On the upper 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
- Proc. Amer. Math. Soc. 44 (1974), 446-448
- DOI: https://doi.org/10.1090/S0002-9939-1974-0438473-5
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Abstract:
Let ${N_n}$ be the number of real roots of $\sum \nolimits _{v = 0}^n {{a_v}{\xi _v}{x^v}} = 0$ where ${\xi _v}$’s are independent random variables identically distributed with a common characteristic function $\exp ( - C|t{|^\alpha });C$ is a positive constant, ${a_0},{a_1}, \cdots ,{a_n}$ are nonzero real numbers such that ${k_n} = {\max _{0 \leqq v \leqq n}}|{a_v}| = O({n^\beta }/\log n)$. Then (i) $\Pr \{ {\operatorname {Sup} _{n > {n_0}}}{N_n}/{(\log n)^2} > \mu \} < \mu ’/n_0^{3\alpha - 2 - \beta },1 \leqq \alpha \leqq 2,0 < \beta < 1$; (ii) $\Pr \{ {N_n}/{(\log n)^2} > \mu \} < \mu ’/n,\alpha \geqq 1$; (iii) $\Pr \{ {N_n}/{(\log n)^2} > \mu \} < \mu ’/{n^{3\alpha - 1 - \beta }},1 \leqq \alpha \leqq 2$.References
- J. E. A. Dunnage, The number of real zeros of a class of random algebraic polynomials. II, Quart. J. Math. Oxford Ser. (2) 21 (1970), 309–319. MR 275485, DOI 10.1093/qmath/21.3.309
- G. Samal and M. N. Mishra, On the upper bound of the number of real roots of a random algebraic equation with infinite variance, J. London Math. Soc. (2) 6 (1973), 598–604. MR 438472, DOI 10.1112/jlms/s2-6.4.598
- G. Samal and M. N. Mishra, On the lower bound of the number of real roots of a random algebraic equation with infinite variance. III, Proc. Amer. Math. Soc. 39 (1973), 184–189. MR 315786, DOI 10.1090/S0002-9939-1973-0315786-8
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 44 (1974), 446-448
- MSC: Primary 60G55; Secondary 92A10
- DOI: https://doi.org/10.1090/S0002-9939-1974-0438473-5
- MathSciNet review: 0438473