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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


On the upper bound of the number of real roots of a random algebraic equation with infinite variance. II

Authors: G. Samal and M. N. Mishra
Journal: Proc. Amer. Math. Soc. 44 (1974), 446-448
MSC: Primary 60G55; Secondary 92A10
MathSciNet review: 0438473
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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\vert t{\vert^\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}}\vert{a_v}\vert = 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$.

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Additional Information

PII: S 0002-9939(1974)0438473-5
Keywords: Random variables, infinite variance, real roots, random algebraic equations
Article copyright: © Copyright 1974 American Mathematical Society

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