Kolmogorov, Limit distributions for sums of independent random variables, Addison-Wesley Publishing Co., Inc., Cambridge, Mass., 1954. Translated and annotated by K. L. Chung. With an Appendix by J. L. Doob. MR 0062975 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.
- G. Samal, On the number of real roots of a random algebraic equation, Proc. Cambridge Philos. Soc. 58 (1962), 433–442. MR 139221
- G. Samal and M. N. Mishra, On the lower bound of the number of real roots of a random algebraic equation with infinite variance, Proc. Amer. Math. Soc. 33 (1972), 523–528. MR 295411, DOI 10.1090/S0002-9939-1972-0295411-4
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 36 (1972), 557-563
- MSC: Primary 60G99
- DOI: https://doi.org/10.1090/S0002-9939-1972-0315785-5
- MathSciNet review: 0315785