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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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

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
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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